Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/PVS/groups/   (Beweissystem der NASA Version 6.0.9©)  Datei vom 28.9.2014 mit Größe 355 kB image not shown  

SSL isomorphism_theorems.prf

  Interaktion und
PortierbarkeitLisp
 

(isomorphism_theorems
 (G_TCC1 0
  (G_TCC1-1 nil 3530909168 ("" (rewrite "T1_is_group"nil nil)
   ((T1_is_group formula-decl nil isomorphism_theorems nil)) nil))
 (GP_TCC1 0
  (GP_TCC1-1 nil 3530909168 ("" (rewrite "T2_is_group"nil nil)
   ((T2_is_group formula-decl nil isomorphism_theorems nil)) nil))
 (quotient_subgroup_TCC1 0
  (quotient_subgroup_TCC1-1 nil 3524781199
   ("" (skosimp*)
    (("" (lemma "normal_subgroup_tran[T1, *, one1]")
      (("1" (inst -1 "G!1" "M!1" "N!1")
        (("1" (assert)
          (("1" (typepred "M!1")
            (("1" (expand "normal_subgroup?") (("1" (propax) nil nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (rewrite "G_TCC1"nil nil))
      nil))
    nil)
   ((one1 formal-const-decl "T1" isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (normal_subgroup_tran formula-decl nil groups_scaf nil)
    (fullset const-decl "set" sets nil)
    (group? const-decl "bool" group_def "algebra/")
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (G_TCC1 assuming-tcc nil isomorphism_theorems nil))
   nil))
 (quotient_subgroup_TCC2 0
  (quotient_subgroup_TCC2-1 nil 3524781199
   ("" (skosimp*)
    (("" (prop)
      (("1" (skosimp)
        (("1" (inst 1 "a!1")
          (("1" (hide -)
            (("1" (typepred "a!1" "M!1")
              (("1" (expand"normal_subgroup?" "subgroup?")
                (("1" (flatten)
                  (("1" (expand"subset?" "member")
                    (("1" (inst?) (("1" (assertnil nil)) nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (hide -1)
        (("2" (inst 1 "lc_gen(M!1, N!1, x!1)") (("2" (assertnil nil))
          nil))
        nil))
      nil))
    nil)
   ((T1 formal-type-decl nil isomorphism_theorems nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (set type-eq-decl nil sets nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (one1 formal-const-decl "T1" isomorphism_theorems nil)
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (a!1 skolem-const-decl "(M!1)" isomorphism_theorems nil)
    (M!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (subset? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (subgroup? const-decl "bool" group_def "algebra/")
    (N!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (subgroup type-eq-decl nil group "algebra/")
    (= const-decl "[T, T -> boolean]" equalities nil)
    (* const-decl "set[T]" cosets "algebra/")
    (left_cosets type-eq-decl nil cosets "algebra/")
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (lc_gen const-decl "{a: T | G(a) AND lc = a * H}" cosets
     "algebra/"))
   nil))
 (quotient_subgroup_TCC3 0
  (quotient_subgroup_TCC3-1 nil 3524781199
   ("" (skosimp*)
    (("" (inst 1 "one1")
      (("1" (rewrite "left_coset_one[T1,*,one1]"nil nil)
       ("2" (typepred "G!1")
        (("2" (expand"group?" "monoid?" "monad?" "member")
          (("2" (flatten) nil nil)) nil))
        nil))
      nil))
    nil)
   ((G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (group nonempty-type-eq-decl nil group "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (one1 formal-const-decl "T1" isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (left_coset_one formula-decl nil cosets "algebra/")
    (monoid? const-decl "bool" monoid_def "algebra/")
    (member const-decl "bool" sets nil)
    (monad? const-decl "bool" monad_def "algebra/")
    (NOT const-decl "[bool -> bool]" booleans nil))
   nil))
 (quotient_subgroup_TCC4 0
  (quotient_subgroup_TCC4-1 nil 3524781199
   ("" (skosimp*)
    (("" (inst 1 "N!1")
      (("" (inst 1 "one1")
        (("1" (rewrite "left_coset_one[T1,*,one1]"nil nil)
         ("2" (typepred "G!1")
          (("2" (expand"group?" "monoid?" "monad?" "member")
            (("2" (flatten) nil nil)) nil))
          nil))
        nil))
      nil))
    nil)
   ((T1 formal-type-decl nil isomorphism_theorems nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (set type-eq-decl nil sets nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (one1 formal-const-decl "T1" isomorphism_theorems nil)
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (= const-decl "[T, T -> boolean]" equalities nil)
    (* const-decl "set[T]" cosets "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (left_cosets type-eq-decl nil cosets "algebra/")
    (N!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (monad? const-decl "bool" monad_def "algebra/")
    (member const-decl "bool" sets nil)
    (monoid? const-decl "bool" monoid_def "algebra/")
    (left_coset_one formula-decl nil cosets "algebra/"))
   nil))
 (quotient_subgroup 0
  (quotient_subgroup-1 nil 3524784283
   ("" (skosimp*)
    (("" (expand "subgroup?")
      (("" (prop)
        (("1" (hide -1)
          (("1" (expand"subset?" "member")
            (("1" (skosimp*)
              (("1" (expand "/")
                (("1" (expand "restrict")
                  (("1" (expand "left_cosets")
                    (("1" (skosimp)
                      (("1" (inst?)
                        (("1" (hide -1)
                          (("1" (typepred "a!1" "M!1")
                            (("1"
                              (expand"normal_subgroup?" "subgroup?"
                               "subset?" "member")
                              (("1"
                                (flatten)
                                (("1"
                                  (hide (-2 -4))
                                  (("1"
                                    (inst?)
                                    (("1" (assertnil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil)
         ("2" (hide -1)
          (("2" (expand "group?")
            (("2" (prop)
              (("1" (expand "monoid?")
                (("1" (prop)
                  (("1" (expand "monad?")
                    (("1" (prop)
                      (("1" (expand "star_closed?")
                        (("1" (skosimp*)
                          (("1" (expand "member")
                            (("1" (expand "/")
                              (("1"
                                (expand "restrict")
                                (("1"
                                  (expand "left_cosets")
                                  (("1"
                                    (typepred "x!1" "y!1")
                                    (("1"
                                      (hide (-2 -4))
                                      (("1"
                                        (skosimp*)
                                        (("1"
                                          (inst 1 "a!1 * a!2")
                                          (("1"
                                            (replaces -1)
                                            (("1"
                                              (replaces -1)
                                              (("1"
                                                (lemma
                                                 "mult_lem[T1, *, one1]")
                                                (("1"
                                                  (inst?)
                                                  (("1"
                                                    (hide 2)
                                                    (("1"
                                                      (typepred
                                                       "a!2"
                                                       "M!1")
                                                      (("1"
                                                        (expand*
                                                         "normal_subgroup?"
                                                         "subgroup?"
                                                         "subset?"
                                                         "member")
                                                        (("1"
                                                          (flatten)
                                                          (("1"
                                                            (hide
                                                             (-2 -4))
                                                            (("1"
                                                              (inst?)
                                                              (("1"
                                                                (assert)
                                                                nil
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (hide 2)
                                                    (("2"
                                                      (typepred
                                                       "a!1"
                                                       "M!1")
                                                      (("2"
                                                        (expand*
                                                         "normal_subgroup?"
                                                         "subgroup?"
                                                         "subset?"
                                                         "member")
                                                        (("2"
                                                          (flatten)
                                                          (("2"
                                                            (hide
                                                             (-2 -4))
                                                            (("2"
                                                              (inst?)
                                                              (("2"
                                                                (assert)
                                                                nil
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil)
                                           ("2"
                                            (hide (-1 -2))
                                            (("2"
                                              (rewrite
                                               "product_in[T1, *, one1]")
                                              nil
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (expand"member" "/" "left_cosets")
                        (("2" (expand "restrict")
                          (("2" (typepred "M!1")
                            (("2"
                              (expand"group?" "monoid?" "monad?"
                               "member")
                              (("2"
                                (flatten)
                                (("2"
                                  (hide-all-but (-2 1))
                                  (("2"
                                    (inst 1 "one1")
                                    (("2"
                                      (rewrite "left_coset_one")
                                      nil
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("3" (expand "identity?")
                        (("3" (skosimp*)
                          (("3" (expand "restrict")
                            (("3" (lemma "N_is_identity[T1, *, one1]")
                              (("3"
                                (inst?)
                                (("1"
                                  (flatten)
                                  (("1" (assertnil nil))
                                  nil)
                                 ("2"
                                  (hide 2)
                                  (("2"
                                    (typepred "x!1")
                                    (("2"
                                      (skosimp*)
                                      (("2"
                                        (inst?)
                                        (("2"
                                          (hide -)
                                          (("2"
                                            (typepred "a!1" "M!1")
                                            (("2"
                                              (expand*
                                               "normal_subgroup?"
                                               "subgroup?"
                                               "subset?"
                                               "member")
                                              (("2"
                                                (flatten)
                                                (("2"
                                                  (inst?)
                                                  (("2"
                                                    (assert)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (expand "associative?")
                    (("2" (expand "restrict")
                      (("2" (skosimp*)
                        (("2" (typepred "z!1")
                          (("2" (typepred "y!1")
                            (("2" (typepred "x!1")
                              (("2"
                                (skosimp*)
                                (("2"
                                  (replace -1)
                                  (("2"
                                    (hide -1)
                                    (("2"
                                      (replace -2)
                                      (("2"
                                        (hide -2)
                                        (("2"
                                          (replace -3)
                                          (("2"
                                            (hide -3)
                                            (("2"
                                              (rewrite
                                               "mult_lem[T1, *, one1]")
                                              (("1"
                                                (rewrite
                                                 "mult_lem[T1, *, one1]")
                                                (("1"
                                                  (rewrite
                                                   "mult_lem[T1, *, one1]")
                                                  (("1"
                                                    (rewrite
                                                     "mult_lem[T1, *, one1]")
                                                    (("1"
                                                      (rewrite
                                                       "assoc[T1, *, one1]")
                                                      nil
                                                      nil)
                                                     ("2"
                                                      (hide-all-but 1)
                                                      (("2"
                                                        (typepred
                                                         "a!2"
                                                         "a!3"
                                                         "G!1"
                                                         "M!1")
                                                        (("2"
                                                          (expand
                                                           "normal_subgroup?")
                                                          (("2"
                                                            (flatten)
                                                            (("2"
                                                              (expand
                                                               "subgroup?")
                                                              (("2"
                                                                (expand
                                                                 "subset?")
                                                                (("2"
                                                                  (hide
                                                                   (-4
                                                                    -6))
                                                                  (("2"
                                                                    (copy
                                                                     -4)
                                                                    (("2"
                                                                      (inst
                                                                       -1
                                                                       "a!2")
                                                                      (("2"
                                                                        (inst
                                                                         -5
                                                                         "a!3")
                                                                        (("2"
                                                                          (expand
                                                                           "member")
                                                                          (("2"
                                                                            (expand*
                                                                             "group?"
                                                                             "monoid?"
                                                                             "monad?")
                                                                            (("2"
                                                                              (flatten)
                                                                              (("2"
                                                                                (hide-all-but
                                                                                 (-1
                                                                                  -4
                                                                                  -9
                                                                                  1))
                                                                                (("2"
                                                                                  (expand
                                                                                   "star_closed?")
                                                                                  (("2"
                                                                                    (inst?)
                                                                                    (("2"
                                                                                      (expand
                                                                                       "member")
                                                                                      (("2"
                                                                                        (propax)
                                                                                        nil
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("3"
                                                      (hide-all-but 1)
                                                      (("3"
                                                        (typepred
                                                         "a!1"
                                                         "M!1")
                                                        (("3"
                                                          (expand*
                                                           "normal_subgroup?"
                                                           "subgroup?"
                                                           "subset?"
                                                           "member")
                                                          (("3"
                                                            (flatten)
                                                            (("3"
                                                              (hide
                                                               (-2 -4))
                                                              (("3"
                                                                (inst?)
                                                                (("3"
                                                                  (assert)
                                                                  nil
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (hide-all-but 1)
                                                    (("2"
                                                      (typepred
                                                       "a!3"
                                                       "M!1")
                                                      (("2"
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                                                         "normal_subgroup?"
                                                         "subgroup?"
                                                         "subset?"
                                                         "member")
                                                        (("2"
                                                          (flatten)
                                                          (("2"
                                                            (hide
                                                             (-2 -4))
                                                            (("2"
                                                              (inst?)
                                                              (("2"
                                                                (assert)
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                                                                nil))
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                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("3"
                                                    (hide-all-but 1)
                                                    (("3"
                                                      (typepred
                                                       "a!2"
                                                       "M!1")
                                                      (("3"
                                                        (expand*
                                                         "normal_subgroup?"
                                                         "subgroup?"
                                                         "subset?"
                                                         "member")
                                                        (("3"
                                                          (flatten)
                                                          (("3"
                                                            (hide
                                                             (-2 -4))
                                                            (("3"
                                                              (inst?)
                                                              (("3"
                                                                (assert)
                                                                nil
                                                                nil))
                                                              nil))
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                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (hide-all-but 1)
                                                  (("2"
                                                    (typepred
                                                     "a!3"
                                                     "M!1")
                                                    (("2"
                                                      (expand*
                                                       "normal_subgroup?"
                                                       "subgroup?"
                                                       "subset?"
                                                       "member")
                                                      (("2"
                                                        (flatten)
                                                        (("2"
                                                          (hide
                                                           (-2 -4))
                                                          (("2"
                                                            (inst?)
                                                            (("2"
                                                              (assert)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil)
                                                 ("3"
                                                  (hide-all-but 1)
                                                  (("3"
                                                    (typepred
                                                     "a!1"
                                                     "a!2"
                                                     "G!1"
                                                     "M!1")
                                                    (("3"
                                                      (expand
                                                       "normal_subgroup?")
                                                      (("3"
                                                        (flatten)
                                                        (("3"
                                                          (expand
                                                           "subgroup?")
                                                          (("3"
                                                            (expand
                                                             "subset?")
                                                            (("3"
                                                              (hide
                                                               (-4 -6))
                                                              (("3"
                                                                (copy
                                                                 -4)
                                                                (("3"
                                                                  (inst
                                                                   -1
                                                                   "a!1")
                                                                  (("3"
                                                                    (inst
                                                                     -5
                                                                     "a!2")
                                                                    (("3"
                                                                      (expand
                                                                       "member")
                                                                      (("3"
                                                                        (expand*
                                                                         "group?"
                                                                         "monoid?"
                                                                         "monad?")
                                                                        (("3"
                                                                          (flatten)
                                                                          (("3"
                                                                            (hide-all-but
                                                                             (-1
                                                                              -4
                                                                              -9
                                                                              1))
                                                                            (("3"
                                                                              (expand
                                                                               "star_closed?")
                                                                              (("3"
                                                                                (inst?)
                                                                                (("3"
                                                                                  (expand
                                                                                   "member")
                                                                                  (("3"
                                                                                    (propax)
                                                                                    nil
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2"
                                                (hide-all-but 1)
                                                (("2"
                                                  (typepred
                                                   "a!2"
                                                   "M!1")
                                                  (("2"
                                                    (expand*
                                                     "normal_subgroup?"
                                                     "subgroup?"
                                                     "subset?"
                                                     "member")
                                                    (("2"
                                                      (flatten)
                                                      (("2"
                                                        (hide (-2 -4))
                                                        (("2"
                                                          (inst?)
                                                          (("2"
                                                            (assert)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("3"
                                                (hide-all-but 1)
                                                (("3"
                                                  (typepred
                                                   "a!1"
                                                   "M!1")
                                                  (("3"
                                                    (expand*
                                                     "normal_subgroup?"
                                                     "subgroup?"
                                                     "subset?"
                                                     "member")
                                                    (("3"
                                                      (flatten)
                                                      (("3"
                                                        (hide (-2 -4))
                                                        (("3"
                                                          (inst?)
                                                          (("3"
                                                            (assert)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (expand "inv_exists?")
                (("2" (skosimp*)
                  (("2" (typepred "x!1")
                    (("2" (skosimp*)
                      (("2"
                        (inst 1
                         "*[T1, *, one1](inv[T1, *, one1](a!1), N!1)")
                        (("1" (replace -1)
                          (("1" (hide -1)
                            (("1" (rewrite "mult_lem[T1, *, one1]")
                              (("1"
                                (rewrite "mult_lem[T1, *, one1]")
                                (("1"
                                  (rewrite "inv_right[T1, *, one1]")
                                  (("1"
                                    (rewrite "inv_left[T1, *, one1]")
                                    (("1"
                                      (rewrite "left_coset_one")
                                      nil
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (hide-all-but 1)
                                  (("2"
                                    (typepred "a!1" "M!1")
                                    (("2"
                                      (expand*
                                       "normal_subgroup?"
                                       "subgroup?"
                                       "subset?"
                                       "member")
                                      (("2"
                                        (flatten)
                                        (("2"
                                          (hide (-2 -4))
                                          (("2"
                                            (inst?)
                                            (("2" (assertnil nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("3"
                                  (hide-all-but 1)
                                  (("3"
                                    (typepred "a!1" "M!1")
                                    (("3"
                                      (expand*
                                       "normal_subgroup?"
