SET007 Axioms: SET007+366.ax


%------------------------------------------------------------------------------
% File     : SET007+366 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Monoid of Multisets and Subsets
% Version  : [Urb08] axioms.
% English  :

% Refs     : [Mat90] Matuszewski (1990), Formalized Mathematics
%          : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
%          : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb08]
% Names    : monoid_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  111 (   5 unit)
%            Number of atoms       :  701 (  79 equality)
%            Maximal formula depth :   20 (   9 average)
%            Number of connectives :  750 ( 160 ~  ;   0  |; 324  &)
%                                         (   8 <=>; 258 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   41 (   1 propositional; 0-4 arity)
%            Number of functors    :   78 (   8 constant; 0-8 arity)
%            Number of variables   :  374 (   0 singleton; 371 !;   3 ?)
%            Maximal term depth    :    5 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(fc1_monoid_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ~ v3_struct_0(k9_monoid_1(A,B)) ) )).

fof(fc2_monoid_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( ~ v3_struct_0(k9_monoid_1(A,B))
        & v1_monoid_0(k9_monoid_1(A,B)) ) ) )).

fof(fc3_monoid_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(C,u1_struct_0(k9_monoid_1(A,B))) )
     => ~ v1_xboole_0(k2_relat_1(C)) ) )).

fof(fc4_monoid_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_finset_1(B) )
     => ( v1_relat_1(k8_relat_1(A,B))
        & v1_finset_1(k8_relat_1(A,B)) ) ) )).

fof(fc5_monoid_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ~ v3_struct_0(k22_monoid_1(A)) ) )).

fof(d1_monoid_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C,D] : k1_monoid_1(A,B,C,D) = k5_funct_6(A,k4_tarski(B,C),D) ) )).

fof(t1_monoid_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C,D,E] :
              ( ( r2_hidden(k4_tarski(C,D),k1_relat_1(A))
                & B = k1_binop_1(A,C,D)
                & r2_hidden(E,k1_relat_1(B)) )
             => k1_monoid_1(A,C,D,E) = k1_funct_1(B,E) ) ) ) )).

fof(t2_monoid_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] : k5_relat_1(B,k7_relat_1(A,C)) = k5_relat_1(k8_relat_1(C,B),A) ) ) )).

fof(d2_monoid_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,B),C)
                    & m2_relset_1(D,k2_zfmisc_1(A,B),C) )
                 => ! [E,F] :
                      ( ( v1_funct_1(F)
                        & v1_funct_2(F,k2_zfmisc_1(k1_fraenkel(E,A),k1_fraenkel(E,B)),k1_fraenkel(E,C))
                        & m2_relset_1(F,k2_zfmisc_1(k1_fraenkel(E,A),k1_fraenkel(E,B)),k1_fraenkel(E,C)) )
                     => ( F = k8_monoid_1(A,B,C,D,E)
                      <=> ! [G] :
                            ( m2_fraenkel(G,E,A,k1_fraenkel(E,A))
                           => ! [H] :
                                ( m2_fraenkel(H,E,B,k1_fraenkel(E,B))
                               => k2_binop_1(k1_fraenkel(E,A),k1_fraenkel(E,B),k1_fraenkel(E,C),F,G,H) = k3_monoid_1(E,A,B,C,D,G,H) ) ) ) ) ) ) ) ) )).

fof(t3_monoid_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,B),C)
                    & m2_relset_1(D,k2_zfmisc_1(A,B),C) )
                 => ! [E,F] :
                      ( ( v1_funct_1(F)
                        & v1_funct_2(F,E,A)
                        & m2_relset_1(F,E,A) )
                     => ! [G] :
                          ( ( v1_funct_1(G)
                            & v1_funct_2(G,E,B)
                            & m2_relset_1(G,E,B) )
                         => ! [H] :
                              ( r2_hidden(H,E)
                             => k1_monoid_1(k8_monoid_1(A,B,C,D,E),F,G,H) = k1_binop_1(D,k1_funct_1(F,H),k1_funct_1(G,H)) ) ) ) ) ) ) ) )).

fof(t4_monoid_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(B,B),B)
            & m2_relset_1(C,k2_zfmisc_1(B,B),B) )
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,A,B)
                & m2_relset_1(D,A,B) )
             => ! [E] :
                  ( ( v1_funct_1(E)
                    & v1_funct_2(E,A,B)
                    & m2_relset_1(E,A,B) )
                 => ( v1_binop_1(C,B)
                   => k3_monoid_1(A,B,B,B,C,D,E) = k3_monoid_1(A,B,B,B,C,E,D) ) ) ) ) ) )).

fof(t5_monoid_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(B,B),B)
            & m2_relset_1(C,k2_zfmisc_1(B,B),B) )
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,A,B)
                & m2_relset_1(D,A,B) )
             => ! [E] :
                  ( ( v1_funct_1(E)
                    & v1_funct_2(E,A,B)
                    & m2_relset_1(E,A,B) )
                 => ! [F] :
                      ( ( v1_funct_1(F)
                        & v1_funct_2(F,A,B)
                        & m2_relset_1(F,A,B) )
                     => ( v2_binop_1(C,B)
                       => k3_monoid_1(A,B,B,B,C,k3_monoid_1(A,B,B,B,C,D,E),F) = k3_monoid_1(A,B,B,B,C,D,k3_monoid_1(A,B,B,B,C,E,F)) ) ) ) ) ) ) )).

