SET007 Axioms: SET007+365.ax


%------------------------------------------------------------------------------
% File     : SET007+365 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Monoids
% 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_0 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  227 (  22 unit)
%            Number of atoms       : 1278 ( 148 equality)
%            Maximal formula depth :   21 (   7 average)
%            Number of connectives : 1270 ( 219 ~  ;   3  |; 573  &)
%                                         (  60 <=>; 415 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   65 (   1 propositional; 0-3 arity)
%            Number of functors    :   65 (  25 constant; 0-6 arity)
%            Number of variables   :  442 (   0 singleton; 409 !;  33 ?)
%            Maximal term depth    :    4 (   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(rc1_monoid_0,axiom,(
    ? [A] :
      ( l1_struct_0(A)
      & v1_monoid_0(A) ) )).

fof(rc2_monoid_0,axiom,(
    ? [A] :
      ( l1_struct_0(A)
      & v2_monoid_0(A) ) )).

fof(cc1_monoid_0,axiom,(
    ! [A] :
      ( ( v1_monoid_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v1_relat_1(B)
            & v1_funct_1(B) ) ) ) )).

fof(cc2_monoid_0,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ( v2_monoid_0(A)
       => v1_monoid_0(A) ) ) )).

fof(cc3_monoid_0,axiom,(
    ! [A] :
      ( l1_group_1(A)
     => ( v2_monoid_0(A)
       => v1_monoid_0(A) ) ) )).

fof(cc4_monoid_0,axiom,(
    ! [A] :
      ( ( v2_monoid_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finset_1(B)
            & v1_finseq_1(B) ) ) ) )).

fof(rc3_monoid_0,axiom,(
    ? [A] :
      ( l1_group_1(A)
      & ~ v3_struct_0(A)
      & v1_group_1(A)
      & v2_group_1(A)
      & v4_group_1(A)
      & v7_group_1(A)
      & v1_monoid_0(A)
      & v2_monoid_0(A)
      & v10_monoid_0(A)
      & v13_monoid_0(A)
      & v16_monoid_0(A)
      & v17_monoid_0(A) ) )).

fof(cc5_monoid_0,axiom,(
    ! [A] :
      ( l1_group_1(A)
     => ( ( ~ v3_struct_0(A)
          & v3_group_1(A)
          & v4_group_1(A) )
       => ( ~ v3_struct_0(A)
          & v13_monoid_0(A) ) ) ) )).

fof(cc6_monoid_0,axiom,(
    ! [A] :
      ( l1_group_1(A)
     => ( ( ~ v3_struct_0(A)
          & v4_group_1(A)
          & v13_monoid_0(A) )
       => ( ~ v3_struct_0(A)
          & v2_group_1(A)
          & v3_group_1(A) ) ) ) )).

fof(cc7_monoid_0,axiom,(
    ! [A] :
      ( l1_group_1(A)
     => ( ( ~ v3_struct_0(A)
          & v13_monoid_0(A) )
       => ( ~ v3_struct_0(A)
          & v11_monoid_0(A)
          & v12_monoid_0(A) ) ) ) )).

fof(cc8_monoid_0,axiom,(
    ! [A] :
      ( l1_group_1(A)
     => ( ( ~ v3_struct_0(A)
          & v11_monoid_0(A)
          & v12_monoid_0(A) )
       => ( ~ v3_struct_0(A)
          & v13_monoid_0(A) ) ) ) )).

fof(cc9_monoid_0,axiom,(
    ! [A] :
      ( l1_group_1(A)
     => ( ( ~ v3_struct_0(A)
          & v16_monoid_0(A) )
       => ( ~ v3_struct_0(A)
          & v14_monoid_0(A)
          & v15_monoid_0(A) ) ) ) )).

fof(cc10_monoid_0,axiom,(
    ! [A] :
      ( l1_group_1(A)
     => ( ( ~ v3_struct_0(A)
          & v14_monoid_0(A)
          & v15_monoid_0(A) )
       => ( ~ v3_struct_0(A)
          & v16_monoid_0(A) ) ) ) )).

fof(cc11_monoid_0,axiom,(
    ! [A] :
      ( l1_group_1(A)
     => ( ( ~ v3_struct_0(A)
          & v4_group_1(A)
          & v13_monoid_0(A) )
       => ( ~ v3_struct_0(A)
          & v2_group_1(A)
          & v16_monoid_0(A) ) ) ) )).

fof(cc12_monoid_0,axiom,(
    ! [A] :
      ( l1_vectsp_1(A)
     => ( ( ~ v3_struct_0(A)
          & v1_vectsp_2(A) )
       => ( ~ v3_struct_0(A)
          & v2_group_1(A)
          & v6_vectsp_1(A)
          & v8_vectsp_1(A) ) ) ) )).

fof(fc1_monoid_0,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),A)
        & m1_relset_1(B,k2_zfmisc_1(A,A),A)
        & m1_subset_1(C,A) )
     => ( ~ v3_struct_0(g1_vectsp_1(A,B,C))
        & v1_vectsp_1(g1_vectsp_1(A,B,C)) ) ) )).

fof(rc4_monoid_0,axiom,(
    ? [A] :
      ( l1_vectsp_1(A)
      & ~ v3_struct_0(A)
      & v2_group_1(A)
      & v3_group_1(A)
      & v4_group_1(A)
      & v7_group_1(A)
      & v1_vectsp_1(A)
      & v6_vectsp_1(A)
      & v8_vectsp_1(A)
      & v1_vectsp_2(A)
      & v1_monoid_0(A)
      & v2_monoid_0(A)
      & v10_monoid_0(A)
      & v11_monoid_0(A)
      & v12_monoid_0(A)
      & v13_monoid_0(A)
      & v14_monoid_0(A)
      & v15_monoid_0(A)
      & v16_monoid_0(A)
      & v17_monoid_0(A) ) )).

fof(cc13_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_monoid_0(B,A)
         => ~ v3_struct_0(B) ) ) )).

fof(rc5_monoid_0,axiom,(
    ! [A] :
      ( l1_group_1(A)
     => ? [B] :
          ( m1_monoid_0(B,A)
          & v1_vectsp_1(B) ) ) )).

fof(cc14_monoid_0,axiom,(
    ! [A] :
      ( ( v1_monoid_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_monoid_0(B,A)
         => v1_monoid_0(B) ) ) )).

fof(cc15_monoid_0,axiom,(
    ! [A] :
      ( ( v2_monoid_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_monoid_0(B,A)
         => ( v1_monoid_0(B)
            & v2_monoid_0(B) ) ) ) )).

fof(cc16_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_monoid_0(B,A)
         => ( ~ v3_struct_0(B)
            & v2_group_1(B)
            & v6_vectsp_1(B)
            & v8_vectsp_1(B) ) ) ) )).

fof(cc17_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_monoid_0(B,A)
         => ( ~ v3_struct_0(B)
            & v4_group_1(B) ) ) ) )).

fof(cc18_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v7_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_monoid_0(B,A)
         => ( ~ v3_struct_0(B)
            & v7_group_1(B) ) ) ) )).

fof(cc19_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v13_monoid_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_monoid_0(B,A)
         => ( ~ v3_struct_0(B)
            & v11_monoid_0(B)
            & v12_monoid_0(B)
            & v13_monoid_0(B) ) ) ) )).

fof(cc20_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v16_monoid_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_monoid_0(B,A)
         => ( ~ v3_struct_0(B)
            & v14_monoid_0(B)
            & v15_monoid_0(B)
            & v16_monoid_0(B) ) ) ) )).