                                       "subgroup?"
                                       "subset?"
                                       "member")
                                      (("3"
                                        (flatten)
                                        (("3"
                                          (hide (-2 -4))
                                          (("3"
                                            (inst?)
                                            (("3"
                                              (assert)
                                              (("3"
                                                (rewrite
                                                 "inv_in[T1, *, one1]")
                                                nil
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil)
                               ("2"
                                (hide-all-but 1)
                                (("2"
                                  (typepred "a!1" "M!1")
                                  (("2"
                                    (expand*
                                     "normal_subgroup?"
                                     "subgroup?"
                                     "subset?"
                                     "member")
                                    (("2"
                                      (flatten)
                                      (("2"
                                        (hide (-2 -4))
                                        (("2"
                                          (inst?)
                                          (("2"
                                            (assert)
                                            (("2"
                                              (rewrite
                                               "inv_in[T1, *, one1]")
                                              nil
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil)
                               ("3"
                                (hide-all-but 1)
                                (("3"
                                  (typepred "a!1" "M!1")
                                  (("3"
                                    (expand*
                                     "normal_subgroup?"
                                     "subgroup?"
                                     "subset?"
                                     "member")
                                    (("3"
                                      (flatten)
                                      (("3"
                                        (hide (-2 -4))
                                        (("3"
                                          (inst?)
                                          (("3" (assertnil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (prop)
                          (("1" (expand "*")
                            (("1" (inst + "inv[T1, *, one1](a!1)")
                              (("1"
                                (rewrite "inv_in[T1, *, one1]")
                                nil
                                nil))
                              nil))
                            nil)
                           ("2" (expand "/")
                            (("2" (hide -2)
                              (("2"
                                (expand "restrict")
                                (("2"
                                  (expand "left_cosets")
                                  (("2"
                                    (inst 1 "inv[T1, *, one1](a!1)")
                                    (("2"
                                      (rewrite "inv_in[T1, *, one1]")
                                      nil
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((subgroup? const-decl "bool" group_def "algebra/")
    (a!1 skolem-const-decl "(M!1)" isomorphism_theorems nil)
    (a!2 skolem-const-decl "(M!1)" isomorphism_theorems nil)
    (mult_lem formula-decl nil factor_groups "algebra/")
    (product_in formula-decl nil group "algebra/")
    (= const-decl "[T, T -> boolean]" equalities nil)
    (* const-decl "set[T]" cosets "algebra/")
    (left_cosets type-eq-decl nil cosets "algebra/")
    (mult const-decl "left_cosets(G, H)" factor_groups "algebra/")
    (star_closed? const-decl "bool" groupoid_def "algebra/")
    (left_coset_one formula-decl nil cosets "algebra/")
    (N_is_identity formula-decl nil factor_groups "algebra/")
    (a!1 skolem-const-decl "(M!1)" isomorphism_theorems nil)
    (N!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (x!1 skolem-const-decl "(M!1 / N!1)" isomorphism_theorems nil)
    (identity? const-decl "bool" operator_defs nil)
    (monad? const-decl "bool" monad_def "algebra/")
    (a!1 skolem-const-decl "(M!1)" isomorphism_theorems nil)
    (a!3 skolem-const-decl "(M!1)" isomorphism_theorems nil)
    (a!2 skolem-const-decl "(M!1)" isomorphism_theorems nil)
    (assoc formula-decl nil group "algebra/")
    (associative? const-decl "bool" operator_defs nil)
    (monoid? const-decl "bool" monoid_def "algebra/")
    (inv_in formula-decl nil group "algebra/")
    (inv_right formula-decl nil group "algebra/")
    (inv_left formula-decl nil group "algebra/")
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (inv const-decl "{y | x * y = one AND y * x = one}" group
     "algebra/")
    (a!1 skolem-const-decl "(M!1)" isomorphism_theorems nil)
    (inv_exists? const-decl "bool" group_def "algebra/")
    (restrict const-decl "R" restrict nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (M!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (a!1 skolem-const-decl "(M!1)" isomorphism_theorems nil)
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (one1 formal-const-decl "T1" isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (left_cosets const-decl "setofsets[T]" right_left_cosets nil)
    (/ const-decl "group[left_cosets(G, N), mult(G, N), N]"
       right_left_cosets nil)
    (subset? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil))
   shostak))
 (second_isomorphism_th_aux_TCC1 0
  (second_isomorphism_th_aux_TCC1-1 nil 3524841142
   ("" (skosimp*)
    (("" (replaces -1)
      (("" (lemma "HK_subgroup[T1, *, one1]")
        (("" (inst -1 "G!1" "H!1" "N!1")
          (("1" (assert)
            (("1" (lemma "HK_subgroup_permute[T1, *, one1]")
              (("1" (inst?)
                (("1" (assert)
                  (("1" (expand "subgroup?") (("1" (propax) nil nil))
                    nil))
                  nil)
                 ("2" (hide (-1 2))
                  (("2" (typepred "N!1")
                    (("2" (expand "normal_subgroup?")
                      (("2" (assertnil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (typepred "N!1")
            (("2" (expand "normal_subgroup?") (("2" (assertnil nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((N!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (subgroup type-eq-decl nil group "algebra/")
    (HK_subgroup_permute formula-decl nil products_subgroups nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (HK_subgroup formula-decl nil products_subgroups nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (one1 formal-const-decl "T1" isomorphism_theorems nil))
   nil))
 (second_isomorphism_th_aux_TCC2 0
  (second_isomorphism_th_aux_TCC2-1 nil 3524841142
   ("" (skosimp*)
    (("" (replaces -2)
      (("" (replaces -1)
        (("" (lemma "HK_subgroup[T1, *, one1]")
          (("" (inst -1 "G!1" "H!1" "N!1")
            (("1" (assert)
              (("1" (lemma "HK_subgroup_permute[T1, *, one1]")
                (("1" (inst?)
                  (("1" (assert)
                    (("1" (replaces -1)
                      (("1" (lemma "H_K_are_subgroups[T1, *, one1]")
                        (("1" (inst?)
                          (("1" (prop)
                            (("1" (hide -2)
                              (("1"
                                (lemma
                                 "normal_subgroup_tran[T1, *, one1]")
                                (("1"
                                  (inst
                                   -1
                                   "G!1"
                                   "prod[T1, *, one1](N!1, H!1)"
                                   "N!1")
                                  (("1" (assertnil nil))
                                  nil)
                                 ("2" (rewrite "G_TCC1"nil nil))
                                nil))
                              nil)
                             ("2" (expand "subgroup?")
                              (("2" (assertnil nil)) nil))
                            nil)
                           ("2" (typepred "N!1")
                            (("2" (expand "normal_subgroup?" -2)
                              (("2" (assertnil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (hide 2)
                    (("2" (typepred "N!1")
                      (("2" (expand "normal_subgroup?")
                        (("2" (assertnil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (hide 2)
              (("2" (typepred "N!1")
                (("2" (expand "normal_subgroup?")
                  (("2" (assertnil nil)) nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((N!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (subgroup type-eq-decl nil group "algebra/")
    (HK_subgroup_permute formula-decl nil products_subgroups nil)
    (H_K_are_subgroups formula-decl nil products_subgroups nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (normal_subgroup_tran formula-decl nil groups_scaf nil)
    (fullset const-decl "set" sets nil)
    (prod const-decl "set[T]" products_subgroups nil)
    (G_TCC1 assuming-tcc nil isomorphism_theorems nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (HK_subgroup formula-decl nil products_subgroups nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (one1 formal-const-decl "T1" isomorphism_theorems nil))
   nil))
 (second_isomorphism_th_aux_TCC3 0
  (second_isomorphism_th_aux_TCC3-1 nil 3524841142
   ("" (skosimp*)
    (("" (inst 1 "one1")
      (("1" (rewrite "left_coset_one"nil nil)
       ("2" (replaces -1)
        (("2" (lemma "HK_subgroup[T1, *, one1]")
          (("2" (inst -1 "G!1" "H!1" "N!1")
            (("1" (assert)
              (("1" (lemma "HK_subgroup_permute[T1, *, one1]")
                (("1" (inst?)
                  (("1" (assert)
                    (("1" (replaces -1)
                      (("1"
                        (expand"subgroup?" "group?" "monoid?"
                         "monad?" "member")
                        nil nil))
                      nil))
                    nil)
                   ("2" (typepred "N!1")
                    (("2" (expand "normal_subgroup?")
                      (("2" (assertnil nil)) nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (typepred "N!1")
              (("2" (expand "normal_subgroup?")
                (("2" (assertnil nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((one1 formal-const-decl "T1" isomorphism_theorems nil)
    (NH!1 skolem-const-decl "set[T1]" isomorphism_theorems nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (left_coset_one formula-decl nil cosets "algebra/")
    (HK_subgroup formula-decl nil products_subgroups nil)
    (member const-decl "bool" sets nil)
    (monad? const-decl "bool" monad_def "algebra/")
    (monoid? const-decl "bool" monoid_def "algebra/")
    (NOT const-decl "[bool -> bool]" booleans nil)
    (HK_subgroup_permute formula-decl nil products_subgroups nil)
    (subgroup type-eq-decl nil group "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/")
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (N!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil))
   nil))
 (second_isomorphism_th_aux_TCC4 0
  (second_isomorphism_th_aux_TCC4-1 nil 3524841142
   ("" (skosimp*)
    (("" (rewrite "left_cosets_group[T1, *, one1]")
      (("1" (hide 2)
        (("1" (replaces -1)
          (("1" (lemma "HK_subgroup[T1, *, one1]")
            (("1" (inst -1 "G!1" "H!1" "N!1")
              (("1" (assert)
                (("1" (lemma "HK_subgroup_permute[T1, *, one1]")
                  (("1" (inst?)
                    (("1" (assert)
                      (("1" (replaces -1)
                        (("1" (lemma "H_K_are_subgroups[T1, *, one1]")
                          (("1" (inst?)
                            (("1" (prop)
                              (("1"
                                (hide -2)
                                (("1"
                                  (lemma
                                   "normal_subgroup_tran[T1, *, one1]")
                                  (("1"
                                    (inst
                                     -1
                                     "G!1"
                                     "prod[T1, *, one1](N!1, H!1)"
                                     "N!1")
                                    (("1" (assertnil nil))
                                    nil)
                                   ("2" (rewrite "G_TCC1"nil nil))
                                  nil))
                                nil)
                               ("2"
                                (expand "subgroup?")
                                (("2" (assertnil nil))
                                nil))
                              nil)
                             ("2" (hide 2)
                              (("2"
                                (typepred "N!1")
                                (("2"
                                  (expand "normal_subgroup?")
                                  (("2" (assertnil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (hide 2)
                      (("2" (typepred "N!1")
                        (("2" (expand "normal_subgroup?")
                          (("2" (assertnil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (hide 2)
                (("2" (typepred "N!1")
                  (("2" (expand "normal_subgroup?")
                    (("2" (assertnil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (hide 2)
        (("2" (replaces -1)
          (("2" (lemma "HK_subgroup[T1, *, one1]")
            (("2" (inst -1 "G!1" "H!1" "N!1")
              (("1" (assert)
                (("1" (lemma "HK_subgroup_permute[T1, *, one1]")
                  (("1" (inst?)
                    (("1" (assert)
                      (("1" (replaces -1)
                        (("1"
                          (expand"subgroup?" "group?" "monoid?"
                           "monad?" "member" -1)
                          nil nil))
                        nil))
                      nil)
                     ("2" (hide-all-but 1)
                      (("2" (typepred "N!1")
                        (("2" (expand "normal_subgroup?")
                          (("2" (assertnil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (hide 2)
                (("2" (typepred "N!1")
                  (("2" (expand "normal_subgroup?")
                    (("2" (assertnil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((left_cosets_group formula-decl nil factor_groups "algebra/")
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (set type-eq-decl nil sets nil)
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (one1 formal-const-decl "T1" isomorphism_theorems nil)
    (N!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (subgroup? const-decl "bool" group_def "algebra/")
    (subgroup type-eq-decl nil group "algebra/")
    (HK_subgroup_permute formula-decl nil products_subgroups nil)
    (H_K_are_subgroups formula-decl nil products_subgroups nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (normal_subgroup_tran formula-decl nil groups_scaf nil)
    (fullset const-decl "set" sets nil)
    (prod const-decl "set[T]" products_subgroups nil)
    (G_TCC1 assuming-tcc nil isomorphism_theorems nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (HK_subgroup formula-decl nil products_subgroups nil))
   nil))
 (second_isomorphism_th_aux 0
  (second_isomorphism_th_aux-1 nil 3524852278
   ("" (skosimp*)
    (("" (expand "homomorphism?")
      (("" (inst 1 "(LAMBDA (x: (H!1)): *[T1, *, one1](x, N!1))")
        (("1" (prop)
          (("1" (skosimp*)
            (("1" (rewrite "mult_lem[T1, *, one1]")
              (("1" (hide 2)
                (("1" (typepred "b!1")
                  (("1" (expand "prod")
                    (("1" (inst 1 "one1" "b!1")
                      (("1" (rewrite "one_left[T1, *, one1]"nil nil)
                       ("2" (typepred "N!1")
                        (("2"
                          (expand"group?" "monoid?" "monad?"
                           "member")
                          (("2" (flatten) nil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (hide 2)
                (("2" (typepred "a!1")
                  (("2" (expand "prod")
                    (("2" (inst 1 "one1" "a!1")
                      (("1" (rewrite "one_left[T1, *, one1]"nil nil)
                       ("2" (typepred "N!1")
                        (("2"
                          (expand"group?" "monoid?" "monad?"
                           "member")
                          (("2" (assertnil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (expand "surjective?")
            (("2" (skosimp*)
              (("2" (typepred "y!1")
                (("2" (skosimp)
                  (("2" (typepred "a!1")
                    (("2" (hide -3)
                      (("2" (expand "prod" -1)
                        (("2" (skosimp)
                          (("2" (typepred "k!1")
                            (("2" (typepred "h!1")
                              (("2"
                                (replaces -4)
                                (("2"
                                  (lemma
                                   "normal_left_is_right[T1, *, one1]")
                                  (("2"
                                    (inst -1 "G!1" "N!1")
                                    (("2"
                                      (assert)
                                      (("2"
                                        (flatten)
                                        (("2"
                                          (inst -2 "a!1")
                                          (("1"
                                            (replaces -2)
                                            (("1"
                                              (replaces -4)
                                              (("1"
                                                (inst 1 "k!1")
                                                (("1"
                                                  (lemma
                                                   "normal_left_is_right[T1, *, one1]")
                                                  (("1"
                                                    (inst
                                                     -1
                                                     "G!1"
                                                     "N!1")
                                                    (("1"
                                                      (assert)
                                                      (("1"
                                                        (inst -1 "k!1")
                                                        (("1"
                                                          (replaces -1)
                                                          (("1"
                                                            (hide-all-but
                                                             1)
                                                            (("1"
                                                              (lemma
                                                               "rc_is_eq[T1, *, one1]")
                                                              (("1"
                                                                (inst
                                                                 -1
                                                                 "G!1"
                                                                 "N!1"
                                                                 "one1"
                                                                 "h!1")
                                                                (("1"
                                                                  (assert)
                                                                  (("1"
                                                                    (prop)
                                                                    (("1"
                                                                      (rewrite
                                                                       "right_coset_one")
                                                                      (("1"
                                                                        (lemma
                                                                         "right_coset_assoc[T1, *, one1]")
                                                                        (("1"
                                                                          (inst?)
                                                                          (("1"
                                                                            (assert)
                                                                            nil
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil)
                                                                     ("2"
                                                                      (inst
                                                                       1
                                                                       "inv[T1, *, one1](h!1)")
                                                                      (("1"
                                                                        (rewrite
                                                                         "inv_left[T1, *, one1]")
                                                                        nil
                                                                        nil)
                                                                       ("2"
                                                                        (hide
                                                                         2)
                                                                        (("2"
                                                                          (typepred
                                                                           "h!1")
                                                                          (("2"
                                                                            (rewrite
                                                                             "inv_in")
                                                                            nil
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil)
                                                         ("2"
                                                          (typepred
                                                           "H!1")
                                                          (("2"
                                                            (hide-all-but
                                                             (-2 -5 1))
                                                            (("2"
                                                              (expand
                                                               "subgroup?")
                                                              (("2"
                                                                (expand*
                                                                 "subset?"
                                                                 "member")
                                                                (("2"
                                                                  (inst?)
                                                                  (("2"
                                                                    (assert)
                                                                    nil
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil)
                                           ("2"
                                            (replaces -4)
                                            (("2"
                                              (hide (- 2))
                                              (("2"
                                                (typepred
                                                 "h!1"
                                                 "k!1"
                                                 "N!1"
                                                 "H!1")
                                                (("2"
                                                  (expand*
                                                   "normal_subgroup?"
                                                   "subgroup?"
                                                   "subset?"