fof(t6_monoid_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( m1_subset_1(C,B)
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,k2_zfmisc_1(B,B),B)
                & m2_relset_1(D,k2_zfmisc_1(B,B),B) )
             => ! [E] :
                  ( ( v1_funct_1(E)
                    & v1_funct_2(E,A,B)
                    & m2_relset_1(E,A,B) )
                 => ( r3_binop_1(B,C,D)
                   => ( k6_monoid_1(A,B,B,B,D,C,E) = E
                      & k7_monoid_1(A,B,B,B,D,E,C) = E ) ) ) ) ) ) )).

fof(t7_monoid_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(B,B),B)
            & m2_relset_1(C,k2_zfmisc_1(B,B),B) )
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,A,B)
                & m2_relset_1(D,A,B) )
             => ( v3_binop_1(C,B)
               => k3_monoid_1(A,B,B,B,C,D,D) = D ) ) ) ) )).

fof(t8_monoid_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(B,B),B)
            & m2_relset_1(C,k2_zfmisc_1(B,B),B) )
         => ( v1_binop_1(C,B)
           => v1_binop_1(k8_monoid_1(B,B,B,C,A),k1_fraenkel(A,B)) ) ) ) )).

fof(t9_monoid_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(B,B),B)
            & m2_relset_1(C,k2_zfmisc_1(B,B),B) )
         => ( v2_binop_1(C,B)
           => v2_binop_1(k8_monoid_1(B,B,B,C,A),k1_fraenkel(A,B)) ) ) ) )).

fof(t10_monoid_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( m1_subset_1(C,B)
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,k2_zfmisc_1(B,B),B)
                & m2_relset_1(D,k2_zfmisc_1(B,B),B) )
             => ( r3_binop_1(B,C,D)
               => r3_binop_1(k1_fraenkel(A,B),k5_monoid_1(B,A,C),k8_monoid_1(B,B,B,D,A)) ) ) ) ) )).

fof(t11_monoid_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(B,B),B)
            & m2_relset_1(C,k2_zfmisc_1(B,B),B) )
         => ( v1_setwiseo(C,B)
           => ( k3_binop_1(k1_fraenkel(A,B),k8_monoid_1(B,B,B,C,A)) = k5_monoid_1(B,A,k3_binop_1(B,C))
              & v1_setwiseo(k8_monoid_1(B,B,B,C,A),k1_fraenkel(A,B)) ) ) ) ) )).

fof(t12_monoid_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(B,B),B)
            & m2_relset_1(C,k2_zfmisc_1(B,B),B) )
         => ( v3_binop_1(C,B)
           => v3_binop_1(k8_monoid_1(B,B,B,C,A),k1_fraenkel(A,B)) ) ) ) )).

fof(t13_monoid_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(B,B),B)
            & m2_relset_1(C,k2_zfmisc_1(B,B),B) )
         => ( v5_monoid_0(C,B)
           => v5_monoid_0(k8_monoid_1(B,B,B,C,A),k1_fraenkel(A,B)) ) ) ) )).

fof(t14_monoid_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(B,B),B)
            & m2_relset_1(C,k2_zfmisc_1(B,B),B) )
         => ( v8_monoid_0(C,B)
           => v8_monoid_0(k8_monoid_1(B,B,B,C,A),k1_fraenkel(A,B)) ) ) ) )).

fof(t15_monoid_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(B,B),B)
            & m2_relset_1(C,k2_zfmisc_1(B,B),B) )
         => ( v9_monoid_0(C,B)
           => v9_monoid_0(k8_monoid_1(B,B,B,C,A),k1_fraenkel(A,B)) ) ) ) )).

fof(t16_monoid_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(B,B),B)
            & m2_relset_1(C,k2_zfmisc_1(B,B),B) )
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,k2_zfmisc_1(B,B),B)
                & m2_relset_1(D,k2_zfmisc_1(B,B),B) )
             => ( r1_lattice2(B,C,D)
               => r1_lattice2(k1_fraenkel(A,B),k8_monoid_1(B,B,B,C,A),k8_monoid_1(B,B,B,D,A)) ) ) ) ) )).

fof(t17_monoid_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( ~ v1_xboole_0(D)
             => ! [E] :
                  ( ~ v1_xboole_0(E)
                 => ! [F] :
                      ( ~ v1_xboole_0(F)
                     => ! [G] :
                          ( ~ v1_xboole_0(G)
                         => ! [H] :
                              ( ( v1_funct_1(H)
                                & v1_funct_2(H,k2_zfmisc_1(B,C),D)
                                & m2_relset_1(H,k2_zfmisc_1(B,C),D) )
                             => ! [I] :
                                  ( ( v1_funct_1(I)
                                    & v1_funct_2(I,k2_zfmisc_1(E,F),G)
                                    & m2_relset_1(I,k2_zfmisc_1(E,F),G) )
                                 => ( r1_tarski(H,I)
                                   => r1_tarski(k8_monoid_1(B,C,D,H,A),k8_monoid_1(E,F,G,I,A)) ) ) ) ) ) ) ) ) ) )).