fof(cc21_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v17_monoid_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_monoid_0(B,A)
         => ( ~ v3_struct_0(B)
            & v17_monoid_0(B) ) ) ) )).

fof(rc6_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l1_group_1(A) )
     => ? [B] :
          ( m1_monoid_0(B,A)
          & ~ v3_struct_0(B)
          & v2_group_1(B)
          & v1_vectsp_1(B)
          & v6_vectsp_1(B)
          & v8_vectsp_1(B)
          & v1_vectsp_2(B) ) ) )).

fof(rc7_monoid_0,axiom,(
    ! [A] :
      ( l1_group_1(A)
     => ? [B] :
          ( m2_monoid_0(B,A)
          & v1_group_1(B) ) ) )).

fof(rc8_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ? [B] :
          ( m2_monoid_0(B,A)
          & ~ v3_struct_0(B)
          & v1_group_1(B) ) ) )).

fof(rc9_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l1_group_1(A) )
     => ? [B] :
          ( m2_monoid_0(B,A)
          & ~ v3_struct_0(B)
          & v1_group_1(B)
          & v2_group_1(B)
          & v3_group_1(B)
          & v4_group_1(B)
          & v7_group_1(B)
          & v10_monoid_0(B)
          & v11_monoid_0(B)
          & v12_monoid_0(B)
          & v13_monoid_0(B)
          & v14_monoid_0(B)
          & v15_monoid_0(B)
          & v16_monoid_0(B)
          & v17_monoid_0(B) ) ) )).

fof(rc10_monoid_0,axiom,(
    ! [A] :
      ( l1_group_1(A)
     => ? [B] :
          ( m3_monoid_0(B,A)
          & v1_vectsp_1(B) ) ) )).

fof(rc11_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ? [B] :
          ( m3_monoid_0(B,A)
          & ~ v3_struct_0(B)
          & v1_vectsp_1(B) ) ) )).

fof(rc12_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_vectsp_2(A)
        & l1_vectsp_1(A) )
     => ? [B] :
          ( m3_monoid_0(B,A)
          & ~ v3_struct_0(B)
          & v2_group_1(B)
          & v3_group_1(B)
          & v4_group_1(B)
          & v7_group_1(B)
          & v1_vectsp_1(B)
          & v6_vectsp_1(B)
          & v8_vectsp_1(B)
          & v1_vectsp_2(B)
          & v10_monoid_0(B)
          & v11_monoid_0(B)
          & v12_monoid_0(B)
          & v13_monoid_0(B)
          & v14_monoid_0(B)
          & v15_monoid_0(B)
          & v16_monoid_0(B)
          & v17_monoid_0(B) ) ) )).

fof(cc22_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_monoid_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m2_monoid_0(B,A)
         => ( ~ v3_struct_0(B)
           => ( ~ v3_struct_0(B)
              & v1_monoid_0(B) ) ) ) ) )).

fof(cc23_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_monoid_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m3_monoid_0(B,A)
         => ( ~ v3_struct_0(B)
           => ( ~ v3_struct_0(B)
              & v1_monoid_0(B) ) ) ) ) )).

fof(cc24_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_monoid_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m2_monoid_0(B,A)
         => ( ~ v3_struct_0(B)
           => ( ~ v3_struct_0(B)
              & v1_monoid_0(B)
              & v2_monoid_0(B) ) ) ) ) )).

fof(cc25_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_monoid_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m3_monoid_0(B,A)
         => ( ~ v3_struct_0(B)
           => ( ~ v3_struct_0(B)
              & v1_monoid_0(B)
              & v2_monoid_0(B) ) ) ) ) )).

fof(cc26_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_vectsp_2(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( m3_monoid_0(B,A)
         => ( ~ v3_struct_0(B)
           => ( ~ v3_struct_0(B)
              & v2_group_1(B)
              & v6_vectsp_1(B)
              & v8_vectsp_1(B)
              & v1_vectsp_2(B) ) ) ) ) )).

fof(cc27_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v7_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m2_monoid_0(B,A)
         => ( ~ v3_struct_0(B)
           => ( ~ v3_struct_0(B)
              & v7_group_1(B) ) ) ) ) )).

fof(cc28_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v7_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m3_monoid_0(B,A)
         => ( ~ v3_struct_0(B)
           => ( ~ v3_struct_0(B)
              & v7_group_1(B) ) ) ) ) )).

fof(cc29_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m2_monoid_0(B,A)
         => ( ~ v3_struct_0(B)
           => ( ~ v3_struct_0(B)
              & v4_group_1(B) ) ) ) ) )).

fof(cc30_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m3_monoid_0(B,A)
         => ( ~ v3_struct_0(B)
           => ( ~ v3_struct_0(B)
              & v4_group_1(B) ) ) ) ) )).

fof(cc31_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_monoid_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m2_monoid_0(B,A)
         => ( ~ v3_struct_0(B)
           => ( ~ v3_struct_0(B)
              & v10_monoid_0(B) ) ) ) ) )).

fof(cc32_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_monoid_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m3_monoid_0(B,A)
         => ( ~ v3_struct_0(B)
           => ( ~ v3_struct_0(B)
              & v10_monoid_0(B) ) ) ) ) )).

fof(cc33_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v16_monoid_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m2_monoid_0(B,A)
         => ( ~ v3_struct_0(B)
           => ( ~ v3_struct_0(B)
              & v14_monoid_0(B)
              & v15_monoid_0(B)
              & v16_monoid_0(B) ) ) ) ) )).

fof(cc34_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v16_monoid_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m3_monoid_0(B,A)
         => ( ~ v3_struct_0(B)
           => ( ~ v3_struct_0(B)
              & v14_monoid_0(B)
              & v15_monoid_0(B)
              & v16_monoid_0(B) ) ) ) ) )).

fof(cc35_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_vectsp_2(A)
        & v17_monoid_0(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( m3_monoid_0(B,A)
         => ( ~ v3_struct_0(B)
           => ( ~ v3_struct_0(B)
              & v2_group_1(B)
              & v6_vectsp_1(B)
              & v8_vectsp_1(B)
              & v1_vectsp_2(B)
              & v17_monoid_0(B) ) ) ) ) )).

fof(cc36_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m2_monoid_0(B,A)
         => ( ( ~ v3_struct_0(B)
              & v4_group_1(B)
              & v13_monoid_0(B) )
           => ( ~ v3_struct_0(B)
              & v2_group_1(B)
              & v3_group_1(B)
              & v14_monoid_0(B)
              & v15_monoid_0(B)
              & v16_monoid_0(B) ) ) ) ) )).

fof(d1_monoid_0,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ( v1_monoid_0(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ( v1_relat_1(B)
              & v1_funct_1(B) ) ) ) ) )).

fof(d2_monoid_0,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ( v2_monoid_0(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ( v1_relat_1(B)
              & v1_funct_1(B)
              & v1_finseq_1(B) ) ) ) ) )).

fof(d3_monoid_0,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) )
         => ( v3_monoid_0(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,A)
               => ! [D] :
                    ( m1_subset_1(D,A)
                   => ? [E] :
                        ( m1_subset_1(E,A)
                        & k2_binop_1(A,A,A,B,E,C) = D ) ) ) ) ) ) )).

fof(d4_monoid_0,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) )
         => ( v4_monoid_0(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,A)
               => ! [D] :
                    ( m1_subset_1(D,A)
                   => ? [E] :
                        ( m1_subset_1(E,A)
                        & k2_binop_1(A,A,A,B,C,E) = D ) ) ) ) ) ) )).