                                                   "member")
                                                  (("2"
                                                    (flatten)
                                                    (("2"
                                                      (hide (-3 -5 -6))
                                                      (("2"
                                                        (inst -3 "h!1")
                                                        (("2"
                                                          (inst
                                                           -4
                                                           "k!1")
                                                          (("2"
                                                            (assert)
                                                            (("2"
                                                              (rewrite
                                                               "product_in[T1, *, one1]")
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("3" (decompose-equality 1)
            (("1" (iff)
              (("1" (prop)
                (("1" (expand"kernel" "intersection")
                  (("1" (expand "member")
                    (("1" (prop)
                      (("1" (lemma "lc_eq[T1, *, one1]")
                        (("1" (inst -1 "G!1" "N!1" "x!1" "one1")
                          (("1" (assert)
                            (("1" (rewrite "left_coset_one")
                              (("1"
                                (assert)
                                (("1"
                                  (prop)
                                  (("1"
                                    (skosimp)
                                    (("1"
                                      (rewrite "one_left[T1, *, one1]")
                                      (("1" (assertnil nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (hide-all-but 1)
                                    (("2"
                                      (typepred "N!1")
                                      (("2"
                                        (expand "normal_subgroup?")
                                        (("2" (assertnil nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (expand"intersection" "kernel")
                  (("2" (expand "member")
                    (("2" (flatten)
                      (("2" (assert)
                        (("2" (lemma "lc_is_eq[T1, *, one1]")
                          (("2" (inst -1 "G!1" "N!1" "x!1" "one1")
                            (("2" (prop)
                              (("1" (rewrite "left_coset_one"nil nil)
                               ("2"
                                (hide-all-but 1)
                                (("2"
                                  (typepred "N!1")
                                  (("2"
                                    (expand "normal_subgroup?")
                                    (("2" (assertnil nil))
                                    nil))
                                  nil))
                                nil)
                               ("3"
                                (inst 1 "x!1")
                                (("3"
                                  (rewrite "one_left[T1, *, one1]")
                                  nil
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (rewrite "left_cosets_group[T1, *, one1]")
              (("1" (hide 2)
                (("1" (lemma "HK_subgroup[T1, *, one1]")
                  (("1" (inst -1 "G!1" "H!1" "N!1")
                    (("1" (assert)
                      (("1" (lemma "HK_subgroup_permute[T1, *, one1]")
                        (("1" (inst?)
                          (("1" (assert)
                            (("1" (replaces -1)
                              (("1"
                                (lemma
                                 "normal_subgroup_tran[T1, *, one1]")
                                (("1"
                                  (inst
                                   -1
                                   "G!1"
                                   "prod[T1, *, one1](N!1, H!1)"
                                   "N!1")
                                  (("1"
                                    (assert)
                                    (("1"
                                      (expand "subgroup?" -1)
                                      (("1"
                                        (flatten)
                                        (("1"
                                          (hide -1)
                                          (("1"
                                            (lemma
                                             "H_K_are_subgroups[T1, *, one1]")
                                            (("1"
                                              (inst?)
                                              (("1" (assertnil nil)
                                               ("2"
                                                (hide-all-but 1)
                                                (("2"
                                                  (typepred "N!1")
                                                  (("2"
                                                    (expand
                                                     "normal_subgroup?")
                                                    (("2"
                                                      (assert)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2" (rewrite "T1_is_group"nil nil))
                                nil))
                              nil))
                            nil)
                           ("2" (hide (-1 2))
                            (("2" (typepred "N!1")
                              (("2"
                                (expand "normal_subgroup?")
                                (("2" (assertnil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (hide-all-but 1)
                      (("2" (typepred "N!1")
                        (("2" (expand "normal_subgroup?")
                          (("2" (assertnil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (hide 2)
                (("2" (lemma "HK_subgroup[T1, *, one1]")
                  (("2" (inst -1 "G!1" "H!1" "N!1")
                    (("1" (assert)
                      (("1" (lemma "HK_subgroup_permute[T1, *, one1]")
                        (("1" (inst?)
                          (("1" (assert)
                            (("1" (replaces -1)
                              (("1"
                                (expand "subgroup?")
                                (("1" (propax) nil nil))
                                nil))
                              nil))
                            nil)
                           ("2" (hide-all-but 1)
                            (("2" (typepred "N!1")
                              (("2"
                                (expand "normal_subgroup?")
                                (("2" (assertnil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (hide-all-but 1)
                      (("2" (typepred "N!1")
                        (("2" (expand "normal_subgroup?")
                          (("2" (assertnil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("3" (rewrite "T1_is_group"nil nil)
             ("4" (inst 1 "one1")
              (("1" (rewrite "left_coset_one"nil nil)
               ("2" (lemma "HK_subgroup[T1, *, one1]")
                (("2" (inst -1 "G!1" "H!1" "N!1")
                  (("1" (assert)
                    (("1" (lemma "HK_subgroup_permute[T1, *, one1]")
                      (("1" (inst?)
                        (("1" (assert)
                          (("1" (replaces -1)
                            (("1" (expand "subgroup?")
                              (("1"
                                (flatten)
                                (("1"
                                  (hide -1)
                                  (("1"
                                    (expand*
                                     "group?"
                                     "monoid?"
                                     "monad?"
                                     "member")
                                    nil
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (hide-all-but 1)
                          (("2" (typepred "N!1")
                            (("2" (expand "normal_subgroup?")
                              (("2" (assertnil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (hide-all-but 1)
                    (("2" (typepred "N!1")
                      (("2" (expand "normal_subgroup?")
                        (("2" (assertnil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("5" (expand "homomorphism?")
              (("5" (skosimp*)
                (("5" (rewrite "mult_lem[T1, *, one1]")
                  (("1" (hide 2)
                    (("1" (typepred "b!1")
                      (("1" (lemma "HK_subgroup[T1, *, one1]")
                        (("1" (inst -1 "G!1" "H!1" "N!1")
                          (("1" (assert)
                            (("1" (expand "subgroup?")
                              (("1"
                                (flatten)
                                (("1"
                                  (lemma
                                   "H_K_are_subgroups[T1, *, one1]")
                                  (("1"
                                    (inst?)
                                    (("1"
                                      (assert)
                                      (("1"
                                        (flatten)
                                        (("1"
                                          (lemma
                                           "HK_subgroup_permute[T1, *, one1]")
                                          (("1"
                                            (inst?)
                                            (("1"
                                              (assert)
                                              (("1"
                                                (replaces -1)
                                                (("1"
                                                  (hide-all-but
                                                   (-1 -5 1))
                                                  (("1"
                                                    (expand*
                                                     "subgroup?"
                                                     "subset?"
                                                     "member")
                                                    (("1"
                                                      (inst?)
                                                      (("1"
                                                        (assert)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil)
                                             ("2"
                                              (hide-all-but 1)
                                              (("2"
                                                (typepred "N!1")
                                                (("2"
                                                  (expand
                                                   "normal_subgroup?")
                                                  (("2"
                                                    (assert)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil)
                                     ("2"
                                      (hide-all-but 1)
                                      (("2"
                                        (typepred "N!1")
                                        (("2"
                                          (expand "normal_subgroup?")
                                          (("2" (assertnil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (hide-all-but 1)
                            (("2" (typepred "N!1")
                              (("2"
                                (expand "normal_subgroup?")
                                (("2" (assertnil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (hide 2)
                    (("2" (typepred "a!1")
                      (("2" (lemma "HK_subgroup[T1, *, one1]")
                        (("2" (inst -1 "G!1" "H!1" "N!1")
                          (("1" (assert)
                            (("1" (expand "subgroup?")
                              (("1"
                                (flatten)
                                (("1"
                                  (hide -1)
                                  (("1"
                                    (lemma
                                     "H_K_are_subgroups[T1, *, one1]")
                                    (("1"
                                      (inst?)
                                      (("1"
                                        (assert)
                                        (("1"
                                          (flatten)
                                          (("1"
                                            (hide -2)
                                            (("1"
                                              (lemma
                                               "HK_subgroup_permute[T1, *, one1]")
                                              (("1"
                                                (inst?)
                                                (("1"
                                                  (assert)
                                                  (("1"
                                                    (replaces -1)
                                                    (("1"
                                                      (expand*
                                                       "subgroup?"
                                                       "subset?"
                                                       "member")
                                                      (("1"
                                                        (inst?)
                                                        (("1"
                                                          (assert)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (hide-all-but 1)
                                                  (("2"
                                                    (typepred "N!1")
                                                    (("2"
                                                      (expand
                                                       "normal_subgroup?")
                                                      (("2"
                                                        (assert)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil)
                                       ("2"
                                        (hide-all-but 1)
                                        (("2"
                                          (typepred "N!1")
                                          (("2"
                                            (expand "normal_subgroup?")
                                            (("2" (assertnil nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (hide-all-but 1)
                            (("2" (typepred "N!1")
                              (("2"
                                (expand "normal_subgroup?")
                                (("2"
                                  (hide-all-but 1)
                                  (("2"
                                    (typepred "N!1")
                                    (("2"
                                      (expand "normal_subgroup?")
                                      (("2" (assertnil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("6" (skosimp*)
              (("6" (prop)
                (("1" (inst 1 "x!1")
                  (("1" (typepred "x!1")
                    (("1" (lemma "HK_subgroup[T1, *, one1]")
                      (("1" (inst -1 "G!1" "H!1" "N!1")
                        (("1" (assert)
                          (("1" (expand "subgroup?")
                            (("1" (flatten)
                              (("1"
                                (hide -1)
                                (("1"
                                  (lemma
                                   "H_K_are_subgroups[T1, *, one1]")
                                  (("1"
                                    (inst?)
                                    (("1"
                                      (assert)
                                      (("1"
                                        (flatten)
                                        (("1"
                                          (hide -2)
                                          (("1"
                                            (lemma
                                             "HK_subgroup_permute[T1, *, one1]")
                                            (("1"
                                              (inst?)
                                              (("1"
                                                (assert)
                                                (("1"
                                                  (replaces -1)
                                                  (("1"
                                                    (expand*
                                                     "subgroup?"
                                                     "subset?"
                                                     "member")
                                                    (("1"
                                                      (inst?)
                                                      (("1"
                                                        (assert)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2"
                                                (hide-all-but 1)
                                                (("2"
                                                  (typepred "N!1")
                                                  (("2"
                                                    (expand
                                                     "normal_subgroup?")
                                                    (("2"
                                                      (assert)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil)
                                     ("2"
                                      (hide-all-but 1)
                                      (("2"
                                        (typepred "N!1")
                                        (("2"
                                          (expand "normal_subgroup?")
                                          (("2" (assertnil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (hide-all-but 1)
                          (("2" (typepred "N!1")
                            (("2" (expand "normal_subgroup?")
                              (("2" (assertnil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (expand "/")
                  (("2" (expand "restrict")
                    (("2" (expand "left_cosets")
                      (("2" (inst 1 "x!1")
                        (("2" (lemma "HK_subgroup[T1, *, one1]")
                          (("2" (inst -1 "G!1" "H!1" "N!1")
                            (("1" (assert)
                              (("1"
                                (expand "subgroup?")
                                (("1"
                                  (flatten)
                                  (("1"
                                    (hide -1)
                                    (("1"
                                      (lemma
                                       "H_K_are_subgroups[T1, *, one1]")
                                      (("1"
                                        (inst?)
                                        (("1"
                                          (assert)
                                          (("1"
                                            (flatten)
                                            (("1"
                                              (hide -2)
                                              (("1"
                                                (lemma
                                                 "HK_subgroup_permute[T1, *, one1]")
                                                (("1"
                                                  (inst?)
                                                  (("1"
                                                    (assert)
                                                    (("1"
                                                      (replaces -1)
                                                      (("1"
                                                        (expand*
                                                         "subgroup?"
                                                         "subset?"
                                                         "member")
                                                        (("1"
                                                          (inst?)
                                                          (("1"
                                                            (assert)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (hide-all-but 1)
                                                    (("2"
                                                      (typepred "N!1")
                                                      (("2"
                                                        (expand
                                                         "normal_subgroup?")
                                                        (("2"
                                                          (assert)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil)
                                         ("2"
                                          (hide-all-but 1)
                                          (("2"
                                            (typepred "N!1")
                                            (("2"
                                              (expand
                                               "normal_subgroup?")
                                              (("2" (assertnil nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil)
                             ("2" (hide-all-but 1)
                              (("2"
                                (typepred "N!1")
                                (("2"
                                  (expand "normal_subgroup?")
                                  (("2" (assertnil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil)
         ("2" (skosimp*)
          (("2" (prop)
            (("1" (inst 1 "x!1")
              (("1" (lemma "HK_subgroup[T1, *, one1]")
                (("1" (inst -1 "G!1" "H!1" "N!1")
                  (("1" (assert)
                    (("1" (expand "subgroup?")
                      (("1" (flatten)
                        (("1" (hide -1)
                          (("1"
                            (lemma "H_K_are_subgroups[T1, *, one1]")
                            (("1" (inst?)
                              (("1"
                                (assert)
                                (("1"
                                  (flatten)
                                  (("1"
                                    (hide -2)
                                    (("1"
                                      (lemma
                                       "HK_subgroup_permute[T1, *, one1]")
                                      (("1"
                                        (inst?)
                                        (("1"
                                          (assert)
                                          (("1"
                                            (replaces -1)
                                            (("1"
                                              (expand*
                                               "subgroup?"
                                               "subset?"
                                               "member")
                                              (("1"
                                                (inst?)
                                                (("1"
                                                  (assert)
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil)
                                         ("2"
                                          (hide-all-but 1)
                                          (("2"
                                            (typepred "N!1")
                                            (("2"
                                              (expand
                                               "normal_subgroup?")
                                              (("2" (assertnil nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil)
                               ("2"
                                (hide-all-but 1)
                                (("2"
                                  (typepred "N!1")
                                  (("2"
                                    (expand "normal_subgroup?")
                                    (("2" (assertnil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (hide-all-but 1)
                    (("2" (typepred "N!1")
                      (("2" (expand "normal_subgroup?")
                        (("2" (assertnil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (expand "/")
              (("2" (expand "restrict")
                (("2" (expand "left_cosets")
                  (("2" (inst 1 "x!1")
                    (("2" (lemma "HK_subgroup[T1, *, one1]")
                      (("2" (inst -1 "G!1" "H!1" "N!1")
                        (("1" (assert)
                          (("1" (expand "subgroup?")
                            (("1" (flatten)
                              (("1"
                                (hide -1)
                                (("1"
                                  (lemma
                                   "H_K_are_subgroups[T1, *, one1]")
                                  (("1"
                                    (inst?)
                                    (("1"
                                      (assert)
                                      (("1"
                                        (flatten)
                                        (("1"
                                          (hide -2)
                                          (("1"
                                            (lemma
                                             "HK_subgroup_permute[T1, *, one1]")
                                            (("1"
                                              (inst?)
                                              (("1"
                                                (assert)
                                                (("1"
                                                  (replaces -1)
                                                  (("1"
                                                    (expand*
                                                     "subgroup?"
                                                     "subset?"
                                                     "member")
                                                    (("1"
                                                      (inst?)
                                                      (("1"
                                                        (assert)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2"
                                                (hide-all-but 1)
                                                (("2"
                                                  (typepred "N!1")
                                                  (("2"
                                                    (expand
                                                     "normal_subgroup?")
                                                    (("2"
                                                      (assert)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil)
                                     ("2"
                                      (hide-all-but 1)
                                      (("2"
                                        (typepred "N!1")
                                        (("2"
                                          (expand "normal_subgroup?")
                                          (("2" (assertnil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (hide-all-but 1)
                          (("2" (typepred "N!1")
                            (("2" (expand "normal_subgroup?")