fof(d3_monoid_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( v2_group_1(A)
           => k9_monoid_1(A,B) = g1_vectsp_1(k1_fraenkel(B,u1_struct_0(A)),k8_monoid_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u1_group_1(A),B),k5_monoid_1(u1_struct_0(A),B,k3_binop_1(u1_struct_0(A),u1_group_1(A)))) )
          & ( ~ v2_group_1(A)
           => k9_monoid_1(A,B) = g1_group_1(k1_fraenkel(B,u1_struct_0(A)),k8_monoid_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u1_group_1(A),B)) ) ) ) )).

fof(t18_monoid_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l1_group_1(B) )
     => ( u1_struct_0(k9_monoid_1(B,A)) = k1_fraenkel(A,u1_struct_0(B))
        & u1_group_1(k9_monoid_1(B,A)) = k8_monoid_1(u1_struct_0(B),u1_struct_0(B),u1_struct_0(B),u1_group_1(B),A) ) ) )).

fof(t19_monoid_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(C)
        & l1_group_1(C) )
     => ( m1_subset_1(A,u1_struct_0(k9_monoid_1(C,B)))
      <=> ( v1_funct_1(A)
          & v1_funct_2(A,B,u1_struct_0(C))
          & m2_relset_1(A,B,u1_struct_0(C)) ) ) ) )).

fof(t20_monoid_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l1_group_1(B) )
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(k9_monoid_1(B,A)))
         => ( k1_relat_1(C) = A
            & r1_tarski(k2_relat_1(C),u1_struct_0(B)) ) ) ) )).

fof(t21_monoid_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l1_group_1(B) )
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(k9_monoid_1(B,A)))
         => ! [D] :
              ( m1_subset_1(D,u1_struct_0(k9_monoid_1(B,A)))
             => ( ! [E] :
                    ( r2_hidden(E,A)
                   => k1_funct_1(C,E) = k1_funct_1(D,E) )
               => C = D ) ) ) ) )).

fof(t22_monoid_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_group_1(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k9_monoid_1(B,A)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k9_monoid_1(B,A)))
                 => ! [E] :
                      ( m1_subset_1(E,A)
                     => k10_monoid_1(B,A,k1_group_1(k9_monoid_1(B,A),C,D),E) = k1_group_1(B,k10_monoid_1(B,A,C,E),k10_monoid_1(B,A,D,E)) ) ) ) ) ) )).

fof(t23_monoid_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v2_group_1(B)
        & l1_group_1(B) )
     => u1_vectsp_1(k11_monoid_1(B,A)) = k5_monoid_1(u1_struct_0(B),A,k3_binop_1(u1_struct_0(B),u1_group_1(B))) ) )).

fof(t24_monoid_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( v7_group_1(A)
           => v7_group_1(k9_monoid_1(A,B)) )
          & ( v4_group_1(A)
           => v4_group_1(k9_monoid_1(A,B)) )
          & ( v10_monoid_0(A)
           => v10_monoid_0(k9_monoid_1(A,B)) )
          & ( v13_monoid_0(A)
           => v13_monoid_0(k9_monoid_1(A,B)) )
          & ( v16_monoid_0(A)
           => v16_monoid_0(k9_monoid_1(A,B)) )
          & ( v17_monoid_0(A)
           => v17_monoid_0(k9_monoid_1(A,B)) ) ) ) )).

fof(t25_monoid_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l1_group_1(B) )
     => ! [C] :
          ( ( ~ v3_struct_0(C)
            & m2_monoid_0(C,B) )
         => m2_monoid_0(k9_monoid_1(C,A),k9_monoid_1(B,A)) ) ) )).

fof(t26_monoid_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v2_group_1(B)
        & l1_group_1(B) )
     => ! [C] :
          ( ( ~ v3_struct_0(C)
            & m2_monoid_0(C,B) )
         => ( r2_hidden(k3_binop_1(u1_struct_0(B),u1_group_1(B)),u1_struct_0(C))
           => m3_monoid_0(k9_monoid_1(C,A),k11_monoid_1(B,A)) ) ) ) )).

fof(d4_monoid_1,axiom,(
    ! [A] : k13_monoid_1(A) = k12_monoid_1(k8_monoid_0,A) )).

fof(t27_monoid_1,axiom,(
    ! [A] :
      ( u1_struct_0(k13_monoid_1(A)) = k1_fraenkel(A,k5_numbers)
      & u1_group_1(k13_monoid_1(A)) = k8_monoid_1(k5_numbers,k5_numbers,k5_numbers,k47_binop_2,A)
      & u1_vectsp_1(k13_monoid_1(A)) = k5_monoid_1(k5_numbers,A,np__0) ) )).

fof(t28_monoid_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(A,u1_struct_0(k13_monoid_1(B)))
    <=> ( v1_funct_1(A)
        & v1_funct_2(A,B,k5_numbers)
        & m2_relset_1(A,B,k5_numbers) ) ) )).

fof(t29_monoid_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,u1_struct_0(k13_monoid_1(A)))
     => ( k1_relat_1(B) = A
        & r1_tarski(k2_relat_1(B),k5_numbers) ) ) )).