fof(d5_monoid_0,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) )
         => ( v5_monoid_0(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,A)
               => ! [D] :
                    ( m1_subset_1(D,A)
                   => ? [E] :
                        ( m1_subset_1(E,A)
                        & ? [F] :
                            ( m1_subset_1(F,A)
                            & k2_binop_1(A,A,A,B,C,E) = D
                            & k2_binop_1(A,A,A,B,F,C) = D ) ) ) ) ) ) ) )).

fof(d6_monoid_0,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) )
         => ( v6_monoid_0(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,A)
               => ! [D] :
                    ( m1_subset_1(D,A)
                   => ! [E] :
                        ( m1_subset_1(E,A)
                       => ( k2_binop_1(A,A,A,B,C,D) = k2_binop_1(A,A,A,B,C,E)
                         => D = E ) ) ) ) ) ) ) )).

fof(d7_monoid_0,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) )
         => ( v7_monoid_0(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,A)
               => ! [D] :
                    ( m1_subset_1(D,A)
                   => ! [E] :
                        ( m1_subset_1(E,A)
                       => ( k2_binop_1(A,A,A,B,D,C) = k2_binop_1(A,A,A,B,E,C)
                         => D = E ) ) ) ) ) ) ) )).

fof(d8_monoid_0,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) )
         => ( v8_monoid_0(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,A)
               => ! [D] :
                    ( m1_subset_1(D,A)
                   => ! [E] :
                        ( m1_subset_1(E,A)
                       => ( ( k2_binop_1(A,A,A,B,C,D) = k2_binop_1(A,A,A,B,C,E)
                            | k2_binop_1(A,A,A,B,D,C) = k2_binop_1(A,A,A,B,E,C) )
                         => D = E ) ) ) ) ) ) ) )).

fof(d9_monoid_0,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)
          <=> ( v1_setwiseo(B,A)
              & ! [C] :
                  ( m1_subset_1(C,A)
                 => ! [D] :
                      ( m1_subset_1(D,A)
                     => ( k2_binop_1(A,A,A,B,C,D) = k3_binop_1(A,B)
                       => ( C = D
                          & D = k3_binop_1(A,B) ) ) ) ) ) ) ) ) )).

fof(t1_monoid_0,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) )
         => ( v5_monoid_0(B,A)
          <=> ( v3_monoid_0(B,A)
              & v4_monoid_0(B,A) ) ) ) ) )).

fof(t2_monoid_0,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) )
         => ( v8_monoid_0(B,A)
          <=> ( v6_monoid_0(B,A)
              & v7_monoid_0(B,A) ) ) ) ) )).

fof(t3_monoid_0,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(k1_tarski(A),k1_tarski(A)),k1_tarski(A))
        & m2_relset_1(B,k2_zfmisc_1(k1_tarski(A),k1_tarski(A)),k1_tarski(A)) )
     => ( B = k2_funcop_1(k1_tarski(k4_tarski(A,A)),A)
        & v1_setwiseo(B,k1_tarski(A))
        & v1_binop_1(B,k1_tarski(A))
        & v2_binop_1(B,k1_tarski(A))
        & v3_binop_1(B,k1_tarski(A))
        & v5_monoid_0(B,k1_tarski(A))
        & v8_monoid_0(B,k1_tarski(A))
        & v9_monoid_0(B,k1_tarski(A)) ) ) )).

fof(d10_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v2_group_1(A)
      <=> v1_setwiseo(u1_group_1(A),u1_struct_0(A)) ) ) )).

fof(d11_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v7_group_1(A)
      <=> v1_binop_1(u1_group_1(A),u1_struct_0(A)) ) ) )).

fof(d12_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v4_group_1(A)
      <=> v2_binop_1(u1_group_1(A),u1_struct_0(A)) ) ) )).

fof(d13_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v10_monoid_0(A)
      <=> v3_binop_1(u1_group_1(A),u1_struct_0(A)) ) ) )).

fof(d14_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v11_monoid_0(A)
      <=> v3_monoid_0(u1_group_1(A),u1_struct_0(A)) ) ) )).

fof(d15_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v12_monoid_0(A)
      <=> v4_monoid_0(u1_group_1(A),u1_struct_0(A)) ) ) )).

fof(d16_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v13_monoid_0(A)
      <=> v5_monoid_0(u1_group_1(A),u1_struct_0(A)) ) ) )).

fof(d17_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v14_monoid_0(A)
      <=> v6_monoid_0(u1_group_1(A),u1_struct_0(A)) ) ) )).

fof(d18_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v15_monoid_0(A)
      <=> v7_monoid_0(u1_group_1(A),u1_struct_0(A)) ) ) )).

fof(d19_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v16_monoid_0(A)
      <=> v8_monoid_0(u1_group_1(A),u1_struct_0(A)) ) ) )).

fof(d20_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v17_monoid_0(A)
      <=> v9_monoid_0(u1_group_1(A),u1_struct_0(A)) ) ) )).

fof(t4_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v2_group_1(A)
       => r3_binop_1(u1_struct_0(A),k3_binop_1(u1_struct_0(A),u1_group_1(A)),u1_group_1(A)) ) ) )).

fof(t5_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v2_group_1(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ( k1_group_1(A,k3_binop_1(u1_struct_0(A),u1_group_1(A)),B) = B
              & k1_group_1(A,B,k3_binop_1(u1_struct_0(A),u1_group_1(A))) = B ) ) ) ) )).

fof(t6_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v2_group_1(A)
      <=> ? [B] :
            ( m1_subset_1(B,u1_struct_0(A))
            & ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ( k1_group_1(A,B,C) = C
                  & k1_group_1(A,C,B) = C ) ) ) ) ) )).

fof(t7_monoid_0,axiom,(
    $true )).

fof(t8_monoid_0,axiom,(
    $true )).

fof(t9_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v10_monoid_0(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => k1_group_1(A,B,B) = B ) ) ) )).

fof(t10_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v11_monoid_0(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ? [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                    & k1_group_1(A,D,B) = C ) ) ) ) ) )).

fof(t11_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v12_monoid_0(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ? [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                    & k1_group_1(A,B,D) = C ) ) ) ) ) )).

fof(t12_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v13_monoid_0(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ? [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                    & ? [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                        & k1_group_1(A,B,D) = C
                        & k1_group_1(A,E,B) = C ) ) ) ) ) ) )).

fof(t13_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v14_monoid_0(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( k1_group_1(A,B,C) = k1_group_1(A,B,D)
                     => C = D ) ) ) ) ) ) )).

fof(t14_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v15_monoid_0(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( k1_group_1(A,C,B) = k1_group_1(A,D,B)
                     => C = D ) ) ) ) ) ) )).

fof(t15_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v16_monoid_0(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( ( k1_group_1(A,B,C) = k1_group_1(A,B,D)
                        | k1_group_1(A,C,B) = k1_group_1(A,D,B) )
                     => C = D ) ) ) ) ) ) )).

fof(t16_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v17_monoid_0(A)
      <=> ( v1_setwiseo(u1_group_1(A),u1_struct_0(A))
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ( k1_group_1(A,B,C) = k3_binop_1(u1_struct_0(A),u1_group_1(A))
                   => ( B = C
                      & C = k3_binop_1(u1_struct_0(A),u1_group_1(A)) ) ) ) ) ) ) ) )).

fof(t17_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v4_group_1(A)
       => ( v13_monoid_0(A)
        <=> ( v2_group_1(A)
            & v1_finseqop(u1_group_1(A),u1_struct_0(A)) ) ) ) ) )).

fof(d21_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ( v1_vectsp_2(A)
      <=> r3_binop_1(u1_struct_0(A),u1_vectsp_1(A),u1_group_1(A)) ) ) )).