                              (("2" (assertnil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((homomorphism? const-decl "bool" homomorphisms "algebra/")
    (x!1 skolem-const-decl "(H!1)" isomorphism_theorems nil)
    (mult_lem formula-decl nil factor_groups "algebra/")
    (NOT const-decl "[bool -> bool]" booleans nil)
    (one_left formula-decl nil group "algebra/")
    (monoid? const-decl "bool" monoid_def "algebra/")
    (member const-decl "bool" sets nil)
    (monad? const-decl "bool" monad_def "algebra/")
    (normal_left_is_right formula-decl nil normal_subgroups "algebra/")
    (a!1 skolem-const-decl "(prod[T1, *, one1](N!1, H!1))"
     isomorphism_theorems nil)
    (subset? const-decl "bool" sets nil)
    (rc_is_eq formula-decl nil cosets "algebra/")
    (inv const-decl "{y | x * y = one AND y * x = one}" group
     "algebra/")
    (h!1 skolem-const-decl "(N!1)" isomorphism_theorems nil)
    (inv_left formula-decl nil group "algebra/")
    (inv_in formula-decl nil group "algebra/")
    (right_coset_one formula-decl nil cosets "algebra/")
    (right_coset_assoc formula-decl nil cosets "algebra/")
    (k!1 skolem-const-decl "(H!1)" isomorphism_theorems nil)
    (product_in formula-decl nil group "algebra/")
    (surjective? const-decl "bool" functions nil)
    (left_cosets const-decl "setofsets[T]" right_left_cosets nil)
    (restrict const-decl "R" restrict nil)
    (x!1 skolem-const-decl "(H!1)" isomorphism_theorems nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (left_cosets_group formula-decl nil factor_groups "algebra/")
    (HK_subgroup formula-decl nil products_subgroups nil)
    (T1_is_group formula-decl nil isomorphism_theorems nil)
    (H_K_are_subgroups formula-decl nil products_subgroups nil)
    (normal_subgroup_tran formula-decl nil groups_scaf nil)
    (HK_subgroup_permute formula-decl nil products_subgroups nil)
    (lc_is_eq formula-decl nil cosets "algebra/")
    (left_coset_one formula-decl nil cosets "algebra/")
    (lc_eq formula-decl nil cosets "algebra/")
    (homomorphism type-eq-decl nil homomorphisms "algebra/")
    (kernel const-decl "subgroup[T1, *, one1](G)" homomorphisms
     "algebra/")
    (intersection const-decl "set" sets nil)
    (fullset const-decl "set" sets nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (set type-eq-decl nil sets nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (one1 formal-const-decl "T1" isomorphism_theorems nil)
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/")
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (subgroup type-eq-decl nil group "algebra/")
    (H!1 skolem-const-decl "subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (prod const-decl "set[T]" products_subgroups nil)
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (N!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (* const-decl "set[T]" cosets "algebra/")
    (left_cosets type-eq-decl nil cosets "algebra/")
    (mult const-decl "left_cosets(G, H)" factor_groups "algebra/")
    (/ const-decl "group[left_cosets(G, N), mult(G, N), N]"
       right_left_cosets nil))
   shostak))
 (second_isomorphism_th_TCC1 0
  (second_isomorphism_th_TCC1-1 nil 3524769565
   ("" (skosimp*)
    (("" (replaces -1)
      (("" (lemma "subgroup_intersection[T1, *, one1]")
        (("1" (inst -1 "G!1" "N!1" "H!1")
          (("1" (assert)
            (("1" (expand "subgroup?")
              (("1" (hide (-1 2))
                (("1" (typepred "N!1")
                  (("1" (expand "normal_subgroup?")
                    (("1" (flatten)
                      (("1" (hide -3)
                        (("1" (expand "subgroup?")
                          (("1" (propax) nil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil)
         ("2" (rewrite "G_TCC1"nil nil))
        nil))
      nil))
    nil)
   ((G_TCC1 assuming-tcc nil isomorphism_theorems nil)
    (group nonempty-type-eq-decl nil group "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/")
    (subgroup type-eq-decl nil group "algebra/")
    (NOT const-decl "[bool -> bool]" booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (set type-eq-decl nil sets nil)
    (group? const-decl "bool" group_def "algebra/")
    (fullset const-decl "set" sets nil)
    (subgroup_intersection formula-decl nil groups_scaf nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (one1 formal-const-decl "T1" isomorphism_theorems nil))
   nil))
 (second_isomorphism_th_TCC2 0
  (second_isomorphism_th_TCC2-1 nil 3524769565
   ("" (skosimp*)
    (("" (replaces -3)
      (("" (lemma "HK_subgroup[T1, *, one1]")
        (("" (inst -1 "G!1" "H!1" "N!1")
          (("1" (assert)
            (("1" (lemma "HK_subgroup_permute[T1, *, one1]")
              (("1" (inst?)
                (("1" (assert)
                  (("1" (expand "subgroup?") (("1" (propax) nil nil))
                    nil))
                  nil)
                 ("2" (hide-all-but 1)
                  (("2" (typepred "N!1")
                    (("2" (expand "normal_subgroup?")
                      (("2" (assertnil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (hide-all-but 1)
            (("2" (typepred "N!1")
              (("2" (expand "normal_subgroup?")
                (("2" (assertnil nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((one1 formal-const-decl "T1" isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (HK_subgroup formula-decl nil products_subgroups nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (HK_subgroup_permute formula-decl nil products_subgroups nil)
    (subgroup type-eq-decl nil group "algebra/")
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (set type-eq-decl nil sets nil)
    (group? const-decl "bool" group_def "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (N!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil))
   nil))
 (second_isomorphism_th_TCC3 0
  (second_isomorphism_th_TCC3-1 nil 3524769565
   ("" (skosimp*)
    (("" (replaces -3)
      (("" (hide -1)
        (("" (lemma "HK_subgroup[T1, *, one1]")
          (("" (inst -1 "G!1" "H!1" "N!1")
            (("1" (assert)
              (("1" (lemma "HK_subgroup_permute[T1, *, one1]")
                (("1" (inst?)
                  (("1" (assert)
                    (("1" (replaces -1)
                      (("1" (expand "subgroup?" -1)
                        (("1" (flatten)
                          (("1"
                            (lemma "H_K_are_subgroups[T1, *, one1]")
                            (("1" (inst?)
                              (("1"
                                (assert)
                                (("1"
                                  (prop)
                                  (("1"
                                    (hide -2)
                                    (("1"
                                      (lemma
                                       "normal_subgroup_tran[T1, *, one1]")
                                      (("1"
                                        (inst
                                         -1
                                         "G!1"
                                         "prod[T1, *, one1](N!1, H!1)"
                                         "N!1")
                                        (("1" (assertnil nil))
                                        nil)
                                       ("2"
                                        (rewrite "G_TCC1")
                                        nil
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil)
                               ("2"
                                (hide-all-but 1)
                                (("2"
                                  (typepred "N!1")
                                  (("2"
                                    (expand "normal_subgroup?")
                                    (("2" (assertnil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (hide-all-but 1)
                    (("2" (typepred "N!1")
                      (("2" (expand "normal_subgroup?")
                        (("2" (assertnil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (hide-all-but 1)
              (("2" (typepred "N!1")
                (("2" (expand "normal_subgroup?")
                  (("2" (assertnil nil)) nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((N!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (subgroup type-eq-decl nil group "algebra/")
    (HK_subgroup_permute formula-decl nil products_subgroups nil)
    (H_K_are_subgroups formula-decl nil products_subgroups nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (G_TCC1 assuming-tcc nil isomorphism_theorems nil)
    (prod const-decl "set[T]" products_subgroups nil)
    (fullset const-decl "set" sets nil)
    (normal_subgroup_tran formula-decl nil groups_scaf nil)
    (HK_subgroup formula-decl nil products_subgroups nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (one1 formal-const-decl "T1" isomorphism_theorems nil))
   nil))
 (second_isomorphism_th_TCC4 0
  (second_isomorphism_th_TCC4-1 nil 3524769565
   ("" (skosimp*)
    (("" (inst 1 "one1")
      (("1" (rewrite "left_coset_one"nil nil)
       ("2" (hide-all-but 1)
        (("2" (typepred "H!1")
          (("2" (expand"group?" "monoid?" "monad?" "member")
            (("2" (assertnil nil)) nil))
          nil))
        nil))
      nil))
    nil)
   ((H!1 skolem-const-decl "subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (subgroup type-eq-decl nil group "algebra/")
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (subgroup? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (one1 formal-const-decl "T1" isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (intersection const-decl "set" sets nil)
    (left_coset_one formula-decl nil cosets "algebra/")
    (NOT const-decl "[bool -> bool]" booleans nil)
    (monad? const-decl "bool" monad_def "algebra/")
    (member const-decl "bool" sets nil)
    (monoid? const-decl "bool" monoid_def "algebra/"))
   nil))
 (second_isomorphism_th_TCC5 0
  (second_isomorphism_th_TCC5-1 nil 3524769565
   ("" (skosimp*)
    (("" (inst 1 "one1")
      (("1" (rewrite "left_coset_one"nil nil)
       ("2" (lemma "HK_subgroup[T1, *, one1]")
        (("2" (inst -1 "G!1" "H!1" "N!1")
          (("1" (assert)
            (("1" (expand "subgroup?")
              (("1" (flatten)
                (("1" (lemma "HK_subgroup_permute[T1, *, one1]")
                  (("1" (inst?)
                    (("1" (assert)
                      (("1"
                        (expand"group?" "monoid?" "monad?" "member")
                        nil nil))
                      nil)
                     ("2" (hide-all-but 1)
                      (("2" (typepred "N!1")
                        (("2" (expand "normal_subgroup?")
                          (("2" (assertnil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (hide-all-but 1)
            (("2" (typepred "N!1")
              (("2" (expand "normal_subgroup?")
                (("2" (assertnil nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((one1 formal-const-decl "T1" isomorphism_theorems nil)
    (NH!1 skolem-const-decl "set[T1]" isomorphism_theorems nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (left_coset_one formula-decl nil cosets "algebra/")
    (N!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (subgroup? const-decl "bool" group_def "algebra/")
    (subgroup type-eq-decl nil group "algebra/")
    (HK_subgroup_permute formula-decl nil products_subgroups nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (member const-decl "bool" sets nil)
    (monad? const-decl "bool" monad_def "algebra/")
    (monoid? const-decl "bool" monoid_def "algebra/")
    (HK_subgroup formula-decl nil products_subgroups nil))
   nil))
 (second_isomorphism_th_TCC6 0
  (second_isomorphism_th_TCC6-1 nil 3524769565
   ("" (skosimp*)
    (("" (rewrite "left_cosets_group[T1, *, one1]"nil nil)) nil)
   ((left_cosets_group formula-decl nil factor_groups "algebra/")
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (set type-eq-decl nil sets nil)
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/")
    (subgroup type-eq-decl nil group "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (intersection const-decl "set" sets nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (one1 formal-const-decl "T1" isomorphism_theorems nil))
   nil))
 (second_isomorphism_th_TCC7 0
  (second_isomorphism_th_TCC7-1 nil 3524769565
   ("" (skosimp*)
    (("" (rewrite "left_cosets_group[T1, *, one1]")
      (("1" (hide (-1 2))
        (("1" (replaces -2)
          (("1" (lemma "HK_subgroup[T1, *, one1]")
            (("1" (inst -1 "G!1" "H!1" "N!1")
              (("1" (assert)
                (("1" (lemma "HK_subgroup_permute[T1, *, one1]")
                  (("1" (inst?)
                    (("1" (assert)
                      (("1" (replaces -1)
                        (("1" (lemma "H_K_are_subgroups[T1, *, one1]")
                          (("1" (inst?)
                            (("1" (expand "subgroup?" -2)
                              (("1"
                                (flatten)
                                (("1"
                                  (assert)
                                  (("1"
                                    (flatten)
                                    (("1"
                                      (hide -2)
                                      (("1"
                                        (lemma
                                         "normal_subgroup_tran[T1, *, one1]")
                                        (("1"
                                          (inst
                                           -1
                                           "G!1"
                                           "prod[T1, *, one1](N!1, H!1)"
                                           "N!1")
                                          (("1" (assertnil nil))
                                          nil)
                                         ("2"
                                          (rewrite "G_TCC1")
                                          nil
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil)
                             ("2" (hide-all-but 1)
                              (("2"
                                (typepred "N!1")
                                (("2"
                                  (expand "normal_subgroup?")
                                  (("2" (assertnil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (hide-all-but 1)
                      (("2" (typepred "N!1")
                        (("2" (expand "normal_subgroup?")
                          (("2" (assertnil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (hide-all-but 1)
                (("2" (typepred "N!1")
                  (("2" (expand "normal_subgroup?")
                    (("2" (assertnil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (hide (-1 2))
        (("2" (replaces -2)
          (("2" (lemma "HK_subgroup[T1, *, one1]")
            (("2" (inst -1 "G!1" "H!1" "N!1")
              (("1" (assert)
                (("1" (lemma "HK_subgroup_permute[T1, *, one1]")
                  (("1" (inst?)
                    (("1" (assert)
                      (("1" (replaces -1)
                        (("1" (expand "subgroup?")
                          (("1" (propax) nil nil)) nil))
                        nil))
                      nil)
                     ("2" (hide-all-but 1)
                      (("2" (typepred "N!1")
                        (("2" (expand "normal_subgroup?")
                          (("2" (assertnil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (hide-all-but 1)
                (("2" (typepred "N!1")
                  (("2" (expand "normal_subgroup?")
                    (("2" (assertnil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((left_cosets_group formula-decl nil factor_groups "algebra/")
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (set type-eq-decl nil sets nil)
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (one1 formal-const-decl "T1" isomorphism_theorems nil)
    (HK_subgroup formula-decl nil products_subgroups nil)
    (normal_subgroup_tran formula-decl nil groups_scaf nil)
    (fullset const-decl "set" sets nil)
    (prod const-decl "set[T]" products_subgroups nil)
    (G_TCC1 assuming-tcc nil isomorphism_theorems nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (H_K_are_subgroups formula-decl nil products_subgroups nil)
    (HK_subgroup_permute formula-decl nil products_subgroups nil)
    (subgroup type-eq-decl nil group "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/")
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (N!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil))
   nil))
 (second_isomorphism_th 0
  (second_isomorphism_th-1 nil 3524840934
   ("" (skosimp*)
    (("" (skoletin* 1)
      (("1" (lemma "second_isomorphism_th_aux")
        (("1" (inst -1 "G!1" "H!1" "N!1")
          (("1" (assert)
            (("1" (skosimp)
              (("1"
                (lemma
                 "first_isomorphism_th[T1, *, one1, left_cosets[T1, *, one1](prod[T1, *, one1](N!1, H!1), N!1), mult(prod[T1, *, one1](N!1, H!1), N!1), N!1]")
                (("1"
                  (inst -1 "H!1" "prod[T1, *, one1](N!1, H!1) / N!1"
                   "varphi!1")
                  (("1" (assert)
                    (("1" (replace -4 -1)
                      (("1"
                        (case-replace "extend
                               [left_cosets[T1, *, one1](prod[T1, *, one1](N!1, H!1), N!1),
                                (prod[T1, *, one1](N!1, H!1) / N!1), bool, FALSE]
                               (image(varphi!1, restrict[T1, (H!1), bool](H!1))) = prod[T1, *, one1](N!1, H!1) / N!1")
                        (("1" (prop)
                          (("1" (hide (-1 -2 -4))
                            (("1"
                              (lemma
                               "kernel_normal[T1, *, one1, left_cosets[T1, *, one1](prod[T1, *, one1](N!1, H!1), N!1), mult(prod[T1, *, one1](N!1, H!1), N!1), N!1]")
                              (("1"
                                (inst
                                 -1
                                 "H!1"
                                 "prod[T1, *, one1](N!1, H!1) / N!1"
                                 "varphi!1")
                                (("1" (assertnil nil))
                                nil))
                              nil))
                            nil)
                           ("2" (replaces -6)
                            (("2" (hide-all-but (-2 1))
                              (("2"
                                (expand "isomorphic?")
                                (("2"
                                  (skosimp)
                                  (("2"
                                    (inst 1 "phi!1")
                                    (("2"
                                      (expand "isomorphism?")
                                      (("2"
                                        (prop)
                                        (("2"
                                          (hide -1)
                                          (("2"
                                            (expand "homomorphism?")
                                            (("2"
                                              (skosimp*)
                                              (("2"
                                                (inst -1 "a!1" "b!1")
                                                (("2"
                                                  (assert)
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (hide (-1 -4 2))
                          (("2" (decompose-equality 1)
                            (("2" (expand "extend")
                              (("2"
                                (lift-if)
                                (("2"
                                  (prop)
                                  (("2"
                                    (expand "image")
                                    (("2"
                                      (expand "surjective?")
                                      (("2"
                                        (inst -3 "x!1")
                                        (("2"
                                          (skosimp)
                                          (("2"
                                            (inst 1 "x!2")
                                            (("1" (assertnil nil)
                                             ("2"
                                              (expand "restrict")
                                              (("2" (propax) nil nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (hide-all-but 1)
                  (("2" (rewrite "left_cosets_group[T1, *, one1]")
                    (("1" (hide-all-but 1)
                      (("1" (lemma "HK_subgroup[T1, *, one1]")
                        (("1" (inst -1 "G!1" "H!1" "N!1")
                          (("1" (assert)
                            (("1"
                              (lemma
                               "HK_subgroup_permute[T1, *, one1]")
                              (("1"
                                (inst?)
                                (("1"
                                  (assert)
                                  (("1"
                                    (replaces -1)
                                    (("1"
                                      (lemma
                                       "H_K_are_subgroups[T1, *, one1]")
                                      (("1"
                                        (inst?)
                                        (("1"
                                          (prop)
                                          (("1"
                                            (hide -2)
                                            (("1"
                                              (lemma
                                               "normal_subgroup_tran[T1, *, one1]")
                                              (("1"
                                                (inst
                                                 -1
                                                 "G!1"
                                                 "prod[T1, *, one1](N!1, H!1)"
                                                 "N!1")
                                                (("1"
                                                  (assert)
                                                  nil
                                                  nil))
                                                nil)
                                               ("2"
                                                (rewrite "G_TCC1")
                                                nil
                                                nil))
                                              nil))
                                            nil)
                                           ("2"
                                            (expand "subgroup?")
                                            (("2" (assertnil nil))
                                            nil))
                                          nil)
                                         ("2"
                                          (hide 2)
                                          (("2"
                                            (hide-all-but 1)
                                            (("2"
                                              (typepred "N!1")
                                              (("2"
                                                (expand
                                                 "normal_subgroup?")
                                                (("2"
                                                  (assert)
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (hide-all-but 1)
                                  (("2"
                                    (typepred "N!1")
                                    (("2"
                                      (expand "normal_subgroup?")