fof(t30_monoid_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k13_monoid_1(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k13_monoid_1(A)))
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => k15_monoid_1(A,k4_monoid_0(k13_monoid_1(A),B,C),D) = k1_nat_1(k15_monoid_1(A,B,D),k15_monoid_1(A,C,D)) ) ) ) ) )).

fof(t31_monoid_1,axiom,(
    ! [A,B] : m1_subset_1(k5_funct_3(A,B),u1_struct_0(k13_monoid_1(B))) )).

fof(d5_monoid_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => k18_monoid_1(A,B) = k16_monoid_1(k6_domain_1(A,B),A) ) ) )).

fof(t32_monoid_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ( k15_monoid_1(A,k18_monoid_1(A,B),B) = np__1
                & ( C != B
                 => k15_monoid_1(A,k18_monoid_1(A,B),C) = np__0 ) ) ) ) ) )).

fof(t33_monoid_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k13_monoid_1(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k13_monoid_1(A)))
             => ( ! [D] :
                    ( m1_subset_1(D,A)
                   => k15_monoid_1(A,B,D) = k15_monoid_1(A,C,D) )
               => B = C ) ) ) ) )).

fof(d6_monoid_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v1_vectsp_1(B)
        & m3_monoid_0(B,k13_monoid_1(A)) )
     => ( B = k19_monoid_1(A)
      <=> ! [C] :
            ( m1_subset_1(C,u1_struct_0(k13_monoid_1(A)))
           => ( r2_hidden(C,u1_struct_0(B))
            <=> v1_finset_1(k10_relat_1(C,k4_xboole_0(k5_numbers,k6_domain_1(k5_numbers,np__0)))) ) ) ) ) )).

fof(t34_monoid_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => m1_subset_1(k18_monoid_1(A,B),u1_struct_0(k19_monoid_1(A))) ) ) )).

fof(t35_monoid_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( m2_finseq_1(C,B)
         => k1_relat_1(k8_relat_1(k1_tarski(A),k7_finseq_1(C,k9_finseq_1(A)))) = k2_xboole_0(k1_relat_1(k8_relat_1(k1_tarski(A),C)),k6_domain_1(k5_numbers,k1_nat_1(k3_finseq_1(C),np__1))) ) ) )).

fof(t36_monoid_1,axiom,(
    ! [A,B,C] :
      ( ~ v1_xboole_0(C)
     => ! [D] :
          ( m2_finseq_1(D,C)
         => ( A != B
           => k1_relat_1(k8_relat_1(k1_tarski(A),k7_finseq_1(D,k9_finseq_1(B)))) = k1_relat_1(k8_relat_1(k1_tarski(A),D)) ) ) ) )).

fof(d7_monoid_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k13_monoid_1(A)))
             => ( C = k20_monoid_1(A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,A)
                   => k15_monoid_1(A,C,D) = k4_card_1(k1_relat_1(k8_relat_1(k6_domain_1(A,D),B))) ) ) ) ) ) )).

fof(t37_monoid_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => k15_monoid_1(A,k20_monoid_1(A,k6_finseq_1(A)),B) = np__0 ) ) )).

fof(t38_monoid_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k20_monoid_1(A,k6_finseq_1(A)) = k17_monoid_1(A,np__0) ) )).

fof(t39_monoid_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => k20_monoid_1(A,k12_finseq_1(A,B)) = k18_monoid_1(A,B) ) ) )).

fof(t40_monoid_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => k20_monoid_1(A,k1_monoid_0(A,C,k12_finseq_1(A,B))) = k4_monoid_0(k13_monoid_1(A),k20_monoid_1(A,C),k18_monoid_1(A,B)) ) ) ) )).

fof(t41_monoid_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => k20_monoid_1(A,k1_monoid_0(A,B,C)) = k4_monoid_0(k13_monoid_1(A),k20_monoid_1(A,B),k20_monoid_1(A,C)) ) ) ) )).

fof(t42_monoid_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,B)
             => ( k15_monoid_1(B,k20_monoid_1(B,k4_monoid_1(B,A,C)),C) = A
                & ! [D] :
                    ( m1_subset_1(D,B)
                   => ( D != C
                     => k15_monoid_1(B,k20_monoid_1(B,k4_monoid_1(B,A,C)),D) = np__0 ) ) ) ) ) ) )).

fof(t43_monoid_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => m1_subset_1(k20_monoid_1(A,B),u1_struct_0(k19_monoid_1(A))) ) ) )).

fof(t44_monoid_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ~ ( m1_subset_1(A,u1_struct_0(k19_monoid_1(B)))
          & ! [C] :
              ( m2_finseq_1(C,B)
             => A != k20_monoid_1(B,C) ) ) ) )).

fof(d8_monoid_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,B),C)
                    & m2_relset_1(D,k2_zfmisc_1(A,B),C) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k2_zfmisc_1(k1_zfmisc_1(A),k1_zfmisc_1(B)),k1_zfmisc_1(C))
                        & m2_relset_1(E,k2_zfmisc_1(k1_zfmisc_1(A),k1_zfmisc_1(B)),k1_zfmisc_1(C)) )
                     => ( E = k21_monoid_1(A,B,C,D)
                      <=> ! [F] :
                            ( m1_subset_1(F,k2_zfmisc_1(k1_zfmisc_1(A),k1_zfmisc_1(B)))
                           => k8_funct_2(k2_zfmisc_1(k1_zfmisc_1(A),k1_zfmisc_1(B)),k1_zfmisc_1(C),E,F) = k2_funct_2(k2_zfmisc_1(A,B),C,D,k2_zfmisc_1(k2_domain_1(k1_zfmisc_1(A),k1_zfmisc_1(B),F),k3_domain_1(k1_zfmisc_1(A),k1_zfmisc_1(B),F))) ) ) ) ) ) ) ) )).