fof(t18_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ( v1_vectsp_2(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ( k1_group_1(A,u1_vectsp_1(A),B) = B
              & k1_group_1(A,B,u1_vectsp_1(A)) = B ) ) ) ) )).

fof(t19_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ( v1_vectsp_2(A)
       => u1_vectsp_1(A) = k3_binop_1(u1_struct_0(A),u1_group_1(A)) ) ) )).

fof(d22_monoid_0,axiom,(
    ! [A] :
      ( l1_group_1(A)
     => ! [B] :
          ( l1_vectsp_1(B)
         => ( m1_monoid_0(B,A)
          <=> g1_group_1(u1_struct_0(B),u1_group_1(B)) = g1_group_1(u1_struct_0(A),u1_group_1(A)) ) ) ) )).

fof(t20_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_monoid_0(B,A)
         => ( u1_struct_0(B) = u1_struct_0(A)
            & u1_group_1(B) = u1_group_1(A)
            & ! [C] :
                ( m1_subset_1(C,u1_struct_0(B))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(B))
                   => ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => ! [F] :
                            ( m1_subset_1(F,u1_struct_0(A))
                           => ( ( C = E
                                & D = F )
                             => k1_group_1(B,C,D) = k1_group_1(A,E,F) ) ) ) ) ) ) ) ) )).

fof(t21_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_monoid_0(B,A)
         => ( ( v2_group_1(A)
             => v2_group_1(B) )
            & ( v7_group_1(A)
             => v7_group_1(B) )
            & ( v4_group_1(A)
             => v4_group_1(B) )
            & ( v13_monoid_0(A)
             => v13_monoid_0(B) )
            & ( v17_monoid_0(A)
             => v17_monoid_0(B) )
            & ( v16_monoid_0(A)
             => v16_monoid_0(B) ) ) ) ) )).

fof(t22_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( v1_vectsp_1(B)
            & v1_vectsp_2(B)
            & m1_monoid_0(B,A) )
         => ! [C] :
              ( ( v1_vectsp_1(C)
                & v1_vectsp_2(C)
                & m1_monoid_0(C,A) )
             => B = C ) ) ) )).

fof(d23_monoid_0,axiom,(
    ! [A] :
      ( l1_group_1(A)
     => ! [B] :
          ( l1_group_1(B)
         => ( m2_monoid_0(B,A)
          <=> r1_tarski(u1_group_1(B),u1_group_1(A)) ) ) ) )).

fof(d24_monoid_0,axiom,(
    ! [A] :
      ( l1_group_1(A)
     => ! [B] :
          ( l1_vectsp_1(B)
         => ( m3_monoid_0(B,A)
          <=> ( r1_tarski(u1_group_1(B),u1_group_1(A))
              & ! [C] :
                  ( l1_vectsp_1(C)
                 => ( A = C
                   => u1_vectsp_1(B) = u1_vectsp_1(C) ) ) ) ) ) ) )).

fof(d25_monoid_0,axiom,(
    ! [A] :
      ( l1_vectsp_1(A)
     => ! [B] :
          ( l1_vectsp_1(B)
         => ( m3_monoid_0(B,A)
          <=> ( r1_tarski(u1_group_1(B),u1_group_1(A))
              & u1_vectsp_1(B) = u1_vectsp_1(A) ) ) ) ) )).

fof(t23_monoid_0,axiom,(
    ! [A] :
      ( l1_group_1(A)
     => ! [B] :
          ( m3_monoid_0(B,A)
         => m2_monoid_0(B,A) ) ) )).

fof(t24_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_vectsp_1(B) )
         => ( m2_monoid_0(A,A)
            & m3_monoid_0(B,B) ) ) ) )).

fof(t25_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,A) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & m3_monoid_0(C,A) )
             => ( r1_tarski(u1_struct_0(B),u1_struct_0(A))
                & r1_tarski(u1_struct_0(C),u1_struct_0(A)) ) ) ) ) )).

fof(t26_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,A) )
         => u1_group_1(B) = k1_realset1(u1_group_1(A),u1_struct_0(B)) ) ) )).

fof(t27_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(A))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(A))
                         => ( ( C = E
                              & D = F )
                           => k1_group_1(B,C,D) = k1_group_1(A,E,F) ) ) ) ) ) ) ) )).

fof(t28_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,A) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & m2_monoid_0(C,A) )
             => ( u1_struct_0(B) = u1_struct_0(C)
               => g1_group_1(u1_struct_0(B),u1_group_1(B)) = g1_group_1(u1_struct_0(C),u1_group_1(C)) ) ) ) ) )).

fof(t29_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m3_monoid_0(B,A) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & m3_monoid_0(C,A) )
             => ( u1_struct_0(B) = u1_struct_0(C)
               => g1_vectsp_1(u1_struct_0(B),u1_group_1(B),u1_vectsp_1(B)) = g1_vectsp_1(u1_struct_0(C),u1_group_1(C),u1_vectsp_1(C)) ) ) ) ) )).

fof(t30_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,A) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & m2_monoid_0(C,A) )
             => ( r1_tarski(u1_struct_0(B),u1_struct_0(C))
               => m5_monoid_0(B,A,C) ) ) ) ) )).

fof(t31_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m3_monoid_0(B,A) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & m3_monoid_0(C,A) )
             => ( r1_tarski(u1_struct_0(B),u1_struct_0(C))
               => m7_monoid_0(B,A,C) ) ) ) ) )).

fof(t32_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,A) )
         => ( ( v2_group_1(A)
              & r2_hidden(k3_binop_1(u1_struct_0(A),u1_group_1(A)),u1_struct_0(B)) )
           => ( v2_group_1(B)
              & k3_binop_1(u1_struct_0(A),u1_group_1(A)) = k3_binop_1(u1_struct_0(B),u1_group_1(B)) ) ) ) ) )).

fof(t33_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_vectsp_2(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m3_monoid_0(B,A) )
         => v1_vectsp_2(B) ) ) )).

fof(t34_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,A) )
         => ( v7_group_1(A)
           => v7_group_1(B) ) ) ) )).

fof(t35_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,A) )
         => ( v4_group_1(A)
           => v4_group_1(B) ) ) ) )).

fof(t36_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,A) )
         => ( v10_monoid_0(A)
           => v10_monoid_0(B) ) ) ) )).

fof(t37_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,A) )
         => ( v16_monoid_0(A)
           => v16_monoid_0(B) ) ) ) )).

fof(t38_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,A) )
         => ( ( r2_hidden(k3_binop_1(u1_struct_0(A),u1_group_1(A)),u1_struct_0(B))
              & v17_monoid_0(A) )
           => v17_monoid_0(B) ) ) ) )).

fof(t39_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_vectsp_2(A)
        & v17_monoid_0(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m3_monoid_0(B,A) )
         => v17_monoid_0(B) ) ) )).

fof(d26_monoid_0,axiom,(
    k5_monoid_0 = g1_group_1(k1_numbers,k33_binop_2) )).

fof(t40_monoid_0,axiom,
    ( u1_struct_0(k5_monoid_0) = k1_numbers
    & u1_group_1(k5_monoid_0) = k33_binop_2
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k5_monoid_0))
       => ! [B] :
            ( m1_subset_1(B,u1_struct_0(k5_monoid_0))
           => ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ( ( A = C
                        & B = D )
                     => k10_group_1(k5_monoid_0,A,B) = k9_binop_2(C,D) ) ) ) ) ) )).

fof(t41_monoid_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k5_monoid_0))
    <=> m1_subset_1(A,k1_numbers) ) )).