                                      (("2" (assertnil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (hide-all-but 1)
                            (("2" (typepred "N!1")
                              (("2"
                                (expand "normal_subgroup?")
                                (("2" (assertnil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (hide 2)
                      (("2" (lemma "HK_subgroup[T1, *, one1]")
                        (("2" (inst -1 "G!1" "H!1" "N!1")
                          (("1" (assert)
                            (("1"
                              (lemma
                               "HK_subgroup_permute[T1, *, one1]")
                              (("1"
                                (inst?)
                                (("1"
                                  (assert)
                                  (("1"
                                    (replaces -1)
                                    (("1"
                                      (lemma
                                       "H_K_are_subgroups[T1, *, one1]")
                                      (("1"
                                        (inst?)
                                        (("1"
                                          (prop)
                                          (("1"
                                            (hide -2)
                                            (("1"
                                              (expand "subgroup?")
                                              (("1" (propax) nil nil))
                                              nil))
                                            nil)
                                           ("2"
                                            (expand "subgroup?")
                                            (("2" (propax) nil nil))
                                            nil))
                                          nil)
                                         ("2"
                                          (hide 2)
                                          (("2"
                                            (typepred "N!1")
                                            (("2"
                                              (expand
                                               "normal_subgroup?")
                                              (("2" (assertnil nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (typepred "N!1")
                                  (("2"
                                    (expand "normal_subgroup?")
                                    (("2" (assertnil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (typepred "N!1")
                            (("2" (expand "normal_subgroup?")
                              (("2" (assertnil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("3" (hide-all-but 1)
                  (("3" (rewrite "G_TCC1"nil nil)) nil)
                 ("4" (hide-all-but 1)
                  (("4" (inst 1 "one1")
                    (("1" (rewrite "left_coset_one[T1,*,one1]"nil
                      nil)
                     ("2" (lemma "HK_subgroup[T1, *, one1]")
                      (("2" (inst -1 "G!1" "H!1" "N!1")
                        (("1" (assert)
                          (("1"
                            (lemma "HK_subgroup_permute[T1, *, one1]")
                            (("1" (inst?)
                              (("1"
                                (assert)
                                (("1"
                                  (replaces -1)
                                  (("1"
                                    (expand*
                                     "subgroup?"
                                     "group?"
                                     "monoid?"
                                     "monad?"
                                     "member")
                                    nil
                                    nil))
                                  nil))
                                nil)
                               ("2"
                                (typepred "N!1")
                                (("2"
                                  (expand "normal_subgroup?")
                                  (("2" (assertnil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (typepred "N!1")
                          (("2" (expand "normal_subgroup?")
                            (("2" (assertnil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (skosimp)
        (("2" (rewrite "left_cosets_group[T1, *, one1]")
          (("1" (replaces -1)
            (("1" (replaces -2)
              (("1" (hide (-1 2))
                (("1" (lemma "HK_subgroup[T1, *, one1]")
                  (("1" (inst -1 "G!1" "H!1" "N!1")
                    (("1" (assert)
                      (("1" (lemma "HK_subgroup_permute[T1, *, one1]")
                        (("1" (inst?)
                          (("1" (assert)
                            (("1" (replaces -1)
                              (("1"
                                (lemma
                                 "H_K_are_subgroups[T1, *, one1]")
                                (("1"
                                  (inst?)
                                  (("1"
                                    (prop)
                                    (("1"
                                      (hide -2)
                                      (("1"
                                        (lemma
                                         "normal_subgroup_tran[T1, *, one1]")
                                        (("1"
                                          (inst
                                           -1
                                           "G!1"
                                           "prod[T1, *, one1](N!1, H!1)"
                                           "N!1")
                                          (("1" (assertnil nil))
                                          nil)
                                         ("2"
                                          (rewrite "G_TCC1")
                                          nil
                                          nil))
                                        nil))
                                      nil)
                                     ("2"
                                      (expand "subgroup?")
                                      (("2" (assertnil nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (hide 2)
                                    (("2"
                                      (typepred "N!1")
                                      (("2"
                                        (expand "normal_subgroup?")
                                        (("2" (assertnil nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (typepred "N!1")
                            (("2" (expand "normal_subgroup?")
                              (("2" (assertnil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (typepred "N!1")
                      (("2" (expand "normal_subgroup?")
                        (("2" (assertnil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (replaces -1)
            (("2" (replaces -2)
              (("2" (hide (-1 2))
                (("2" (lemma "HK_subgroup[T1, *, one1]")
                  (("2" (inst -1 "G!1" "H!1" "N!1")
                    (("1" (assert)
                      (("1" (lemma "HK_subgroup_permute[T1, *, one1]")
                        (("1" (inst?)
                          (("1" (assert)
                            (("1" (replaces -1)
                              (("1"
                                (expand "subgroup?")
                                (("1" (propax) nil nil))
                                nil))
                              nil))
                            nil)
                           ("2" (typepred "N!1")
                            (("2" (expand "normal_subgroup?")
                              (("2" (assertnil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (typepred "N!1")
                      (("2" (expand "normal_subgroup?")
                        (("2" (assertnil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("3" (skosimp)
        (("3" (rewrite "left_cosets_group[T1, *, one1]"nil nil)) nil)
       ("4" (skosimp)
        (("4" (hide -2)
          (("4" (inst 1 "one1")
            (("1" (rewrite "left_coset_one[T1,*,one1]"nil nil)
             ("2" (replaces -1)
              (("2" (replaces -1)
                (("2" (lemma "HK_subgroup[T1, *, one1]")
                  (("2" (inst -1 "G!1" "H!1" "N!1")
                    (("1" (assert)
                      (("1" (lemma "HK_subgroup_permute[T1, *, one1]")
                        (("1" (inst?)
                          (("1" (assert)
                            (("1" (replaces -1)
                              (("1"
                                (expand*
                                 "subgroup?"
                                 "group?"
                                 "monoid?"
                                 "monad?"
                                 "member")
                                nil
                                nil))
                              nil))
                            nil)
                           ("2" (typepred "N!1")
                            (("2" (expand "normal_subgroup?")
                              (("2" (assertnil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (typepred "N!1")
                      (("2" (expand "normal_subgroup?")
                        (("2" (assertnil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("5" (skosimp)
        (("5" (hide -2)
          (("5" (inst 1 "one1")
            (("1" (rewrite "left_coset_one[T1,*,one1]"nil nil)
             ("2" (replaces -1)
              (("2" (replaces -1)
                (("2" (typepred "H!1")
                  (("2"
                    (expand"subgroup?" "group?" "monoid?" "monad?"
                     "member")
                    (("2" (assertnil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("6" (skosimp)
        (("6" (replaces -1)
          (("6" (replaces -2)
            (("6" (hide -1)
              (("6" (lemma "HK_subgroup[T1, *, one1]")
                (("6" (inst -1 "G!1" "H!1" "N!1")
                  (("1" (assert)
                    (("1" (lemma "HK_subgroup_permute[T1, *, one1]")
                      (("1" (inst?)
                        (("1" (assert)
                          (("1" (replaces -1)
                            (("1"
                              (lemma "H_K_are_subgroups[T1, *, one1]")
                              (("1"
                                (inst?)
                                (("1"
                                  (prop)
                                  (("1"
                                    (hide -2)
                                    (("1"
                                      (lemma
                                       "normal_subgroup_tran[T1, *, one1]")
                                      (("1"
                                        (inst
                                         -1
                                         "G!1"
                                         "prod[T1, *, one1](N!1, H!1)"
                                         "N!1")
                                        (("1" (assertnil nil))
                                        nil)
                                       ("2"
                                        (rewrite "G_TCC1")
                                        nil
                                        nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (expand "subgroup?")
                                    (("2" (assertnil nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (typepred "N!1")
                                  (("2"
                                    (expand "normal_subgroup?")
                                    (("2" (assertnil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (typepred "N!1")
                          (("2" (expand "normal_subgroup?")
                            (("2" (assertnil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (typepred "N!1")
                    (("2" (expand "normal_subgroup?")
                      (("2" (assertnil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("7" (skosimp)
        (("7" (replaces -1)
          (("7" (replaces -2)
            (("7" (hide -1)
              (("7" (lemma "HK_subgroup[T1, *, one1]")
                (("7" (inst -1 "G!1" "H!1" "N!1")
                  (("1" (assert)
                    (("1" (lemma "HK_subgroup_permute[T1, *, one1]")
                      (("1" (inst?)
                        (("1" (assert)
                          (("1" (replaces -1)
                            (("1"
                              (expand"subgroup?" "group?" "monoid?"
                               "monad?" "member")
                              nil nil))
                            nil))
                          nil)
                         ("2" (typepred "N!1")
                          (("2" (expand "normal_subgroup?")
                            (("2" (assertnil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (typepred "N!1")
                    (("2" (expand "normal_subgroup?")
                      (("2" (assertnil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("8" (skosimp)
        (("8" (hide (-1 -3))
          (("8" (expand "normal_subgroup?")
            (("8" (flatten)
              (("8" (hide -2)
                (("8" (expand "subgroup?") (("8" (assertnil nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("9" (skosimp)
        (("9" (hide (-1 -2))
          (("9" (lemma "subgroup_intersection[T1, *, one1]")
            (("1" (inst -1 "G!1" "N!1" "H!1")
              (("1" (assert)
                (("1" (typepred "N!1")
                  (("1" (expand "normal_subgroup?")
                    (("1" (flatten)
                      (("1" (assert)
                        (("1" (hide (-1 -2 -3))
                          (("1" (expand "subgroup?")
                            (("1" (propax) nil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (rewrite "G_TCC1"nil nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (bijective? const-decl "bool" functions nil)
    (id const-decl "(bijective?[T, T])" identity nil)
    (TRUE const-decl "bool" booleans nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (set type-eq-decl nil sets nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (one1 formal-const-decl "T1" isomorphism_theorems nil)
    (prod const-decl "set[T]" products_subgroups nil)
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/")
    (subgroup type-eq-decl nil group "algebra/")
    (intersection const-decl "set" sets nil)
    (* const-decl "set[T]" cosets "algebra/")
    (left_cosets type-eq-decl nil cosets "algebra/")
    (mult const-decl "left_cosets(G, H)" factor_groups "algebra/")
    (fullset const-decl "set" sets nil)
    (IFF const-decl "[bool, bool -> bool]" booleans nil)
    (isomorphic? const-decl "bool" homomorphisms "algebra/")
    (/ const-decl "group[left_cosets(G, N), mult(G, N), N]"
       right_left_cosets nil)
    (member const-decl "bool" sets nil)
    (monad? const-decl "bool" monad_def "algebra/")
    (monoid? const-decl "bool" monoid_def "algebra/")
    (left_coset_one formula-decl nil cosets "algebra/")
    (HK_subgroup_permute formula-decl nil products_subgroups nil)
    (H_K_are_subgroups formula-decl nil products_subgroups nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (normal_subgroup_tran formula-decl nil groups_scaf nil)
    (G_TCC1 assuming-tcc nil isomorphism_theorems nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (HK_subgroup formula-decl nil products_subgroups nil)
    (left_cosets_group formula-decl nil factor_groups "algebra/")
    (homomorphism type-eq-decl nil homomorphisms "algebra/")
    (homomorphism? const-decl "bool" homomorphisms "algebra/")
    (surjective? const-decl "bool" functions nil)
    (x!2 skolem-const-decl "(H!1)" isomorphism_theorems nil)
    (x!1 skolem-const-decl
     "left_cosets[T1, *, one1](prod[T1, *, one1](N!1, H!1), N!1)"
     isomorphism_theorems nil)
    (H!1 skolem-const-decl "subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (N!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (kernel_normal formula-decl nil homomorphism_lemmas nil)
    (isomorphism? const-decl "bool" homomorphisms "algebra/")
    (FALSE const-decl "bool" booleans nil)
    (extend const-decl "R" extend nil)
    (image const-decl "set[R]" function_image nil)
    (restrict const-decl "R" restrict nil)
    (first_isomorphism_th formula-decl nil homomorphism_lemmas nil)
    (second_isomorphism_th_aux formula-decl nil isomorphism_theorems
     nil))
   shostak))
 (third_isomorphism_th_aux_TCC1 0
  (third_isomorphism_th_aux_TCC1-1 nil 3524797530
   ("" (skosimp*)
    (("" (inst 1 "one1")
      (("1" (rewrite "left_coset_one"nil nil)
       ("2" (hide-all-but 1)
        (("2" (typepred "G!1")
          (("2" (expand"group?" "monoid?" "monad?" "member")
            (("2" (assertnil nil)) nil))
          nil))
        nil))
      nil))
    nil)
   ((G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (group nonempty-type-eq-decl nil group "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (one1 formal-const-decl "T1" isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (left_coset_one formula-decl nil cosets "algebra/")
    (NOT const-decl "[bool -> bool]" booleans nil)
    (monad? const-decl "bool" monad_def "algebra/")
    (member const-decl "bool" sets nil)
    (monoid? const-decl "bool" monoid_def "algebra/"))
   nil))
 (third_isomorphism_th_aux_TCC2 0
  (third_isomorphism_th_aux_TCC2-1 nil 3524797530
   ("" (skosimp*)
    (("" (rewrite "left_cosets_group[T1, *, one1]"nil nil)) nil)
   ((left_cosets_group formula-decl nil factor_groups "algebra/")
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (set type-eq-decl nil sets nil)
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (one1 formal-const-decl "T1" isomorphism_theorems nil))
   nil))
 (third_isomorphism_th_aux_TCC3 0
  (third_isomorphism_th_aux_TCC3-1 nil 3524797530
   ("" (skosimp*)
    (("" (rewrite "left_cosets_group[T1, *, one1]"nil nil)) nil)
   ((left_cosets_group formula-decl nil factor_groups "algebra/")
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (set type-eq-decl nil sets nil)
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (one1 formal-const-decl "T1" isomorphism_theorems nil))
   nil))
 (third_isomorphism_th_aux 0
  (third_isomorphism_th_aux-1 nil 3524915806
   ("" (skosimp*)
    ((""
      (inst 1
       "(LAMBDA (A: left_cosets(G!1,N!1)): *[T1, *, one1](lc_gen(G!1,N!1,A), M!1))")
      (("1" (prop)
        (("1" (expand "homomorphism?")
          (("1" (skosimp*)
            (("1" (expand "restrict")
              (("1" (typepred "a!1" "b!1")
                (("1" (hide (-2 -4))
                  (("1" (skosimp*)
                    (("1" (replace -1 1)
                      (("1" (replace -2 1)
                        (("1" (lemma "lc_gen_normal[T1, *, one1]")
                          (("1" (copy -1)
                            (("1" (inst -1 "G!1" "N!1" "a!2")
                              (("1"
                                (inst -2 "G!1" "N!1" "a!3")
                                (("1"
                                  (assert)
                                  (("1"
                                    (skosimp*)
                                    (("1"
                                      (replaces -1)
                                      (("1"
                                        (replaces -1)
                                        (("1"
                                          (rewrite
                                           "mult_lem[T1, *, one1]")
                                          (("1"
                                            (lemma
                                             "lc_gen_normal[T1, *, one1]")
                                            (("1"
                                              (inst
                                               -1
                                               "G!1"
                                               "N!1"
                                               "a!2 * a!3")
                                              (("1"
                                                (assert)
                                                (("1"
                                                  (prop)
                                                  (("1"
                                                    (skosimp)
                                                    (("1"
                                                      (replaces -1)
                                                      (("1"
                                                        (typepred
                                                         "h!1"
                                                         "h!2"
                                                         "h!3")
                                                        (("1"
                                                          (expand
                                                           "subgroup?")
                                                          (("1"
                                                            (expand*
                                                             "subset?"
                                                             "member")
                                                            (("1"
                                                              (copy -6)
                                                              (("1"
                                                                (copy
                                                                 -7)
                                                                (("1"
                                                                  (inst
                                                                   -1
                                                                   "h!1")
                                                                  (("1"
                                                                    (inst
                                                                     -2
                                                                     "h!2")
                                                                    (("1"
                                                                      (inst
                                                                       -8
                                                                       "h!3")
                                                                      (("1"
                                                                        (assert)
                                                                        (("1"
                                                                          (hide
                                                                           (-3
                                                                            -4
                                                                            -5
                                                                            -6
                                                                            -7))
                                                                          (("1"
                                                                            (lemma
                                                                             "normal_left_is_right[T1, *, one1]")
                                                                            (("1"
                                                                              (copy
                                                                               -1)
                                                                              (("1"
                                                                                (copy
                                                                                 -1)
                                                                                (("1"
                                                                                  (inst
                                                                                   -1
                                                                                   "G!1"
                                                                                   "M!1")
                                                                                  (("1"
                                                                                    (assert)
                                                                                    (("1"
                                                                                      (inst
                                                                                       -2
                                                                                       "G!1"
                                                                                       "M!1")
                                                                                      (("1"
                                                                                        (assert)
                                                                                        (("1"
                                                                                          (inst
                                                                                           -3
                                                                                           "G!1"
                                                                                           "M!1")
                                                                                          (("1"
                                                                                            (assert)
                                                                                            (("1"
                                                                                              (typepred
                                                                                               "M!1")
                                                                                              (("1"
                                                                                                (hide
                                                                                                 -1)
                                                                                                (("1"
                                                                                                  (expand
                                                                                                   "normal_subgroup?")