fof(t45_monoid_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,B),C)
                    & m2_relset_1(D,k2_zfmisc_1(A,B),C) )
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(A))
                     => ! [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(B))
                         => k2_binop_1(k1_zfmisc_1(A),k1_zfmisc_1(B),k1_zfmisc_1(C),k21_monoid_1(A,B,C,D),E,F) = k2_funct_2(k2_zfmisc_1(A,B),C,D,k2_zfmisc_1(E,F)) ) ) ) ) ) ) )).

fof(t46_monoid_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,B),C)
                    & m2_relset_1(D,k2_zfmisc_1(A,B),C) )
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(A))
                     => ! [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(B))
                         => ! [G,H] :
                              ( ( r2_hidden(G,E)
                                & r2_hidden(H,F) )
                             => r2_hidden(k1_binop_1(D,G,H),k2_binop_1(k1_zfmisc_1(A),k1_zfmisc_1(B),k1_zfmisc_1(C),k21_monoid_1(A,B,C,D),E,F)) ) ) ) ) ) ) ) )).

fof(t48_monoid_1,axiom,(
    ! [A,B,C] :
      ( ~ v1_xboole_0(C)
     => ! [D] :
          ( ( v1_funct_1(D)
            & v1_funct_2(D,k2_zfmisc_1(C,C),C)
            & m2_relset_1(D,k2_zfmisc_1(C,C),C) )
         => ( v1_binop_1(D,C)
           => k2_funct_2(k2_zfmisc_1(C,C),C,D,k2_zfmisc_1(A,B)) = k2_funct_2(k2_zfmisc_1(C,C),C,D,k2_zfmisc_1(B,A)) ) ) ) )).

fof(t49_monoid_1,axiom,(
    ! [A,B,C,D] :
      ( ~ v1_xboole_0(D)
     => ! [E] :
          ( ( v1_funct_1(E)
            & v1_funct_2(E,k2_zfmisc_1(D,D),D)
            & m2_relset_1(E,k2_zfmisc_1(D,D),D) )
         => ( v2_binop_1(E,D)
           => k2_funct_2(k2_zfmisc_1(D,D),D,E,k2_zfmisc_1(k2_funct_2(k2_zfmisc_1(D,D),D,E,k2_zfmisc_1(A,B)),C)) = k2_funct_2(k2_zfmisc_1(D,D),D,E,k2_zfmisc_1(A,k2_funct_2(k2_zfmisc_1(D,D),D,E,k2_zfmisc_1(B,C)))) ) ) ) )).

fof(t50_monoid_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),A)
            & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
         => ( v1_binop_1(B,A)
           => v1_binop_1(k21_monoid_1(A,A,A,B),k1_zfmisc_1(A)) ) ) ) )).

fof(t51_monoid_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),A)
            & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
         => ( v2_binop_1(B,A)
           => v2_binop_1(k21_monoid_1(A,A,A,B),k1_zfmisc_1(A)) ) ) ) )).

fof(t52_monoid_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(B,B),B)
            & m2_relset_1(C,k2_zfmisc_1(B,B),B) )
         => ! [D] :
              ( m1_subset_1(D,B)
             => ( r3_binop_1(B,D,C)
               => ( k2_funct_2(k2_zfmisc_1(B,B),B,C,k2_zfmisc_1(k6_domain_1(B,D),A)) = k3_xboole_0(B,A)
                  & k2_funct_2(k2_zfmisc_1(B,B),B,C,k2_zfmisc_1(A,k6_domain_1(B,D))) = k3_xboole_0(B,A) ) ) ) ) ) )).

fof(t53_monoid_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),A)
            & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ( r3_binop_1(A,C,B)
               => ( r3_binop_1(k1_zfmisc_1(A),k6_domain_1(A,C),k21_monoid_1(A,A,A,B))
                  & v1_setwiseo(k21_monoid_1(A,A,A,B),k1_zfmisc_1(A))
                  & k3_binop_1(k1_zfmisc_1(A),k21_monoid_1(A,A,A,B)) = k6_domain_1(A,C) ) ) ) ) ) )).

fof(t54_monoid_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),A)
            & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
         => ( v1_setwiseo(B,A)
           => ( v1_setwiseo(k21_monoid_1(A,A,A,B),k1_zfmisc_1(A))
              & r3_binop_1(k1_zfmisc_1(A),k6_domain_1(A,k3_binop_1(A,B)),k21_monoid_1(A,A,A,B))
              & k3_binop_1(k1_zfmisc_1(A),k21_monoid_1(A,A,A,B)) = k6_domain_1(A,k3_binop_1(A,B)) ) ) ) ) )).