fof(t42_monoid_0,axiom,(
    k3_binop_1(u1_struct_0(k5_monoid_0),u1_group_1(k5_monoid_0)) = np__0 )).

fof(t43_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & m2_monoid_0(A,k5_monoid_0) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ( ( B = D
                          & C = E )
                       => k10_group_1(A,B,C) = k9_binop_2(D,E) ) ) ) ) ) ) )).

fof(t44_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & m2_monoid_0(A,k5_monoid_0) )
     => k3_binop_1(u1_struct_0(A),u1_group_1(A)) = np__0 ) )).

fof(t45_monoid_0,axiom,(
    $true )).

fof(t46_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & m2_monoid_0(A,k5_monoid_0) )
     => ( A = k6_monoid_0
      <=> u1_struct_0(A) = k4_numbers ) ) )).

fof(t47_monoid_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k6_monoid_0))
    <=> v1_int_1(A) ) )).

fof(d27_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v2_group_1(A)
        & v17_monoid_0(A)
        & m5_monoid_0(A,k5_monoid_0,k6_monoid_0) )
     => ( A = k7_monoid_0
      <=> u1_struct_0(A) = k5_numbers ) ) )).

fof(d28_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_vectsp_1(A)
        & v1_vectsp_2(A)
        & m1_monoid_0(A,k7_monoid_0) )
     => A = k8_monoid_0 ) )).

fof(d29_monoid_0,axiom,(
    k47_binop_2 = u1_group_1(k7_monoid_0) )).

fof(t48_monoid_0,axiom,(
    $true )).

fof(t49_monoid_0,axiom,(
    k7_monoid_0 = g1_group_1(k5_numbers,k47_binop_2) )).

fof(t50_monoid_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k8_monoid_0))
    <=> m2_subset_1(A,k1_numbers,k5_numbers) ) )).

fof(t51_monoid_0,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k8_monoid_0))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k8_monoid_0))
                 => ( ( A = C
                      & B = D )
                   => k10_group_1(k8_monoid_0,C,D) = k23_binop_2(A,B) ) ) ) ) ) )).

fof(t52_monoid_0,axiom,(
    k8_monoid_0 = g1_vectsp_1(k5_numbers,k47_binop_2,np__0) )).

fof(t53_monoid_0,axiom,
    ( k47_binop_2 = k1_realset1(k33_binop_2,k5_numbers)
    & k47_binop_2 = k1_realset1(k44_binop_2,k5_numbers) )).

fof(t54_monoid_0,axiom,
    ( r3_binop_1(k5_numbers,np__0,k47_binop_2)
    & v9_monoid_0(k47_binop_2,k5_numbers) )).

fof(d30_monoid_0,axiom,(
    k9_monoid_0 = g1_group_1(k1_numbers,k35_binop_2) )).

fof(t55_monoid_0,axiom,
    ( u1_struct_0(k9_monoid_0) = k1_numbers
    & u1_group_1(k9_monoid_0) = k35_binop_2
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k9_monoid_0))
       => ! [B] :
            ( m1_subset_1(B,u1_struct_0(k9_monoid_0))
           => ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ( ( A = C
                        & B = D )
                     => k10_group_1(k9_monoid_0,A,B) = k11_binop_2(C,D) ) ) ) ) ) )).

fof(t56_monoid_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k9_monoid_0))
    <=> m1_subset_1(A,k1_numbers) ) )).

fof(t57_monoid_0,axiom,(
    k3_binop_1(u1_struct_0(k9_monoid_0),u1_group_1(k9_monoid_0)) = np__1 )).

fof(t58_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & m2_monoid_0(A,k9_monoid_0) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ( ( B = D
                          & C = E )
                       => k10_group_1(A,B,C) = k11_binop_2(D,E) ) ) ) ) ) ) )).

fof(t59_monoid_0,axiom,(
    $true )).

fof(t60_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & m2_monoid_0(A,k9_monoid_0) )
     => ( k3_binop_1(u1_struct_0(A),u1_group_1(A)) = np__0
        | k3_binop_1(u1_struct_0(A),u1_group_1(A)) = np__1 ) ) )).

fof(d31_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v2_group_1(A)
        & v17_monoid_0(A)
        & m2_monoid_0(A,k9_monoid_0) )
     => ( A = k10_monoid_0
      <=> u1_struct_0(A) = k5_numbers ) ) )).

fof(d32_monoid_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_vectsp_1(A)
        & v1_vectsp_2(A)
        & m1_monoid_0(A,k10_monoid_0) )
     => A = k11_monoid_0 ) )).

fof(d33_monoid_0,axiom,(
    k48_binop_2 = u1_group_1(k10_monoid_0) )).

fof(t61_monoid_0,axiom,(
    k10_monoid_0 = g1_group_1(k5_numbers,k48_binop_2) )).

fof(t62_monoid_0,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k10_monoid_0))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k10_monoid_0))
                 => ( ( A = C
                      & B = D )
                   => k10_group_1(k10_monoid_0,C,D) = k24_binop_2(A,B) ) ) ) ) ) )).

fof(t63_monoid_0,axiom,(
    k3_binop_1(u1_struct_0(k10_monoid_0),u1_group_1(k10_monoid_0)) = np__1 )).

fof(t64_monoid_0,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k11_monoid_0))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k11_monoid_0))
                 => ( ( A = C
                      & B = D )
                   => k10_group_1(k11_monoid_0,C,D) = k24_binop_2(A,B) ) ) ) ) ) )).

fof(t65_monoid_0,axiom,(
    k11_monoid_0 = g1_vectsp_1(k5_numbers,k48_binop_2,np__1) )).

fof(t66_monoid_0,axiom,(
    k48_binop_2 = k1_realset1(k35_binop_2,k5_numbers) )).

fof(t67_monoid_0,axiom,
    ( r3_binop_1(k5_numbers,np__1,k48_binop_2)
    & v9_monoid_0(k48_binop_2,k5_numbers) )).

fof(d34_monoid_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_group_1(B)
            & v2_group_1(B)
            & v4_group_1(B)
            & v2_monoid_0(B)
            & v16_monoid_0(B)
            & v17_monoid_0(B)
            & l1_group_1(B) )
         => ( B = k12_monoid_0(A)
          <=> ( u1_struct_0(B) = k3_finseq_2(A)
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(B))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(B))
                     => k4_monoid_0(B,C,D) = k7_finseq_1(C,D) ) ) ) ) ) ) )).

fof(d35_monoid_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_vectsp_1(B)
            & v1_vectsp_2(B)
            & m1_monoid_0(B,k12_monoid_0(A)) )
         => B = k13_monoid_0(A) ) ) )).

fof(d36_monoid_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k14_monoid_0(A) = u1_group_1(k12_monoid_0(A)) ) )).

fof(t68_monoid_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k12_monoid_0(A) = g1_group_1(k3_finseq_2(A),k14_monoid_0(A)) ) )).

fof(t69_monoid_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k3_binop_1(u1_struct_0(k12_monoid_0(A)),u1_group_1(k12_monoid_0(A))) = k1_xboole_0 ) )).

fof(t70_monoid_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( u1_struct_0(k13_monoid_0(A)) = k3_finseq_2(A)
        & u1_group_1(k13_monoid_0(A)) = k14_monoid_0(A)
        & u1_vectsp_1(k13_monoid_0(A)) = k1_xboole_0 ) ) )).

fof(t71_monoid_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k13_monoid_0(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k13_monoid_0(A)))
             => k4_monoid_0(k13_monoid_0(A),B,C) = k7_finseq_1(B,C) ) ) ) )).