                                                                                                  (("1"
                                                                                                    (flatten)
                                                                                                    (("1"
                                                                                                      (hide
                                                                                                       (-1
                                                                                                        -2
                                                                                                        -3
                                                                                                        -5
                                                                                                        -7))
                                                                                                      (("1"
                                                                                                        (inst
                                                                                                         -1
                                                                                                         "h!3 * (a!2 * a!3)")
                                                                                                        (("1"
                                                                                                          (inst
                                                                                                           -2
                                                                                                           "h!1 * a!2")
                                                                                                          (("1"
                                                                                                            (inst
                                                                                                             -3
                                                                                                             "h!2 * a!3")
                                                                                                            (("1"
                                                                                                              (replaces
                                                                                                               -1)
                                                                                                              (("1"
                                                                                                                (replaces
                                                                                                                 -1)
                                                                                                                (("1"
                                                                                                                  (replaces
                                                                                                                   -1)
                                                                                                                  (("1"
                                                                                                                    (lemma
                                                                                                                     "right_coset_assoc[T1, *, one1]")
                                                                                                                    (("1"
                                                                                                                      (copy
                                                                                                                       -1)
                                                                                                                      (("1"
                                                                                                                        (copy
                                                                                                                         -1)
                                                                                                                        (("1"
                                                                                                                          (inst
                                                                                                                           -1
                                                                                                                           "M!1"
                                                                                                                           "h!3"
                                                                                                                           "a!2 * a!3")
                                                                                                                          (("1"
                                                                                                                            (inst
                                                                                                                             -2
                                                                                                                             "M!1"
                                                                                                                             "h!1"
                                                                                                                             "a!2")
                                                                                                                            (("1"
                                                                                                                              (inst
                                                                                                                               -3
                                                                                                                               "M!1"
                                                                                                                               "h!2"
                                                                                                                               "a!3")
                                                                                                                              (("1"
                                                                                                                                (lemma
                                                                                                                                 "rc_is_eq[T1, *, one1]")
                                                                                                                                (("1"
                                                                                                                                  (inst
                                                                                                                                   -1
                                                                                                                                   "G!1"
                                                                                                                                   "M!1"
                                                                                                                                   "h!1"
                                                                                                                                   "one1")
                                                                                                                                  (("1"
                                                                                                                                    (reveal
                                                                                                                                     -8)
                                                                                                                                    (("1"
                                                                                                                                      (assert)
                                                                                                                                      (("1"
                                                                                                                                        (prop)
                                                                                                                                        (("1"
                                                                                                                                          (lemma
                                                                                                                                           "rc_is_eq[T1, *, one1]")
                                                                                                                                          (("1"
                                                                                                                                            (inst
                                                                                                                                             -1
                                                                                                                                             "G!1"
                                                                                                                                             "M!1"
                                                                                                                                             "h!2"
                                                                                                                                             "one1")
                                                                                                                                            (("1"
                                                                                                                                              (assert)
                                                                                                                                              (("1"
                                                                                                                                                (prop)
                                                                                                                                                (("1"
                                                                                                                                                  (lemma
                                                                                                                                                   "rc_is_eq[T1, *, one1]")
                                                                                                                                                  (("1"
                                                                                                                                                    (inst
                                                                                                                                                     -1
                                                                                                                                                     "G!1"
                                                                                                                                                     "M!1"
                                                                                                                                                     "h!3"
                                                                                                                                                     "one1")
                                                                                                                                                    (("1"
                                                                                                                                                      (assert)
                                                                                                                                                      (("1"
                                                                                                                                                        (prop)
                                                                                                                                                        (("1"
                                                                                                                                                          (rewrite
                                                                                                                                                           "right_coset_one")
                                                                                                                                                          (("1"
                                                                                                                                                            (replaces
                                                                                                                                                             -1)
                                                                                                                                                            (("1"
                                                                                                                                                              (replaces
                                                                                                                                                               -1)
                                                                                                                                                              (("1"
                                                                                                                                                                (replaces
                                                                                                                                                                 -1)
                                                                                                                                                                (("1"
                                                                                                                                                                  (replace
                                                                                                                                                                   -2
                                                                                                                                                                   1
                                                                                                                                                                   rl)
                                                                                                                                                                  (("1"
                                                                                                                                                                    (replace
                                                                                                                                                                     -3
                                                                                                                                                                     1
                                                                                                                                                                     rl)
                                                                                                                                                                    (("1"
                                                                                                                                                                      (replace
                                                                                                                                                                       -4
                                                                                                                                                                       1
                                                                                                                                                                       rl)
                                                                                                                                                                      (("1"
                                                                                                                                                                        (lemma
                                                                                                                                                                         "normal_left_is_right[T1, *, one1]")
                                                                                                                                                                        (("1"
                                                                                                                                                                          (inst?)
                                                                                                                                                                          (("1"
                                                                                                                                                                            (assert)
                                                                                                                                                                            (("1"
                                                                                                                                                                              (copy
                                                                                                                                                                               -1)
                                                                                                                                                                              (("1"
                                                                                                                                                                                (copy
                                                                                                                                                                                 -1)
                                                                                                                                                                                (("1"
                                                                                                                                                                                  (inst
                                                                                                                                                                                   -1
                                                                                                                                                                                   "a!2 * a!3")
                                                                                                                                                                                  (("1"
                                                                                                                                                                                    (inst
                                                                                                                                                                                     -2
                                                                                                                                                                                     "a!2")
                                                                                                                                                                                    (("1"
                                                                                                                                                                                      (inst
                                                                                                                                                                                       -3
                                                                                                                                                                                       "a!3")
                                                                                                                                                                                      (("1"
                                                                                                                                                                                        (replace
                                                                                                                                                                                         -1
                                                                                                                                                                                         1
                                                                                                                                                                                         rl)
                                                                                                                                                                                        (("1"
                                                                                                                                                                                          (replace
                                                                                                                                                                                           -2
                                                                                                                                                                                           1
                                                                                                                                                                                           rl)
                                                                                                                                                                                          (("1"
                                                                                                                                                                                            (replace
                                                                                                                                                                                             -3
                                                                                                                                                                                             1
                                                                                                                                                                                             rl)
                                                                                                                                                                                            (("1"
                                                                                                                                                                                              (rewrite
                                                                                                                                                                                               "mult_lem[T1, *, one1]")
                                                                                                                                                                                              nil
                                                                                                                                                                                              nil))
                                                                                                                                                                                            nil))
                                                                                                                                                                                          nil))
                                                                                                                                                                                        nil))
                                                                                                                                                                                      nil))
                                                                                                                                                                                    nil)
                                                                                                                                                                                   ("2"
                                                                                                                                                                                    (hide-all-but
                                                                                                                                                                                     1)
                                                                                                                                                                                    (("2"
                                                                                                                                                                                      (typepred
                                                                                                                                                                                       "a!2"
                                                                                                                                                                                       "a!3"
                                                                                                                                                                                       "G!1")
                                                                                                                                                                                      (("2"
                                                                                                                                                                                        (expand*
                                                                                                                                                                                         "group?"
                                                                                                                                                                                         "monoid?"
                                                                                                                                                                                         "monad?")
                                                                                                                                                                                        (("2"
                                                                                                                                                                                          (flatten)
                                                                                                                                                                                          (("2"
                                                                                                                                                                                            (hide
                                                                                                                                                                                             (-4
                                                                                                                                                                                              -5
                                                                                                                                                                                              -6
                                                                                                                                                                                              -7))
                                                                                                                                                                                            (("2"
                                                                                                                                                                                              (expand
                                                                                                                                                                                               "star_closed?")
                                                                                                                                                                                              (("2"
                                                                                                                                                                                                (inst?)
                                                                                                                                                                                                (("2"
                                                                                                                                                                                                  (expand
                                                                                                                                                                                                   "member")
                                                                                                                                                                                                  (("2"
                                                                                                                                                                                                    (propax)
                                                                                                                                                                                                    nil
                                                                                                                                                                                                    nil))
                                                                                                                                                                                                  nil))
                                                                                                                                                                                                nil))
                                                                                                                                                                                              nil))
                                                                                                                                                                                            nil))
                                                                                                                                                                                          nil))
                                                                                                                                                                                        nil))
                                                                                                                                                                                      nil))
                                                                                                                                                                                    nil))
                                                                                                                                                                                  nil))
                                                                                                                                                                                nil))
                                                                                                                                                                              nil))
                                                                                                                                                                            nil))
                                                                                                                                                                          nil))
                                                                                                                                                                        nil))
                                                                                                                                                                      nil))
                                                                                                                                                                    nil))
                                                                                                                                                                  nil))
                                                                                                                                                                nil))
                                                                                                                                                              nil))
                                                                                                                                                            nil))
                                                                                                                                                          nil)
                                                                                                                                                         ("2"
                                                                                                                                                          (hide-all-but
                                                                                                                                                           1)
                                                                                                                                                          (("2"
                                                                                                                                                            (inst
                                                                                                                                                             1
                                                                                                                                                             "h!3")
                                                                                                                                                            (("2"
                                                                                                                                                              (rewrite
                                                                                                                                                               "one_right[T1, *, one1]")
                                                                                                                                                              nil
                                                                                                                                                              nil))
                                                                                                                                                            nil))
                                                                                                                                                          nil))
                                                                                                                                                        nil))
                                                                                                                                                      nil))
                                                                                                                                                    nil))
                                                                                                                                                  nil)
                                                                                                                                                 ("2"
                                                                                                                                                  (hide-all-but
                                                                                                                                                   1)
                                                                                                                                                  (("2"
                                                                                                                                                    (inst
                                                                                                                                                     1
                                                                                                                                                     "h!2")
                                                                                                                                                    (("2"
                                                                                                                                                      (rewrite
                                                                                                                                                       "one_right[T1, *, one1]")
                                                                                                                                                      nil
                                                                                                                                                      nil))
                                                                                                                                                    nil))
                                                                                                                                                  nil))
                                                                                                                                                nil))
                                                                                                                                              nil))
                                                                                                                                            nil))
                                                                                                                                          nil)
                                                                                                                                         ("2"
                                                                                                                                          (hide-all-but
                                                                                                                                           1)
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                                                                                                                         -1
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                                                                                                                     -1
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                                                                                                                                     -1
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                                                                                                                                         -2
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                (("2" (typepred "a!1")
                  (("2" (expand "restrict")
                    (("2" (inst 1 "*[T1, *, one1](a!1, N!1)")
                      (("1" (replaces -2)
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                          (("1" (inst?)
                            (("1" (assert)
                              (("1"
                                (skosimp)
                                (("1"
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                                                     -3
                                                     "h!1 * a!1")
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                                                           -1
                                                           "M!1"
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                                                             -1
                                                             1
                                                             rl)
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                                                               (-1
                                                                -2
                                                                -4
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                                                              (("1"
                                                                (lemma
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                                                                  (inst
                                                                   -1
                                                                   "G!1"
                                                                   "M!1"
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                                                                           -1)
                                                                          (("1"
                                                                            (lemma
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                                                                              (inst?)
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                                                                          nil))
                                                                        nil)
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                                                                         2)
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                                                                           1
                                                                           "h!1")
                                                                          (("1"
                                                                            (rewrite
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                                                                            nil)
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                                                                                    nil))
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                                                        (("2"
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                                                            (("2"
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                                                                 (-1
                                                                  -3))
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                                                                     -1
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          (("3" (expand "kernel")
            (("3" (decompose-equality 1)
              (("1" (iff)
                (("1" (prop)
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                    (("1"
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                      (("1" (rewrite "left_coset_one")
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                                              (hide -1)
                                              (("1"
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                                                  (lemma
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                                                  (("1"
                                                    (inst?)
                                                    (("1"
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                                                           1)
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                                                             "N!1")
                                                            (("2"
                                                              (expand
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                                                              (("2"
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                                nil))
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                            nil)
                           ("2" (hide-all-but 1)
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                                (("2" (assertnil nil))
                                nil))
                              nil))
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                        nil))
                      nil))
                    nil)
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                        (("2" (skosimp)
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                              (("2"
                                (hide (-2 -3))
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                                  (typepred "M!1")
                                  (("2"
                                    (hide -1)
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                                            (expand "subgroup?")
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                                              (("2"
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                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("3" (expand "/")
                    (("3" (expand "restrict")
                      (("3" (expand "left_cosets")
                        (("3" (skosimp)
                          (("3" (replaces -1)
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                              (("3"
                                (inst -1 "G!1" "N!1" "a!1")
                                (("3"
                                  (prop)
                                  (("1"
                                    (skosimp)
                                    (("1"
                                      (replaces -1)
                                      (("1"
                                        (typepred "h!1" "a!1")
                                        (("1"
                                          (lemma "lc_is_eq[T1,*,one1]")
                                          (("1"
                                            (inst
                                             -1
                                             "G!1"
                                             "M!1"
                                             "h!1 * a!1"
                                             "one1")
                                            (("1"
                                              (prop)
                                              (("1"
                                                (rewrite
                                                 "left_coset_one")
                                                nil
                                                nil)
                                               ("2"
                                                (typepred "M!1")
                                                (("2"
                                                  (expand
                                                   "normal_subgroup?")
                                                  (("2"
                                                    (assert)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil)
                                               ("3"
                                                (inst 1 "h!1 * a!1")
                                                (("1"
                                                  (rewrite
                                                   "one_left[T1,*,one1]")
                                                  nil
                                                  nil)
                                                 ("2"
                                                  (hide 2)
                                                  (("2"
                                                    (expand
                                                     "subgroup?")
                                                    (("2"
                                                      (expand*
                                                       "subset?"
                                                       "member")
                                                      (("2"
                                                        (inst -3 "h!1")
                                                        (("2"
                                                          (assert)
                                                          (("2"
                                                            (rewrite
                                                             "product_in[T1,*,one1]")
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (hide 2)
                                    (("2"
                                      (typepred "N!1")
                                      (("2" (propax) nil nil))
                                      nil))
                                    nil)
                                   ("3"
                                    (typepred "a!1" "M!1")
                                    (("3"
                                      (hide (-2 -4 2))
                                      (("3"
                                        (expand "normal_subgroup?")
                                        (("3"
                                          (expand "subgroup?")
                                          (("3"
                                            (expand*
                                             "subset?"
                                             "member")
                                            (("3"
                                              (flatten)
                                              (("3"
                                                (inst?)
                                                (("3"
                                                  (assert)
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (skosimp)
                (("2" (inst 1 "one1")
                  (("1" (rewrite "left_coset_one"nil nil)
                   ("2" (hide (-1 -2))
                    (("2" (typepred "G!1")
                      (("2"
                        (expand"group?" "monoid?" "monad?" "subset?"
                         "member")
                        (("2" (assertnil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("3" (skosimp)
                (("3" (inst 1 "lc_gen[T1, *, one1](G!1, N!1, a!1)")
                  (("3" (typepred "N!1")
                    (("3" (expand "normal_subgroup?")
                      (("3" (assertnil nil)) nil))
                    nil))
                  nil))
                nil)
               ("4" (skosimp)
                (("4" (typepred "N!1")
                  (("4" (expand "normal_subgroup?")
                    (("4" (assertnil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (skosimp*)
        (("2" (prop)
          (("1" (inst 1 "lc_gen[T1, *, one1](G!1, N!1, x1!1)")
            (("1" (typepred "N!1")
              (("1" (expand "normal_subgroup?")
                (("1" (assertnil nil)) nil))
              nil))
            nil)
           ("2" (expand "/")
            (("2" (expand "restrict")
              (("2" (expand "left_cosets")
                (("2" (inst 1 "lc_gen[T1, *, one1](G!1, N!1, x1!1)")
                  (("2" (typepred "N!1")
                    (("2" (expand "normal_subgroup?")
                      (("2" (assertnil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("3" (skosimp*)
        (("3" (prop)
          (("1" (inst 1 "lc_gen[T1, *, one1](G!1, N!1, A!1)")
            (("1" (typepred "N!1")
              (("1" (expand "normal_subgroup?")
                (("1" (assertnil nil)) nil))
              nil))
            nil)
           ("2" (expand "/")
            (("2" (expand "restrict")
              (("2" (expand "left_cosets")
                (("2" (inst 1 "lc_gen[T1, *, one1](G!1, N!1, A!1)")
                  (("2" (typepred "N!1")
                    (("2" (expand "normal_subgroup?")