fof(t55_monoid_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),A)
            & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
         => ( v9_monoid_0(B,A)
           => v9_monoid_0(k21_monoid_1(A,A,A,B),k1_zfmisc_1(A)) ) ) ) )).

fof(d9_monoid_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( ( v2_group_1(A)
         => k22_monoid_1(A) = g1_vectsp_1(k1_zfmisc_1(u1_struct_0(A)),k21_monoid_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u1_group_1(A)),k6_domain_1(u1_struct_0(A),k3_binop_1(u1_struct_0(A),u1_group_1(A)))) )
        & ( ~ v2_group_1(A)
         => k22_monoid_1(A) = g1_group_1(k1_zfmisc_1(u1_struct_0(A)),k21_monoid_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u1_group_1(A))) ) ) ) )).

fof(t56_monoid_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( u1_struct_0(k22_monoid_1(A)) = k1_zfmisc_1(u1_struct_0(A))
        & u1_group_1(k22_monoid_1(A)) = k21_monoid_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u1_group_1(A)) ) ) )).

fof(t57_monoid_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l1_group_1(A) )
     => u1_vectsp_1(k23_monoid_1(A)) = k6_domain_1(u1_struct_0(A),k3_binop_1(u1_struct_0(A),u1_group_1(A))) ) )).

fof(t58_monoid_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( ( v7_group_1(A)
         => v7_group_1(k22_monoid_1(A)) )
        & ( v4_group_1(A)
         => v4_group_1(k22_monoid_1(A)) )
        & ( v17_monoid_0(A)
         => v17_monoid_0(k22_monoid_1(A)) ) ) ) )).

fof(s1_monoid_1,axiom,
    ( ! [A] :
        ~ ( r2_hidden(A,f1_s1_monoid_1)
          & ! [B] :
              ( m1_subset_1(B,f2_s1_monoid_1)
             => ~ p1_s1_monoid_1(A,B) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & v1_funct_2(A,f1_s1_monoid_1,f2_s1_monoid_1)
        & m2_relset_1(A,f1_s1_monoid_1,f2_s1_monoid_1)
        & ! [B] :
            ( r2_hidden(B,f1_s1_monoid_1)
           => p1_s1_monoid_1(B,k1_funct_1(A,B)) ) ) )).

fof(dt_k1_monoid_1,axiom,(
    $true )).

fof(dt_k2_monoid_1,axiom,(
    ! [A,B,C,D,E,F,G,H] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D)
        & v1_funct_1(E)
        & v1_funct_2(E,k2_zfmisc_1(B,C),k1_fraenkel(A,D))
        & m1_relset_1(E,k2_zfmisc_1(B,C),k1_fraenkel(A,D))
        & m1_subset_1(F,B)
        & m1_subset_1(G,C)
        & m1_subset_1(H,A) )
     => m1_subset_1(k2_monoid_1(A,B,C,D,E,F,G,H),D) ) )).

fof(redefinition_k2_monoid_1,axiom,(
    ! [A,B,C,D,E,F,G,H] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D)
        & v1_funct_1(E)
        & v1_funct_2(E,k2_zfmisc_1(B,C),k1_fraenkel(A,D))
        & m1_relset_1(E,k2_zfmisc_1(B,C),k1_fraenkel(A,D))
        & m1_subset_1(F,B)
        & m1_subset_1(G,C)
        & m1_subset_1(H,A) )
     => k2_monoid_1(A,B,C,D,E,F,G,H) = k1_monoid_1(E,F,G,H) ) )).

fof(dt_k3_monoid_1,axiom,(
    ! [A,B,C,D,E,F,G] :
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D)
        & v1_funct_1(E)
        & v1_funct_2(E,k2_zfmisc_1(B,C),D)
        & m1_relset_1(E,k2_zfmisc_1(B,C),D)
        & v1_funct_1(F)
        & v1_funct_2(F,A,B)
        & m1_relset_1(F,A,B)
        & v1_funct_1(G)
        & v1_funct_2(G,A,C)
        & m1_relset_1(G,A,C) )
     => m2_fraenkel(k3_monoid_1(A,B,C,D,E,F,G),A,D,k1_fraenkel(A,D)) ) )).

fof(redefinition_k3_monoid_1,axiom,(
    ! [A,B,C,D,E,F,G] :
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D)
        & v1_funct_1(E)
        & v1_funct_2(E,k2_zfmisc_1(B,C),D)
        & m1_relset_1(E,k2_zfmisc_1(B,C),D)
        & v1_funct_1(F)
        & v1_funct_2(F,A,B)
        & m1_relset_1(F,A,B)
        & v1_funct_1(G)
        & v1_funct_2(G,A,C)
        & m1_relset_1(G,A,C) )
     => k3_monoid_1(A,B,C,D,E,F,G) = k3_funcop_1(E,F,G) ) )).

fof(dt_k4_monoid_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,A) )
     => m2_finseq_1(k4_monoid_1(A,B,C),A) ) )).

fof(redefinition_k4_monoid_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,A) )
     => k4_monoid_1(A,B,C) = k2_finseq_2(B,C) ) )).

fof(dt_k5_monoid_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(C,A) )
     => m2_fraenkel(k5_monoid_1(A,B,C),B,A,k1_fraenkel(B,A)) ) )).