fof(t72_monoid_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,k12_monoid_0(A)) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => k4_monoid_0(B,C,D) = k7_finseq_1(C,D) ) ) ) ) )).

fof(t73_monoid_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_group_1(B)
            & m2_monoid_0(B,k12_monoid_0(A)) )
         => k3_binop_1(u1_struct_0(B),u1_group_1(B)) = k1_xboole_0 ) ) )).

fof(t74_monoid_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m2_monoid_0(B,k12_monoid_0(A)) )
         => ( m1_subset_1(k1_xboole_0,u1_struct_0(B))
           => ( v2_group_1(B)
              & k3_binop_1(u1_struct_0(B),u1_group_1(B)) = k1_xboole_0 ) ) ) ) )).

fof(t75_monoid_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ( r1_tarski(A,B)
           => m2_monoid_0(k12_monoid_0(A),k12_monoid_0(B)) ) ) ) )).

fof(t76_monoid_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v1_setwiseo(k14_monoid_0(A),k3_finseq_2(A))
        & k3_binop_1(k3_finseq_2(A),k14_monoid_0(A)) = k1_xboole_0
        & v2_binop_1(k14_monoid_0(A),k3_finseq_2(A)) ) ) )).

fof(d37_monoid_0,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v1_group_1(B)
        & v2_group_1(B)
        & v4_group_1(B)
        & v1_monoid_0(B)
        & l1_group_1(B) )
     => ( B = k15_monoid_0(A)
      <=> ( u1_struct_0(B) = k4_partfun1(A,A)
          & ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => k4_monoid_0(B,C,D) = k5_relat_1(D,C) ) ) ) ) ) )).

fof(d38_monoid_0,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v1_vectsp_1(B)
        & v1_vectsp_2(B)
        & m1_monoid_0(B,k15_monoid_0(A)) )
     => B = k16_monoid_0(A) ) )).

fof(d39_monoid_0,axiom,(
    ! [A] : k17_monoid_0(A) = u1_group_1(k15_monoid_0(A)) )).

fof(t77_monoid_0,axiom,(
    ! [A,B] :
      ( m1_subset_1(A,u1_struct_0(k15_monoid_0(B)))
    <=> ( v1_funct_1(A)
        & m2_relset_1(A,B,B) ) ) )).

fof(t78_monoid_0,axiom,(
    ! [A] : k3_binop_1(u1_struct_0(k15_monoid_0(A)),u1_group_1(k15_monoid_0(A))) = k6_partfun1(A) )).

fof(t79_monoid_0,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & m2_monoid_0(B,k15_monoid_0(A)) )
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(B))
         => ! [D] :
              ( m1_subset_1(D,u1_struct_0(B))
             => k4_monoid_0(B,C,D) = k5_relat_1(D,C) ) ) ) )).

fof(t80_monoid_0,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & m2_monoid_0(B,k15_monoid_0(A)) )
     => ( m1_subset_1(k6_partfun1(A),u1_struct_0(B))
       => ( v2_group_1(B)
          & k3_binop_1(u1_struct_0(B),u1_group_1(B)) = k6_partfun1(A) ) ) ) )).

fof(t81_monoid_0,axiom,(
    ! [A,B] :
      ( r1_tarski(A,B)
     => m2_monoid_0(k15_monoid_0(A),k15_monoid_0(B)) ) )).

fof(d40_monoid_0,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v1_group_1(B)
        & v2_group_1(B)
        & m2_monoid_0(B,k15_monoid_0(A)) )
     => ( B = k18_monoid_0(A)
      <=> u1_struct_0(B) = k1_funct_2(A,A) ) ) )).

fof(d41_monoid_0,axiom,(
    ! [A,B] :
      ( ( v1_vectsp_1(B)
        & v1_vectsp_2(B)
        & m1_monoid_0(B,k18_monoid_0(A)) )
     => B = k19_monoid_0(A) ) )).

fof(t82_monoid_0,axiom,(
    ! [A,B] :
      ( m1_subset_1(A,u1_struct_0(k18_monoid_0(B)))
    <=> ( v1_funct_1(A)
        & v1_funct_2(A,B,B)
        & m2_relset_1(A,B,B) ) ) )).

fof(t83_monoid_0,axiom,(
    ! [A] : u1_group_1(k18_monoid_0(A)) = k1_realset1(k17_monoid_0(A),k1_funct_2(A,A)) )).

fof(t84_monoid_0,axiom,(
    ! [A] : k3_binop_1(u1_struct_0(k18_monoid_0(A)),u1_group_1(k18_monoid_0(A))) = k6_partfun1(A) )).

fof(t85_monoid_0,axiom,(
    ! [A] :
      ( u1_struct_0(k19_monoid_0(A)) = k1_funct_2(A,A)
      & u1_group_1(k19_monoid_0(A)) = k1_realset1(k17_monoid_0(A),k1_funct_2(A,A))
      & u1_vectsp_1(k19_monoid_0(A)) = k6_partfun1(A) ) )).

fof(d42_monoid_0,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v1_group_1(B)
        & v2_group_1(B)
        & v13_monoid_0(B)
        & m5_monoid_0(B,k15_monoid_0(A),k18_monoid_0(A)) )
     => ( B = k20_monoid_0(A)
      <=> ! [C] :
            ( m1_subset_1(C,u1_struct_0(k18_monoid_0(A)))
           => ( r2_hidden(C,u1_struct_0(B))
            <=> ( v1_funct_1(C)
                & v1_funct_2(C,A,A)
                & v3_funct_2(C,A,A)
                & m2_relset_1(C,A,A) ) ) ) ) ) )).

fof(t86_monoid_0,axiom,(
    ! [A,B] :
      ( m1_subset_1(A,u1_struct_0(k20_monoid_0(B)))
    <=> ( v1_funct_1(A)
        & v1_funct_2(A,B,B)
        & v3_funct_2(A,B,B)
        & m2_relset_1(A,B,B) ) ) )).

fof(t87_monoid_0,axiom,(
    ! [A] :
      ( k3_binop_1(u1_struct_0(k20_monoid_0(A)),u1_group_1(k20_monoid_0(A))) = k6_partfun1(A)
      & k2_group_1(k20_monoid_0(A)) = k6_partfun1(A) ) )).

fof(t88_monoid_0,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,u1_struct_0(k20_monoid_0(A)))
     => k3_group_1(k20_monoid_0(A),B) = k2_funct_1(B) ) )).

fof(t89_monoid_0,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ( v1_fraenkel(u1_struct_0(A))
       => v1_monoid_0(A) ) ) )).

fof(s1_monoid_0,axiom,
    ( ! [A] :
        ( m1_struct_0(A,f1_s1_monoid_0,f2_s1_monoid_0)
       => ! [B] :
            ( m1_struct_0(B,f1_s1_monoid_0,f2_s1_monoid_0)
           => r2_hidden(k1_group_1(f1_s1_monoid_0,A,B),f2_s1_monoid_0) ) )
   => ? [A] :
        ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & m2_monoid_0(A,f1_s1_monoid_0)
        & u1_struct_0(A) = f2_s1_monoid_0 ) )).

fof(s2_monoid_0,axiom,
    ( ( ! [A] :
          ( m1_subset_1(A,u1_struct_0(f1_s2_monoid_0))
         => ! [B] :
              ( m1_subset_1(B,u1_struct_0(f1_s2_monoid_0))
             => ( ( p1_s2_monoid_0(A)
                  & p1_s2_monoid_0(B) )
               => p1_s2_monoid_0(k1_group_1(f1_s2_monoid_0,A,B)) ) ) )
      & ? [A] :
          ( m1_subset_1(A,u1_struct_0(f1_s2_monoid_0))
          & p1_s2_monoid_0(A) ) )
   => ? [A] :
        ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & m2_monoid_0(A,f1_s2_monoid_0)
        & ! [B] :
            ( m1_subset_1(B,u1_struct_0(f1_s2_monoid_0))
           => ( r2_hidden(B,u1_struct_0(A))
            <=> p1_s2_monoid_0(B) ) ) ) )).