                      (("2" (assertnil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("4" (skosimp*)
        (("4" (typepred "N!1")
          (("4" (expand "normal_subgroup?") (("4" (assertnil nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((subgroup? const-decl "bool" group_def "algebra/")
    (left_cosets type-eq-decl nil cosets "algebra/")
    (N!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (* const-decl "set[T]" cosets "algebra/")
    (= const-decl "[T, T -> boolean]" equalities nil)
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (group nonempty-type-eq-decl nil group "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (one1 formal-const-decl "T1" isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (/ const-decl "group[left_cosets(G, N), mult(G, N), N]"
       right_left_cosets nil)
    (mult const-decl "left_cosets(G, H)" factor_groups "algebra/")
    (M!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (lc_gen const-decl "{a: T | G(a) AND lc = a * H}" cosets
     "algebra/")
    (subgroup type-eq-decl nil group "algebra/")
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (restrict const-decl "R" restrict nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (lc_gen_def formula-decl nil cosets "algebra/")
    (one_left formula-decl nil group "algebra/")
    (left_coset_one formula-decl nil cosets "algebra/")
    (lc_eq formula-decl nil cosets "algebra/")
    (a!1 skolem-const-decl "(M!1)" isomorphism_theorems nil)
    (lc_is_eq formula-decl nil cosets "algebra/")
    (product_in formula-decl nil group "algebra/")
    (a!1 skolem-const-decl "(M!1)" isomorphism_theorems nil)
    (h!1 skolem-const-decl "(N!1)" isomorphism_theorems nil)
    (kernel const-decl "subgroup[T1, *, one1](G)" homomorphisms
     "algebra/")
    (surjective? const-decl "bool" functions nil)
    (a!1 skolem-const-decl "(G!1)" isomorphism_theorems nil)
    (h!1 skolem-const-decl "(N!1)" isomorphism_theorems nil)
    (left_cosets const-decl "setofsets[T]" right_left_cosets nil)
    (homomorphism? const-decl "bool" homomorphisms "algebra/")
    (lc_gen_normal formula-decl nil normal_subgroups "algebra/")
    (mult_lem formula-decl nil factor_groups "algebra/")
    (h!1 skolem-const-decl "(N!1)" isomorphism_theorems nil)
    (one_right formula-decl nil group "algebra/")
    (right_coset_one formula-decl nil cosets "algebra/")
    (star_closed? const-decl "bool" groupoid_def "algebra/")
    (monad? const-decl "bool" monad_def "algebra/")
    (monoid? const-decl "bool" monoid_def "algebra/")
    (rc_is_eq formula-decl nil cosets "algebra/")
    (right_coset_assoc formula-decl nil cosets "algebra/")
    (h!2 skolem-const-decl "(N!1)" isomorphism_theorems nil)
    (a!3 skolem-const-decl "(G!1)" isomorphism_theorems nil)
    (a!2 skolem-const-decl "(G!1)" isomorphism_theorems nil)
    (h!3 skolem-const-decl "(N!1)" isomorphism_theorems nil)
    (normal_left_is_right formula-decl nil normal_subgroups "algebra/")
    (subset? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (NOT const-decl "[bool -> bool]" booleans nil))
   shostak))
 (third_isomorphism_th_TCC1 0
  (third_isomorphism_th_TCC1-1 nil 3524769565
   ("" (skosimp*)
    (("" (prop)
      (("1" (skosimp)
        (("1" (prop)
          (("1" (skosimp)
            (("1" (inst 1 "a!1")
              (("1" (hide -)
                (("1" (typepred "a!1" "M!1")
                  (("1" (expand"normal_subgroup?" "subgroup?")
                    (("1" (flatten)
                      (("1" (expand"subset?" "member")
                        (("1" (inst?) (("1" (assertnil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (hide -1)
            (("2" (inst 1 "lc_gen(M!1, N!1, x!1)")
              (("2" (assertnil nil)) nil))
            nil))
          nil))
        nil)
       ("2" (lemma "quotient_subgroup")
        (("2" (inst?)
          (("2" (assert)
            (("2" (expand "subgroup?") (("2" (propax) nil nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((T1 formal-type-decl nil isomorphism_theorems nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (set type-eq-decl nil sets nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (one1 formal-const-decl "T1" isomorphism_theorems nil)
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (a!1 skolem-const-decl "(M!1)" isomorphism_theorems nil)
    (M!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (subset? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (subgroup? const-decl "bool" group_def "algebra/")
    (N!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (subgroup type-eq-decl nil group "algebra/")
    (= const-decl "[T, T -> boolean]" equalities nil)
    (* const-decl "set[T]" cosets "algebra/")
    (left_cosets type-eq-decl nil cosets "algebra/")
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (lc_gen const-decl "{a: T | G(a) AND lc = a * H}" cosets
     "algebra/")
    (quotient_subgroup formula-decl nil isomorphism_theorems nil))
   nil))
 (third_isomorphism_th_TCC2 0
  (third_isomorphism_th_TCC2-1 nil 3524769565
   ("" (skosimp*)
    (("" (rewrite "left_cosets_group[T1,*,one1]"nil nil)) nil)
   ((left_cosets_group formula-decl nil factor_groups "algebra/")
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (set type-eq-decl nil sets nil)
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (one1 formal-const-decl "T1" isomorphism_theorems nil))
   nil))
 (third_isomorphism_th_TCC3 0
  (third_isomorphism_th_TCC3-1 nil 3524769565
   ("" (skosimp*)
    (("" (prop)
      (("1" (skosimp)
        (("1" (prop)
          (("1" (hide (-2 -3 -4))
            (("1" (skosimp)
              (("1" (inst 1 "a!1")
                (("1" (hide -)
                  (("1" (typepred "a!1" "M!1")
                    (("1" (expand"normal_subgroup?" "subgroup?")
                      (("1" (flatten)
                        (("1" (expand"subset?" "member")
                          (("1" (inst?) (("1" (assertnil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (hide (-1 -3))
            (("2" (inst 1 "lc_gen(M!1, N!1, x!1)")
              (("2" (assertnil nil)) nil))
            nil))
          nil))
        nil)
       ("2" (hide -1)
        (("2" (inst 1 "N!1")
          (("1" (decompose-equality 1)
            (("1" (iff)
              (("1" (prop)
                (("1" (expand "*" 1)
                  (("1" (inst 1 "x!1")
                    (("1" (lemma "N_is_identity[T1, *, one1]")
                      (("1" (inst -1 "G!1" "N!1" "x!1")
                        (("1" (assertnil nil)) nil))
                      nil))
                    nil))
                  nil)
                 ("2" (grind) nil nil))
                nil))
              nil)
             ("2" (skosimp)
              (("2" (prop)
                (("1" (hide -1)
                  (("1" (expand "*" -1)
                    (("1" (skosimp)
                      (("1" (typepred "h!1")
                        (("1" (skosimp)
                          (("1" (lemma "N_is_identity[T1, *, one1]")
                            (("1" (inst -1 "G!1" "N!1" "h!1")
                              (("1"
                                (flatten)
                                (("1"
                                  (replaces -1)
                                  (("1"
                                    (replaces -2)
                                    (("1"
                                      (replaces -3)
                                      (("1" (inst 1 "a!1"nil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (hide-all-but (-1 1))
                  (("2" (skosimp)
                    (("2" (inst 1 "a!1")
                      (("2" (typepred "a!1" "M!1")
                        (("2"
                          (expand"normal_subgroup?" "subgroup?"
                           "subset?" "member")
                          (("2" (flatten)
                            (("2" (inst?) (("2" (assertnil nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("3" (rewrite "left_cosets_group[T1, *, one1]"nil nil))
            nil)
           ("2" (prop)
            (("1" (inst 1 "one1")
              (("1" (rewrite "left_coset_one[T1,*,one1]"nil nil)
               ("2" (typepred "G!1")
                (("2" (expand"group?" "monoid?" "monad?" "member")
                  (("2" (assertnil nil)) nil))
                nil))
              nil)
             ("2" (expand "/")
              (("2" (expand "restrict")
                (("2" (expand "left_cosets")
                  (("2" (inst 1 "one1")
                    (("1" (rewrite "left_coset_one[T1,*,one1]"nil
                      nil)
                     ("2" (typepred "G!1")
                      (("2"
                        (expand"group?" "monoid?" "monad?" "member")
                        (("2" (assertnil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((subgroup? const-decl "bool" group_def "algebra/")
    (member const-decl "bool" sets nil)
    (subset? const-decl "bool" sets nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (M!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (a!1 skolem-const-decl "(M!1)" isomorphism_theorems nil)
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (one1 formal-const-decl "T1" isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (N!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (subgroup type-eq-decl nil group "algebra/")
    (= const-decl "[T, T -> boolean]" equalities nil)
    (* const-decl "set[T]" cosets "algebra/")
    (left_cosets type-eq-decl nil cosets "algebra/")
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (lc_gen const-decl "{a: T | G(a) AND lc = a * H}" cosets
     "algebra/")
    (/ const-decl "group[left_cosets(G, N), mult(G, N), N]"
       right_left_cosets nil)
    (mult const-decl "left_cosets(G, H)" factor_groups "algebra/")
    (left_cosets_group formula-decl nil factor_groups "algebra/")
    (a!1 skolem-const-decl "(M!1)" isomorphism_theorems nil)
    (left_cosets const-decl "setofsets[T]" right_left_cosets nil)
    (restrict const-decl "R" restrict nil)
    (N_is_identity formula-decl nil factor_groups "algebra/")
    (x!1 skolem-const-decl "left_cosets[T1, *, one1](M!1, N!1)"
     isomorphism_theorems nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (IFF const-decl "[bool, bool -> bool]" booleans nil)
    (fullset const-decl "set" sets nil)
    (left_coset_one formula-decl nil cosets "algebra/")
    (monoid? const-decl "bool" monoid_def "algebra/")
    (monad? const-decl "bool" monad_def "algebra/"))
   nil))
 (third_isomorphism_th_TCC4 0
  (third_isomorphism_th_TCC4-1 nil 3524769565
   ("" (skosimp*)
    (("" (inst 1 "one1")
      (("1" (rewrite "left_coset_one[T1,*,one1]"nil nil)
       ("2" (typepred "G!1")
        (("2" (expand"group?" "monoid?" "monad?" "member")
          (("2" (assertnil nil)) nil))
        nil))
      nil))
    nil)
   ((G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (group nonempty-type-eq-decl nil group "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (one1 formal-const-decl "T1" isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (left_coset_one formula-decl nil cosets "algebra/")
    (monoid? const-decl "bool" monoid_def "algebra/")
    (member const-decl "bool" sets nil)
    (monad? const-decl "bool" monad_def "algebra/")
    (NOT const-decl "[bool -> bool]" booleans nil))
   nil))
 (third_isomorphism_th_TCC5 0
  (third_isomorphism_th_TCC5-1 nil 3524769565
   ("" (skosimp*)
    ((""
      (lemma
       "left_cosets_group[left_cosets[T1, *, one1](G!1, N!1), mult[T1, *, one1](G!1, N!1), N!1]")
      (("1" (inst -1 "G!1 / N!1" "M!1 / N!1"nil nil)
       ("2" (hide 2)
        (("2" (inst 1 "N!1")
          (("2" (inst 1 "one1")
            (("1" (rewrite "left_coset_one[T1,*,one1]"nil nil)
             ("2" (typepred "G!1")
              (("2" (expand"group?" "monoid?" "monad?" "member")
                (("2" (assertnil nil)) nil))
              nil))
            nil))
          nil))
        nil)
       ("3" (hide 2)
        (("3" (rewrite "left_cosets_group[T1, *, one1]"nil nil)) nil)
       ("4" (hide 2)
        (("4" (inst 1 "one1")
          (("1" (rewrite "left_coset_one[T1,*,one1]"nil nil)
           ("2" (typepred "G!1")
            (("2" (expand"group?" "monoid?" "monad?" "member")
              (("2" (assertnil nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((mult const-decl "left_cosets(G, H)" factor_groups "algebra/")
    (left_cosets type-eq-decl nil cosets "algebra/")
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (* const-decl "set[T]" cosets "algebra/")
    (= const-decl "[T, T -> boolean]" equalities nil)
    (group nonempty-type-eq-decl nil group "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (one1 formal-const-decl "T1" isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (left_cosets_group formula-decl nil factor_groups "algebra/")
    (fullset const-decl "set" sets nil)
    (TRUE const-decl "bool" booleans nil)
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (N!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (M!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (/ const-decl "group[left_cosets(G, N), mult(G, N), N]"
       right_left_cosets nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (monad? const-decl "bool" monad_def "algebra/")
    (member const-decl "bool" sets nil)
    (monoid? const-decl "bool" monoid_def "algebra/")
    (left_coset_one formula-decl nil cosets "algebra/"))
   nil))
 (third_isomorphism_th_TCC6 0
  (third_isomorphism_th_TCC6-1 nil 3524769565
   ("" (skosimp*)
    (("" (rewrite "left_cosets_group[T1, *, one1]"nil nil)) nil)
   ((left_cosets_group formula-decl nil factor_groups "algebra/")
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (set type-eq-decl nil sets nil)
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (one1 formal-const-decl "T1" isomorphism_theorems nil))
   nil))
 (third_isomorphism_th 0
  (third_isomorphism_th-1 nil 3524796117
   ("" (skosimp*)
    (("" (lemma "third_isomorphism_th_aux")
      (("" (inst?)
        (("" (assert)
          (("" (skosimp)
            ((""
              (lemma
               "first_isomorphism_th[left_cosets[T1, *, one1](G!1, N!1), mult(G!1, N!1), N!1, left_cosets[T1, *, one1](G!1, M!1), mult(G!1, M!1), M!1]")
              (("1" (inst -1 "G!1 / N!1" "G!1 / M!1" "psi!1")
                (("1" (replace -4 -1)
                  (("1" (assert)
                    (("1"
                      (case-replace "extend
               [left_cosets[T1, *, one1](G!1, M!1), (G!1 / M!1), bool,
                FALSE]
               (image(psi!1,
                      restrict
                          [left_cosets[T1, *, one1](G!1, N!1), (G!1 / N!1),
                           bool]
                          (G!1 / N!1))) = G!1 / M!1" :hide? T)
                      (("1" (assert)
                        (("1" (hide (-1 -2 -3))
                          (("1"
                            (lemma
                             "kernel_normal[left_cosets[T1, *, one1](G!1, N!1), mult[T1, *, one1](G!1, N!1),
                   N!1, left_cosets[T1, *, one1](G!1, M!1), mult[T1, *, one1](G!1, M!1),
                   M!1]")
                            (("1"
                              (inst -1 "/[T1, *, one1](G!1, N!1)"
                               "/[T1, *, one1](G!1, M!1)" "psi!1")
                              (("1" (assertnil nil)) nil))
                            nil))
                          nil))
                        nil)
                       ("2" (hide (-1 -4 2))
                        (("2" (decompose-equality 1)
                          (("2" (expand "extend")
                            (("2" (lift-if)
                              (("2"
                                (prop)
                                (("2"
                                  (expand "image")
                                  (("2"
                                    (expand "surjective?")
                                    (("2"
                                      (inst -3 "x!1")
                                      (("2"
                                        (skosimp)
                                        (("2"
                                          (inst 1 "x!2")
                                          (("1" (assertnil nil)
                                           ("2"
                                            (expand "restrict")
                                            (("2" (propax) nil nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (hide-all-but 1)
                (("2" (rewrite "left_cosets_group[T1, *, one1]"nil
                  nil))
                nil)
               ("3" (hide-all-but 1)
                (("3" (rewrite "left_cosets_group[T1, *, one1]"nil
                  nil))
                nil)
               ("4" (hide-all-but 1)
                (("4" (inst 1 "one1")
                  (("1" (rewrite "left_coset_one[T1,*,one1]"nil nil)
                   ("2" (typepred "G!1")
                    (("2"
                      (expand"group?" "monoid?" "monad?" "member")
                      (("2" (flatten) nil nil)) nil))
                    nil))
                  nil))
                nil)
               ("5" (hide-all-but 1)
                (("5" (inst 1 "one1")
                  (("1" (rewrite "left_coset_one[T1,*,one1]"nil nil)
                   ("2" (typepred "G!1")
                    (("2"
                      (expand"group?" "monoid?" "monad?" "member")
                      (("2" (flatten) nil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((third_isomorphism_th_aux formula-decl nil isomorphism_theorems
     nil)
    (mult const-decl "left_cosets(G, H)" factor_groups "algebra/")
    (left_cosets type-eq-decl nil cosets "algebra/")
    (* const-decl "set[T]" cosets "algebra/")
    (= const-decl "[T, T -> boolean]" equalities nil)
    (first_isomorphism_th formula-decl nil homomorphism_lemmas nil)
    (fullset const-decl "set" sets nil)
    (FALSE const-decl "bool" booleans nil)
    (extend const-decl "R" extend nil)
    (image const-decl "set[R]" function_image nil)
    (restrict const-decl "R" restrict nil)
    (kernel_normal formula-decl nil homomorphism_lemmas nil)
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (M!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (x!1 skolem-const-decl "left_cosets[T1, *, one1](G!1, M!1)"
     isomorphism_theorems nil)
    (N!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (x!2 skolem-const-decl "(G!1 / N!1)" isomorphism_theorems nil)
    (surjective? const-decl "bool" functions nil)
    (homomorphism type-eq-decl nil homomorphisms "algebra/")
    (homomorphism? const-decl "bool" homomorphisms "algebra/")
    (/ const-decl "group[left_cosets(G, N), mult(G, N), N]"
       right_left_cosets nil)
    (left_cosets_group formula-decl nil factor_groups "algebra/")
    (left_coset_one formula-decl nil cosets "algebra/")
    (monoid? const-decl "bool" monoid_def "algebra/")
    (member const-decl "bool" sets nil)
    (monad? const-decl "bool" monad_def "algebra/")
    (NOT const-decl "[bool -> bool]" booleans nil)
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (one1 formal-const-decl "T1" isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T1 formal-type-decl nil isomorphism_theorems nil))
   shostak))
 (correspondence_theorem 0
  (correspondence_theorem-1 nil 3530907884
   ("" (skosimp*)
    ((""
      (inst 1
       "LAMBDA (N: subgroup_contain(G!1, kernel(G!1, GP!1)(f!1))): image(f!1, N)")
      (("1" (prop)
        (("1" (expand "bijective?")