fof(redefinition_k5_monoid_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(C,A) )
     => k5_monoid_1(A,B,C) = k2_funcop_1(B,C) ) )).

fof(dt_k6_monoid_1,axiom,(
    ! [A,B,C,D,E,F,G] :
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D)
        & v1_funct_1(E)
        & v1_funct_2(E,k2_zfmisc_1(B,C),D)
        & m1_relset_1(E,k2_zfmisc_1(B,C),D)
        & m1_subset_1(F,B)
        & v1_funct_1(G)
        & v1_funct_2(G,A,C)
        & m1_relset_1(G,A,C) )
     => m2_fraenkel(k6_monoid_1(A,B,C,D,E,F,G),A,D,k1_fraenkel(A,D)) ) )).

fof(redefinition_k6_monoid_1,axiom,(
    ! [A,B,C,D,E,F,G] :
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D)
        & v1_funct_1(E)
        & v1_funct_2(E,k2_zfmisc_1(B,C),D)
        & m1_relset_1(E,k2_zfmisc_1(B,C),D)
        & m1_subset_1(F,B)
        & v1_funct_1(G)
        & v1_funct_2(G,A,C)
        & m1_relset_1(G,A,C) )
     => k6_monoid_1(A,B,C,D,E,F,G) = k5_funcop_1(E,F,G) ) )).

fof(dt_k7_monoid_1,axiom,(
    ! [A,B,C,D,E,F,G] :
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D)
        & v1_funct_1(E)
        & v1_funct_2(E,k2_zfmisc_1(B,C),D)
        & m1_relset_1(E,k2_zfmisc_1(B,C),D)
        & v1_funct_1(F)
        & v1_funct_2(F,A,B)
        & m1_relset_1(F,A,B)
        & m1_subset_1(G,C) )
     => m2_fraenkel(k7_monoid_1(A,B,C,D,E,F,G),A,D,k1_fraenkel(A,D)) ) )).

fof(redefinition_k7_monoid_1,axiom,(
    ! [A,B,C,D,E,F,G] :
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D)
        & v1_funct_1(E)
        & v1_funct_2(E,k2_zfmisc_1(B,C),D)
        & m1_relset_1(E,k2_zfmisc_1(B,C),D)
        & v1_funct_1(F)
        & v1_funct_2(F,A,B)
        & m1_relset_1(F,A,B)
        & m1_subset_1(G,C) )
     => k7_monoid_1(A,B,C,D,E,F,G) = k4_funcop_1(E,F,G) ) )).

fof(dt_k8_monoid_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(A,B),C)
        & m1_relset_1(D,k2_zfmisc_1(A,B),C) )
     => ( v1_funct_1(k8_monoid_1(A,B,C,D,E))
        & v1_funct_2(k8_monoid_1(A,B,C,D,E),k2_zfmisc_1(k1_fraenkel(E,A),k1_fraenkel(E,B)),k1_fraenkel(E,C))
        & m2_relset_1(k8_monoid_1(A,B,C,D,E),k2_zfmisc_1(k1_fraenkel(E,A),k1_fraenkel(E,B)),k1_fraenkel(E,C)) ) ) )).

fof(dt_k9_monoid_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => l1_group_1(k9_monoid_1(A,B)) ) )).

fof(dt_k10_monoid_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(C,u1_struct_0(k9_monoid_1(A,B)))
        & m1_subset_1(D,B) )
     => m1_subset_1(k10_monoid_1(A,B,C,D),u1_struct_0(A)) ) )).

fof(redefinition_k10_monoid_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(C,u1_struct_0(k9_monoid_1(A,B)))
        & m1_subset_1(D,B) )
     => k10_monoid_1(A,B,C,D) = k1_funct_1(C,D) ) )).

fof(dt_k11_monoid_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l1_group_1(A) )
     => ( ~ v3_struct_0(k11_monoid_1(A,B))
        & v1_vectsp_1(k11_monoid_1(A,B))
        & v1_vectsp_2(k11_monoid_1(A,B))
        & v1_monoid_0(k11_monoid_1(A,B))
        & l1_vectsp_1(k11_monoid_1(A,B)) ) ) )).

fof(redefinition_k11_monoid_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l1_group_1(A) )
     => k11_monoid_1(A,B) = k9_monoid_1(A,B) ) )).

fof(dt_k12_monoid_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v16_monoid_0(A)
        & v17_monoid_0(A)
        & l1_group_1(A) )
     => ( ~ v3_struct_0(k12_monoid_1(A,B))
        & v4_group_1(k12_monoid_1(A,B))
        & v7_group_1(k12_monoid_1(A,B))
        & v1_vectsp_1(k12_monoid_1(A,B))
        & v1_vectsp_2(k12_monoid_1(A,B))
        & v1_monoid_0(k12_monoid_1(A,B))
        & v16_monoid_0(k12_monoid_1(A,B))
        & v17_monoid_0(k12_monoid_1(A,B))
        & l1_vectsp_1(k12_monoid_1(A,B)) ) ) )).

fof(redefinition_k12_monoid_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v16_monoid_0(A)
        & v17_monoid_0(A)
        & l1_group_1(A) )
     => k12_monoid_1(A,B) = k9_monoid_1(A,B) ) )).