fof(s3_monoid_0,axiom,
    ( ( ! [A] :
          ( m1_struct_0(A,f1_s3_monoid_0,f2_s3_monoid_0)
         => ! [B] :
              ( m1_struct_0(B,f1_s3_monoid_0,f2_s3_monoid_0)
             => r2_hidden(k1_group_1(f1_s3_monoid_0,A,B),f2_s3_monoid_0) ) )
      & r2_hidden(u1_vectsp_1(f1_s3_monoid_0),f2_s3_monoid_0) )
   => ? [A] :
        ( ~ v3_struct_0(A)
        & v1_vectsp_1(A)
        & m3_monoid_0(A,f1_s3_monoid_0)
        & u1_struct_0(A) = f2_s3_monoid_0 ) )).

fof(s4_monoid_0,axiom,
    ( ( ! [A] :
          ( m1_subset_1(A,u1_struct_0(f1_s4_monoid_0))
         => ! [B] :
              ( m1_subset_1(B,u1_struct_0(f1_s4_monoid_0))
             => ( ( p1_s4_monoid_0(A)
                  & p1_s4_monoid_0(B) )
               => p1_s4_monoid_0(k1_group_1(f1_s4_monoid_0,A,B)) ) ) )
      & p1_s4_monoid_0(u1_vectsp_1(f1_s4_monoid_0)) )
   => ? [A] :
        ( ~ v3_struct_0(A)
        & v1_vectsp_1(A)
        & m3_monoid_0(A,f1_s4_monoid_0)
        & ! [B] :
            ( m1_subset_1(B,u1_struct_0(f1_s4_monoid_0))
           => ( r2_hidden(B,u1_struct_0(A))
            <=> p1_s4_monoid_0(B) ) ) ) )).

fof(dt_m1_monoid_0,axiom,(
    ! [A] :
      ( l1_group_1(A)
     => ! [B] :
          ( m1_monoid_0(B,A)
         => l1_vectsp_1(B) ) ) )).

fof(existence_m1_monoid_0,axiom,(
    ! [A] :
      ( l1_group_1(A)
     => ? [B] : m1_monoid_0(B,A) ) )).

fof(dt_m2_monoid_0,axiom,(
    ! [A] :
      ( l1_group_1(A)
     => ! [B] :
          ( m2_monoid_0(B,A)
         => l1_group_1(B) ) ) )).

fof(existence_m2_monoid_0,axiom,(
    ! [A] :
      ( l1_group_1(A)
     => ? [B] : m2_monoid_0(B,A) ) )).

fof(dt_m3_monoid_0,axiom,(
    ! [A] :
      ( l1_group_1(A)
     => ! [B] :
          ( m3_monoid_0(B,A)
         => l1_vectsp_1(B) ) ) )).

fof(existence_m3_monoid_0,axiom,(
    ! [A] :
      ( l1_group_1(A)
     => ? [B] : m3_monoid_0(B,A) ) )).

fof(dt_m4_monoid_0,axiom,(
    ! [A,B] :
      ( ( l1_group_1(A)
        & m1_monoid_0(B,A) )
     => ! [C] :
          ( m4_monoid_0(C,A,B)
         => m2_monoid_0(C,A) ) ) )).

fof(existence_m4_monoid_0,axiom,(
    ! [A,B] :
      ( ( l1_group_1(A)
        & m1_monoid_0(B,A) )
     => ? [C] : m4_monoid_0(C,A,B) ) )).

fof(redefinition_m4_monoid_0,axiom,(
    ! [A,B] :
      ( ( l1_group_1(A)
        & m1_monoid_0(B,A) )
     => ! [C] :
          ( m4_monoid_0(C,A,B)
        <=> m2_monoid_0(C,B) ) ) )).

fof(dt_m5_monoid_0,axiom,(
    ! [A,B] :
      ( ( l1_group_1(A)
        & m2_monoid_0(B,A) )
     => ! [C] :
          ( m5_monoid_0(C,A,B)
         => m2_monoid_0(C,A) ) ) )).

fof(existence_m5_monoid_0,axiom,(
    ! [A,B] :
      ( ( l1_group_1(A)
        & m2_monoid_0(B,A) )
     => ? [C] : m5_monoid_0(C,A,B) ) )).

fof(redefinition_m5_monoid_0,axiom,(
    ! [A,B] :
      ( ( l1_group_1(A)
        & m2_monoid_0(B,A) )
     => ! [C] :
          ( m5_monoid_0(C,A,B)
        <=> m2_monoid_0(C,B) ) ) )).

fof(dt_m6_monoid_0,axiom,(
    ! [A,B] :
      ( ( l1_group_1(A)
        & m3_monoid_0(B,A) )
     => ! [C] :
          ( m6_monoid_0(C,A,B)
         => m2_monoid_0(C,A) ) ) )).

fof(existence_m6_monoid_0,axiom,(
    ! [A,B] :
      ( ( l1_group_1(A)
        & m3_monoid_0(B,A) )
     => ? [C] : m6_monoid_0(C,A,B) ) )).

fof(redefinition_m6_monoid_0,axiom,(
    ! [A,B] :
      ( ( l1_group_1(A)
        & m3_monoid_0(B,A) )
     => ! [C] :
          ( m6_monoid_0(C,A,B)
        <=> m2_monoid_0(C,B) ) ) )).

fof(dt_m7_monoid_0,axiom,(
    ! [A,B] :
      ( ( l1_group_1(A)
        & m3_monoid_0(B,A) )
     => ! [C] :
          ( m7_monoid_0(C,A,B)
         => m3_monoid_0(C,A) ) ) )).

fof(existence_m7_monoid_0,axiom,(
    ! [A,B] :
      ( ( l1_group_1(A)
        & m3_monoid_0(B,A) )
     => ? [C] : m7_monoid_0(C,A,B) ) )).

fof(redefinition_m7_monoid_0,axiom,(
    ! [A,B] :
      ( ( l1_group_1(A)
        & m3_monoid_0(B,A) )
     => ! [C] :
          ( m7_monoid_0(C,A,B)
        <=> m3_monoid_0(C,B) ) ) )).

fof(dt_k1_monoid_0,axiom,(
    ! [A,B,C] :
      ( ( m1_finseq_1(B,A)
        & m1_finseq_1(C,A) )
     => m2_finseq_2(k1_monoid_0(A,B,C),A,k3_finseq_2(A)) ) )).

fof(redefinition_k1_monoid_0,axiom,(
    ! [A,B,C] :
      ( ( m1_finseq_1(B,A)
        & m1_finseq_1(C,A) )
     => k1_monoid_0(A,B,C) = k7_finseq_1(B,C) ) )).

fof(dt_k2_monoid_0,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,A)
        & m1_relset_1(B,A,A)
        & v1_funct_1(C)
        & v1_funct_2(C,A,A)
        & m1_relset_1(C,A,A) )
     => ( v1_funct_1(k2_monoid_0(A,B,C))
        & v1_funct_2(k2_monoid_0(A,B,C),A,A)
        & m2_relset_1(k2_monoid_0(A,B,C),A,A) ) ) )).