          (("1" (prop)
            (("1" (expand "injective?")
              (("1" (skosimp*)
                (("1" (typepred "x1!1" "x2!1")
                  (("1" (hide (-1 -4))
                    (("1"
                      (lemma
                       "homo_inv_image_image_cor[T1,*,one1,T2,o,one2]")
                      (("1" (assert)
                        (("1" (expand "subgroup?" -1)
                          (("1" (inst-cp -1 "G!1" "GP!1" "f!1" "x2!1")
                            (("1" (inst -1 "G!1" "GP!1" "f!1" "x1!1")
                              (("1"
                                (assert)
                                (("1"
                                  (decompose-equality -1)
                                  (("1"
                                    (decompose-equality -2)
                                    (("1"
                                      (decompose-equality -7)
                                      (("1"
                                        (decompose-equality 1)
                                        (("1"
                                          (iff)
                                          (("1"
                                            (prop)
                                            (("1"
                                              (inst -3 "x!1")
                                              (("1"
                                                (inst -4 "x!1")
                                                (("1"
                                                  (iff)
                                                  (("1"
                                                    (prop)
                                                    (("1"
                                                      (hide-all-but
                                                       (-4 1))
                                                      (("1"
                                                        (expand
                                                         "restrict")
                                                        (("1"
                                                          (propax)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("2"
                                                      (hide-all-but
                                                       (-3 1))
                                                      (("2"
                                                        (expand
                                                         "restrict")
                                                        (("2"
                                                          (propax)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("3"
                                                      (hide (-2 1 3))
                                                      (("3"
                                                        (expand
                                                         "inverse_image")
                                                        (("3"
                                                          (expand
                                                           "member")
                                                          (("3"
                                                            (inst?)
                                                            (("3"
                                                              (expand
                                                               "extend")
                                                              (("3"
                                                                (assert)
                                                                nil
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("4"
                                                      (hide-all-but
                                                       (-1 1))
                                                      (("4"
                                                        (expand
                                                         "restrict")
                                                        (("4"
                                                          (propax)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2"
                                                (hide-all-but
                                                 (-1 -4 1))
                                                (("2"
                                                  (expand*
                                                   "subgroup?"
                                                   "subset?"
                                                   "member")
                                                  (("2"
                                                    (inst?)
                                                    (("2"
                                                      (assert)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil)
                                             ("2"
                                              (inst -3 "x!1")
                                              (("1"
                                                (inst -4 "x!1")
                                                (("1"
                                                  (iff)
                                                  (("1"
                                                    (prop)
                                                    (("1"
                                                      (hide-all-but
                                                       (-2 1))
                                                      (("1"
                                                        (expand
                                                         "restrict")
                                                        (("1"
                                                          (propax)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("2"
                                                      (hide (-2 1 3))
                                                      (("2"
                                                        (expand
                                                         "inverse_image")
                                                        (("2"
                                                          (expand
                                                           "member")
                                                          (("2"
                                                            (inst?)
                                                            (("2"
                                                              (expand
                                                               "extend")
                                                              (("2"
                                                                (assert)
                                                                nil
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("3"
                                                      (hide-all-but
                                                       (-3 1))
                                                      (("3"
                                                        (expand
                                                         "restrict")
                                                        (("3"
                                                          (propax)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("4"
                                                      (hide-all-but
                                                       (-1 3))
                                                      (("4"
                                                        (expand
                                                         "restrict")
                                                        (("4"
                                                          (propax)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2"
                                                (hide-all-but
                                                 (-1 -6 1))
                                                (("2"
                                                  (expand*
                                                   "subgroup?"
                                                   "subset?"
                                                   "member")
                                                  (("2"
                                                    (inst?)
                                                    (("2"
                                                      (assert)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (rewrite "T2_is_group"nil nil)
                       ("3" (rewrite "T1_is_group"nil nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (expand "surjective?")
              (("2" (skosimp*)
                (("2" (inst 1 "inverse_image(f!1, y!1)")
                  (("1" (decompose-equality 1)
                    (("1" (iff)
                      (("1" (prop)
                        (("1" (expand "extend")
                          (("1" (prop)
                            (("1" (expand "image")
                              (("1"
                                (skosimp)
                                (("1"
                                  (typepred "x!2")
                                  (("1"
                                    (expand "restrict")
                                    (("1"
                                      (expand "inverse_image")
                                      (("1"
                                        (expand "member")
                                        (("1" (assertnil nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (expand "extend")
                          (("2" (prop)
                            (("1" (expand "image")
                              (("1"
                                (inst?)
                                (("1"
                                  (skosimp)
                                  (("1"
                                    (inst 1 "x!2")
                                    (("1" (assertnil nil)
                                     ("2"
                                      (expand "restrict")
                                      (("2"
                                        (expand "inverse_image")
                                        (("2"
                                          (expand "member")
                                          (("2" (assertnil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil)
                             ("2" (hide -2)
                              (("2"
                                (typepred "y!1")
                                (("2"
                                  (hide -1)
                                  (("2"
                                    (expand*
                                     "subgroup?"
                                     "subset?"
                                     "member")
                                    (("2"
                                      (inst?)
                                      (("2" (assertnil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (hide -1)
                    (("2" (lemma "homo_inv_image[T1,*,one1,T2,o,one2]")
                      (("1" (inst?)
                        (("1" (assert)
                          (("1" (prop)
                            (("1"
                              (lemma "subgroup_is_group[T1, *, one1]")
                              (("1"
                                (inst?)
                                (("1" (assertnil nil))
                                nil))
                              nil)
                             ("2"
                              (lemma
                               "kernel_in_inv_image[T1,*,one1,T2,o,one2]")
                              (("2"
                                (inst?)
                                (("2" (assertnil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (hide 2)
                        (("2" (rewrite "T2_is_group"nil nil)) nil)
                       ("3" (hide 2)
                        (("3" (rewrite "T1_is_group"nil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil)
         ("2" (skosimp*)
          (("2" (prop)
            (("1" (lemma "homo_image[T1,*,one1,T2,o,one2]")
              (("1" (inst?) nil nil)
               ("2" (hide-all-but 1)
                (("2" (rewrite "T2_is_group"nil nil)) nil)
               ("3" (hide-all-but 1)
                (("3" (rewrite "T1_is_group"nil nil)) nil))
              nil)
             ("2" (lemma "homo_inv_image[T1,*,one1,T2,o,one2]")
              (("1"
                (inst -1 "G!1" "GP!1" "f!1"
                 "image(f!1, restrict[T1, (G!1), boolean](N!1))")
                (("1"
                  (lemma
                   "homo_inv_image_image_cor[T1, *, one1,T2, o, one2]")
                  (("1" (inst?) (("1" (assertnil nil)) nil)) nil)
                 ("2" (prop)
                  (("2" (hide (-2 2))
                    (("2" (lemma "subgroup_is_group[T2, o, one2]")
                      (("2" (inst?) (("2" (assertnil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (hide-all-but 1)
                (("2" (rewrite "T2_is_group"nil nil)) nil)
               ("3" (hide-all-but 1)
                (("3" (rewrite "T1_is_group"nil nil)) nil))
              nil)
             ("3" (lemma "homo_image_normal[T1,*,one1,T2,o,one2]")
              (("1" (inst?) (("1" (assertnil nil)) nil)
               ("2" (hide-all-but 1)
                (("2" (rewrite "T2_is_group"nil nil)) nil)
               ("3" (hide-all-but 1)
                (("3" (rewrite "T1_is_group"nil nil)) nil))
              nil)
             ("4" (lemma "homo_inv_image_normal[T1,*,one1,T2,o,one2]")
              (("1"
                (inst -1 "G!1" "GP!1" "f!1"
                 "image(f!1, restrict[T1, (G!1), boolean](N!1))")
                (("1"
                  (lemma
                   "homo_inv_image_image_cor[T1, *, one1,T2, o, one2]")
                  (("1" (inst?)
                    (("1" (assert)
                      (("1" (prop)
                        (("1" (hide (-2 -4 -5))
                          (("1" (expand "normal_subgroup?")
                            (("1" (flatten)
                              (("1"
                                (skosimp)
                                (("1"
                                  (hide -2)
                                  (("1"
                                    (expand"subset?" "member")
                                    (("1"
                                      (skosimp)
                                      (("1"
                                        (decompose-equality -1)
                                        (("1"
                                          (inst -1 "x!1")
                                          (("1"
                                            (iff)
                                            (("1"
                                              (prop)
                                              (("1"
                                                (hide-all-but (-2 1))
                                                (("1"
                                                  (expand "restrict")
                                                  (("1"
                                                    (propax)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2"
                                                (hide (1 3))
                                                (("2"
                                                  (expand
                                                   "inverse_image")
                                                  (("2"
                                                    (expand "member")
                                                    (("2"
                                                      (expand "image")
                                                      (("2"
                                                        (inst -1 "a!1")
                                                        (("2"
                                                          (inst
                                                           -1
                                                           "x!1")
                                                          (("2"
                                                            (prop)
                                                            (("1"
                                                              (expand
                                                               "extend")
                                                              (("1"
                                                                (prop)
                                                                nil
                                                                nil))
                                                              nil)
                                                             ("2"
                                                              (hide 2)
                                                              (("2"
                                                                (expand
                                                                 "*")
                                                                (("2"
                                                                  (skosimp)
                                                                  (("2"
                                                                    (typepred
                                                                     "h!1")
                                                                    (("2"
                                                                      (skosimp)
                                                                      (("2"
                                                                        (replaces
                                                                         -1)
                                                                        (("2"
                                                                          (inst
                                                                           1
                                                                           "inv[T1, *, one1](a!1) * h!2")
                                                                          (("2"
                                                                            (inst
                                                                             1
                                                                             "h!2")
                                                                            (("2"
                                                                              (expand
                                                                               "extend")
                                                                              (("2"
                                                                                (prop)
                                                                                (("1"
                                                                                  (inst?)
                                                                                  (("1"
                                                                                    (assert)
                                                                                    (("1"
                                                                                      (expand
                                                                                       "restrict")
                                                                                      (("1"
                                                                                        (propax)
                                                                                        nil
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil)
                                                                                 ("2"
                                                                                  (hide
                                                                                   -1)
                                                                                  (("2"
                                                                                    (typepred
                                                                                     "h!2"
                                                                                     "N!1")
                                                                                    (("2"
                                                                                      (hide
                                                                                       (-2
                                                                                        -4))
                                                                                      (("2"
                                                                                        (expand*
                                                                                         "subgroup?"
                                                                                         "subset?"
                                                                                         "member")
                                                                                        (("2"
                                                                                          (inst?)
                                                                                          (("2"
                                                                                            (assert)
                                                                                            nil
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil)
                                           ("2"
                                            (hide (-1 2))
                                            (("2"
                                              (expand "*")
                                              (("2"
                                                (skosimp)
                                                (("2"
                                                  (typepred "h!1")
                                                  (("2"
                                                    (skosimp)
                                                    (("2"
                                                      (replaces -1)
                                                      (("2"
                                                        (replaces -1)
                                                        (("2"
                                                          (rewrite
                                                           "product_in[T1, *, one1]")
                                                          (("2"
                                                            (hide 2)
                                                            (("2"
                                                              (rewrite
                                                               "product_in[T1, *, one1]")
                                                              (("1"
                                                                (hide
                                                                 2)
                                                                (("1"
                                                                  (rewrite
                                                                   "inv_in")
                                                                  nil
                                                                  nil))
                                                                nil)
                                                               ("2"
                                                                (hide
                                                                 2)
                                                                (("2"
                                                                  (typepred
                                                                   "h!2"
                                                                   "N!1")
                                                                  (("2"
                                                                    (hide
                                                                     (-2
                                                                      -4))
                                                                    (("2"
                                                                      (expand*
                                                                       "subgroup?"
                                                                       "subset?"
                                                                       "member")
                                                                      (("2"
                                                                        (inst?)
                                                                        (("2"
                                                                          (assert)
                                                                          nil
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (hide-all-but 1)
                          (("2" (typepred "N!1")
                            (("2" (hide (-1 -2))
                              (("2"
                                (expand "subgroup?")
                                (("2" (propax) nil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (hide-all-but 1)
                (("2" (rewrite "T2_is_group"nil nil)) nil)
               ("3" (hide-all-but 1)
                (("3" (rewrite "T1_is_group"nil nil)) nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (skosimp*)
        (("2" (lemma "homo_image[T1,*,one1,T2,o,one2]")
          (("1" (inst?)
            (("1" (assert)
              (("1" (hide -2)
                (("1" (lemma "subgroup_is_group[T2, o, one2]")
                  (("1" (inst?) (("1" (assertnil nil)) nil)) nil))
                nil))
              nil))
            nil)
           ("2" (hide 2) (("2" (rewrite "T2_is_group"nil nil)) nil)
           ("3" (hide 2) (("3" (rewrite "T1_is_group"nil nil)) nil))
          nil))
        nil)
       ("3" (rewrite "T1_is_group"nil nil))
      nil))
    nil)
   ((fullset const-decl "set" sets nil)
    (group? const-decl "bool" group_def "algebra/")
    (one1 formal-const-decl "T1" isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (restrict const-decl "R" restrict nil)
    (image const-decl "set[R]" function_image nil)
    (extend const-decl "R" extend nil)
    (FALSE const-decl "bool" booleans nil)
    (subgroup_contain type-eq-decl nil groups_scaf nil)
    (f!1 skolem-const-decl
     "homomorphism[T1, *, one1, T2, o, one2](G!1, GP!1)"
     isomorphism_theorems nil)
    (GP!1 skolem-const-decl "group[T2, o, one2]" isomorphism_theorems
     nil)
    (kernel const-decl "subgroup[T1, *, one1](G)" homomorphisms
     "algebra/")
    (subgroup type-eq-decl nil group "algebra/")
    (homomorphism type-eq-decl nil homomorphisms "algebra/")
    (homomorphism? const-decl "bool" homomorphisms "algebra/")
    (one2 formal-const-decl "T2" isomorphism_theorems nil)
    (O formal-const-decl "[T2, T2 -> T2]" isomorphism_theorems nil)
    (T2 formal-type-decl nil isomorphism_theorems nil)
    (subset? const-decl "bool" sets nil)
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (subgroup? const-decl "bool" group_def "algebra/")
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (group nonempty-type-eq-decl nil group "algebra/")
    (homo_inv_image_normal formula-decl nil homomorphism_lemmas nil)
    (x!1 skolem-const-decl "T1" isomorphism_theorems nil)
    (a!1 skolem-const-decl "(G!1)" isomorphism_theorems nil)
    (h!2 skolem-const-decl "(N!1)" isomorphism_theorems nil)
    (inv const-decl "{y | x * y = one AND y * x = one}" group
     "algebra/")
    (* const-decl "set[T]" cosets "algebra/")
    (* const-decl "set[T]" cosets "algebra/")
    (product_in formula-decl nil group "algebra/")
    (inv_in formula-decl nil group "algebra/")
    (homo_image_normal formula-decl nil homomorphism_lemmas nil)
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (N!1 skolem-const-decl
     "subgroup_contain[T1, *, one1](G!1, kernel(G!1, GP!1)(f!1))"
     isomorphism_theorems nil)
    (homo_image formula-decl nil homomorphism_lemmas nil)
    (bijective? const-decl "bool" functions nil)
    (surjective? const-decl "bool" functions nil)
    (y!1 skolem-const-decl "subgroup[T2, o, one2](GP!1)"
     isomorphism_theorems nil)
    (x!2 skolem-const-decl "(G!1)" isomorphism_theorems nil)
    (x!1 skolem-const-decl "T2" isomorphism_theorems nil)
    (IF const-decl "[boolean, T, T -> T]" if_def nil)
    (homo_inv_image formula-decl nil homomorphism_lemmas nil)
    (kernel_in_inv_image formula-decl nil homomorphism_lemmas nil)
    (subgroup_is_group formula-decl nil group "algebra/")
    (injective? const-decl "bool" functions nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (homo_inv_image_image_cor formula-decl nil homomorphism_lemmas nil)
    (inverse_image const-decl "set[D]" function_image nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (x!1 skolem-const-decl "T1" isomorphism_theorems nil)
    (member const-decl "bool" sets nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (T2_is_group formula-decl nil isomorphism_theorems nil)
    (T1_is_group formula-decl nil isomorphism_theorems nil))
   shostak)))


Messung V0.5 in Prozent
C=100 H=100 G=100

¤ Dauer der Verarbeitung: 0.441 Sekunden  (vorverarbeitet am  2026-04-27) ¤

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