fof(dt_k13_monoid_1,axiom,(
    ! [A] :
      ( ~ v3_struct_0(k13_monoid_1(A))
      & v4_group_1(k13_monoid_1(A))
      & v7_group_1(k13_monoid_1(A))
      & v1_vectsp_1(k13_monoid_1(A))
      & v1_vectsp_2(k13_monoid_1(A))
      & v1_monoid_0(k13_monoid_1(A))
      & v16_monoid_0(k13_monoid_1(A))
      & v17_monoid_0(k13_monoid_1(A))
      & l1_vectsp_1(k13_monoid_1(A)) ) )).

fof(dt_k14_monoid_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,u1_struct_0(k13_monoid_1(A))) )
     => ( ~ v1_xboole_0(k14_monoid_1(A,B))
        & m1_subset_1(k14_monoid_1(A,B),k1_zfmisc_1(k5_numbers)) ) ) )).

fof(redefinition_k14_monoid_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,u1_struct_0(k13_monoid_1(A))) )
     => k14_monoid_1(A,B) = k2_relat_1(B) ) )).

fof(dt_k15_monoid_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,u1_struct_0(k13_monoid_1(A)))
        & m1_subset_1(C,A) )
     => m2_subset_1(k15_monoid_1(A,B,C),k1_numbers,k5_numbers) ) )).

fof(redefinition_k15_monoid_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,u1_struct_0(k13_monoid_1(A)))
        & m1_subset_1(C,A) )
     => k15_monoid_1(A,B,C) = k1_funct_1(B,C) ) )).

fof(dt_k16_monoid_1,axiom,(
    ! [A,B] : m1_subset_1(k16_monoid_1(A,B),u1_struct_0(k13_monoid_1(B))) )).

fof(redefinition_k16_monoid_1,axiom,(
    ! [A,B] : k16_monoid_1(A,B) = k4_funct_3(A,B) )).

fof(dt_k17_monoid_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k5_numbers)
     => m1_subset_1(k17_monoid_1(A,B),u1_struct_0(k13_monoid_1(A))) ) )).

fof(redefinition_k17_monoid_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k5_numbers)
     => k17_monoid_1(A,B) = k2_funcop_1(A,B) ) )).

fof(dt_k18_monoid_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A) )
     => m1_subset_1(k18_monoid_1(A,B),u1_struct_0(k13_monoid_1(A))) ) )).

fof(dt_k19_monoid_1,axiom,(
    ! [A] :
      ( ~ v3_struct_0(k19_monoid_1(A))
      & v1_vectsp_1(k19_monoid_1(A))
      & m3_monoid_0(k19_monoid_1(A),k13_monoid_1(A)) ) )).

fof(dt_k20_monoid_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(B,A) )
     => m1_subset_1(k20_monoid_1(A,B),u1_struct_0(k13_monoid_1(A))) ) )).

fof(dt_k21_monoid_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(A,B),C)
        & m1_relset_1(D,k2_zfmisc_1(A,B),C) )
     => ( v1_funct_1(k21_monoid_1(A,B,C,D))
        & v1_funct_2(k21_monoid_1(A,B,C,D),k2_zfmisc_1(k1_zfmisc_1(A),k1_zfmisc_1(B)),k1_zfmisc_1(C))
        & m2_relset_1(k21_monoid_1(A,B,C,D),k2_zfmisc_1(k1_zfmisc_1(A),k1_zfmisc_1(B)),k1_zfmisc_1(C)) ) ) )).

fof(dt_k22_monoid_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => l1_group_1(k22_monoid_1(A)) ) )).

fof(dt_k23_monoid_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l1_group_1(A) )
     => ( ~ v3_struct_0(k23_monoid_1(A))
        & v1_vectsp_1(k23_monoid_1(A))
        & v1_vectsp_2(k23_monoid_1(A))
        & l1_vectsp_1(k23_monoid_1(A)) ) ) )).

fof(redefinition_k23_monoid_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l1_group_1(A) )
     => k23_monoid_1(A) = k22_monoid_1(A) ) )).

fof(t47_monoid_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,B),C)
                    & m2_relset_1(D,k2_zfmisc_1(A,B),C) )
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(A))
                     => ! [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(B))
                         => k2_binop_1(k1_zfmisc_1(A),k1_zfmisc_1(B),k1_zfmisc_1(C),k21_monoid_1(A,B,C,D),E,F) = a_6_0_monoid_1(A,B,C,D,E,F) ) ) ) ) ) ) )).

fof(fraenkel_a_6_0_monoid_1,axiom,(
    ! [A,B,C,D,E,F,G] :
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D)
        & v1_funct_1(E)
        & v1_funct_2(E,k2_zfmisc_1(B,C),D)
        & m2_relset_1(E,k2_zfmisc_1(B,C),D)
        & m1_subset_1(F,k1_zfmisc_1(B))
        & m1_subset_1(G,k1_zfmisc_1(C)) )
     => ( r2_hidden(A,a_6_0_monoid_1(B,C,D,E,F,G))
      <=> ? [H,I] :
            ( m1_subset_1(H,B)
            & m1_subset_1(I,C)
            & A = k2_binop_1(B,C,D,E,H,I)
            & r2_hidden(H,F)
            & r2_hidden(I,G) ) ) ) )).
%------------------------------------------------------------------------------