fof(redefinition_k2_monoid_0,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,A)
        & m1_relset_1(B,A,A)
        & v1_funct_1(C)
        & v1_funct_2(C,A,A)
        & m1_relset_1(C,A,A) )
     => k2_monoid_0(A,B,C) = k5_relat_1(B,C) ) )).

fof(dt_k3_monoid_0,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,A)
        & v3_funct_2(B,A,A)
        & m1_relset_1(B,A,A)
        & v1_funct_1(C)
        & v1_funct_2(C,A,A)
        & v3_funct_2(C,A,A)
        & m1_relset_1(C,A,A) )
     => ( v1_funct_1(k3_monoid_0(A,B,C))
        & v1_funct_2(k3_monoid_0(A,B,C),A,A)
        & v3_funct_2(k3_monoid_0(A,B,C),A,A)
        & m2_relset_1(k3_monoid_0(A,B,C),A,A) ) ) )).

fof(redefinition_k3_monoid_0,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,A)
        & v3_funct_2(B,A,A)
        & m1_relset_1(B,A,A)
        & v1_funct_1(C)
        & v1_funct_2(C,A,A)
        & v3_funct_2(C,A,A)
        & m1_relset_1(C,A,A) )
     => k3_monoid_0(A,B,C) = k5_relat_1(B,C) ) )).

fof(dt_k4_monoid_0,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k4_monoid_0(A,B,C),u1_struct_0(A)) ) )).

fof(redefinition_k4_monoid_0,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => k4_monoid_0(A,B,C) = k1_group_1(A,B,C) ) )).

fof(dt_k5_monoid_0,axiom,
    ( ~ v3_struct_0(k5_monoid_0)
    & v1_group_1(k5_monoid_0)
    & v2_group_1(k5_monoid_0)
    & v4_group_1(k5_monoid_0)
    & v7_group_1(k5_monoid_0)
    & v13_monoid_0(k5_monoid_0)
    & v16_monoid_0(k5_monoid_0)
    & l1_group_1(k5_monoid_0) )).

fof(dt_k6_monoid_0,axiom,
    ( ~ v3_struct_0(k6_monoid_0)
    & v1_group_1(k6_monoid_0)
    & v2_group_1(k6_monoid_0)
    & v13_monoid_0(k6_monoid_0)
    & m2_monoid_0(k6_monoid_0,k5_monoid_0) )).

fof(redefinition_k6_monoid_0,axiom,(
    k6_monoid_0 = k3_gr_cy_1 )).

fof(dt_k7_monoid_0,axiom,
    ( ~ v3_struct_0(k7_monoid_0)
    & v1_group_1(k7_monoid_0)
    & v2_group_1(k7_monoid_0)
    & v17_monoid_0(k7_monoid_0)
    & m5_monoid_0(k7_monoid_0,k5_monoid_0,k6_monoid_0) )).

fof(dt_k8_monoid_0,axiom,
    ( ~ v3_struct_0(k8_monoid_0)
    & v1_vectsp_1(k8_monoid_0)
    & v1_vectsp_2(k8_monoid_0)
    & m1_monoid_0(k8_monoid_0,k7_monoid_0) )).

fof(dt_k9_monoid_0,axiom,
    ( ~ v3_struct_0(k9_monoid_0)
    & v1_group_1(k9_monoid_0)
    & v2_group_1(k9_monoid_0)
    & v4_group_1(k9_monoid_0)
    & v7_group_1(k9_monoid_0)
    & l1_group_1(k9_monoid_0) )).

fof(dt_k10_monoid_0,axiom,
    ( ~ v3_struct_0(k10_monoid_0)
    & v1_group_1(k10_monoid_0)
    & v2_group_1(k10_monoid_0)
    & v17_monoid_0(k10_monoid_0)
    & m2_monoid_0(k10_monoid_0,k9_monoid_0) )).

fof(dt_k11_monoid_0,axiom,
    ( ~ v3_struct_0(k11_monoid_0)
    & v1_vectsp_1(k11_monoid_0)
    & v1_vectsp_2(k11_monoid_0)
    & m1_monoid_0(k11_monoid_0,k10_monoid_0) )).

fof(dt_k12_monoid_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v3_struct_0(k12_monoid_0(A))
        & v1_group_1(k12_monoid_0(A))
        & v2_group_1(k12_monoid_0(A))
        & v4_group_1(k12_monoid_0(A))
        & v2_monoid_0(k12_monoid_0(A))
        & v16_monoid_0(k12_monoid_0(A))
        & v17_monoid_0(k12_monoid_0(A))
        & l1_group_1(k12_monoid_0(A)) ) ) )).

fof(dt_k13_monoid_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v3_struct_0(k13_monoid_0(A))
        & v1_vectsp_1(k13_monoid_0(A))
        & v1_vectsp_2(k13_monoid_0(A))
        & m1_monoid_0(k13_monoid_0(A),k12_monoid_0(A)) ) ) )).

fof(dt_k14_monoid_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v1_funct_1(k14_monoid_0(A))
        & v1_funct_2(k14_monoid_0(A),k2_zfmisc_1(k3_finseq_2(A),k3_finseq_2(A)),k3_finseq_2(A))
        & m2_relset_1(k14_monoid_0(A),k2_zfmisc_1(k3_finseq_2(A),k3_finseq_2(A)),k3_finseq_2(A)) ) ) )).

fof(dt_k15_monoid_0,axiom,(
    ! [A] :
      ( ~ v3_struct_0(k15_monoid_0(A))
      & v1_group_1(k15_monoid_0(A))
      & v2_group_1(k15_monoid_0(A))
      & v4_group_1(k15_monoid_0(A))
      & v1_monoid_0(k15_monoid_0(A))
      & l1_group_1(k15_monoid_0(A)) ) )).

fof(dt_k16_monoid_0,axiom,(
    ! [A] :
      ( ~ v3_struct_0(k16_monoid_0(A))
      & v1_vectsp_1(k16_monoid_0(A))
      & v1_vectsp_2(k16_monoid_0(A))
      & m1_monoid_0(k16_monoid_0(A),k15_monoid_0(A)) ) )).

fof(dt_k17_monoid_0,axiom,(
    ! [A] :
      ( v1_funct_1(k17_monoid_0(A))
      & v1_funct_2(k17_monoid_0(A),k2_zfmisc_1(k4_partfun1(A,A),k4_partfun1(A,A)),k4_partfun1(A,A))
      & m2_relset_1(k17_monoid_0(A),k2_zfmisc_1(k4_partfun1(A,A),k4_partfun1(A,A)),k4_partfun1(A,A)) ) )).

fof(dt_k18_monoid_0,axiom,(
    ! [A] :
      ( ~ v3_struct_0(k18_monoid_0(A))
      & v1_group_1(k18_monoid_0(A))
      & v2_group_1(k18_monoid_0(A))
      & m2_monoid_0(k18_monoid_0(A),k15_monoid_0(A)) ) )).

fof(dt_k19_monoid_0,axiom,(
    ! [A] :
      ( v1_vectsp_1(k19_monoid_0(A))
      & v1_vectsp_2(k19_monoid_0(A))
      & m1_monoid_0(k19_monoid_0(A),k18_monoid_0(A)) ) )).

fof(dt_k20_monoid_0,axiom,(
    ! [A] :
      ( ~ v3_struct_0(k20_monoid_0(A))
      & v1_group_1(k20_monoid_0(A))
      & v2_group_1(k20_monoid_0(A))
      & v13_monoid_0(k20_monoid_0(A))
      & m5_monoid_0(k20_monoid_0(A),k15_monoid_0(A),k18_monoid_0(A)) ) )).
%------------------------------------------------------------------------------