SET007 Axioms: SET007+512.ax


%------------------------------------------------------------------------------
% File     : SET007+512 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Equations in Many Sorted Algebras
% 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    : equation [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   60 (   3 unit)
%            Number of atoms       :  500 (  22 equality)
%            Maximal formula depth :   25 (  10 average)
%            Number of connectives :  535 (  95 ~  ;   1  |; 215  &)
%                                         (   6 <=>; 218 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   43 (   1 propositional; 0-4 arity)
%            Number of functors    :   41 (   2 constant; 0-6 arity)
%            Number of variables   :  240 (   0 singleton; 231 !;   9 ?)
%            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_equation,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & l3_msualg_1(B,A)
        & m1_msualg_2(C,A,B) )
     => ~ v1_xboole_0(k2_equation(A,B,C)) ) )).

fof(rc1_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ? [B] :
          ( l3_msualg_1(B,A)
          & v4_msualg_1(B,A)
          & v5_msualg_1(B,A)
          & v2_msafree(B,A)
          & v1_msualg_6(B,A) ) ) )).

fof(fc2_equation,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v5_msualg_1(B,A)
        & l3_msualg_1(B,A)
        & v2_relat_1(C)
        & v1_pre_circ(C,u1_struct_0(A))
        & m4_pboole(C,u1_struct_0(A),u4_msualg_1(A,B)) )
     => ( v4_msualg_1(k12_msualg_2(A,B,C),A)
        & v5_msualg_1(k12_msualg_2(A,B,C),A)
        & v3_msafree2(k12_msualg_2(A,B,C),A)
        & v1_msualg_6(k12_msualg_2(A,B,C),A) ) ) )).

fof(rc2_equation,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v5_msualg_1(B,A)
        & l3_msualg_1(B,A) )
     => ? [C] :
          ( m1_msualg_2(C,A,B)
          & v4_msualg_1(C,A)
          & v5_msualg_1(C,A)
          & v3_msafree2(C,A)
          & v1_msualg_6(C,A) ) ) )).

fof(rc3_equation,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v1_msualg_6(B,A)
        & l3_msualg_1(B,A) )
     => ? [C] :
          ( m1_msualg_2(C,A,B)
          & v1_msualg_6(C,A) ) ) )).

fof(fc3_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ( v4_msualg_1(k3_equation(A),A)
        & v5_msualg_1(k3_equation(A),A)
        & v2_msafree(k3_equation(A),A)
        & v1_msualg_6(k3_equation(A),A) ) ) )).

fof(fc4_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ( v1_relat_1(k4_equation(A))
        & v2_relat_1(k4_equation(A))
        & ~ v3_relat_1(k4_equation(A))
        & v1_funct_1(k4_equation(A)) ) ) )).

fof(t1_equation,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [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,B,C)
                    & m2_relset_1(E,B,C) )
                 => ( k2_relat_1(k7_funct_2(A,B,C,D,E)) = C
                   => k2_relat_1(E) = C ) ) ) ) ) )).

fof(t2_equation,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( ( v2_relat_1(C)
            & m1_pboole(C,A) )
         => ! [D] :
              ( ( v2_relat_1(D)
                & m1_pboole(D,A) )
             => ! [E] :
                  ( m3_pboole(E,A,B,C)
                 => ! [F] :
                      ( m3_pboole(F,A,C,D)
                     => ( v2_msualg_3(k3_msualg_3(A,B,C,D,E,F),A,B,D)
                       => v2_msualg_3(F,A,C,D) ) ) ) ) ) ) )).

fof(t3_equation,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C,D,E] :
              ( ( v1_relat_1(E)
                & v1_funct_1(E) )
             => ( ( r2_hidden(E,k1_funct_2(A,k1_funct_2(B,C)))
                  & r2_hidden(D,B) )
               => ( k1_relat_1(k1_funct_1(k10_funct_6(E),D)) = A
                  & r1_tarski(k2_relat_1(k1_funct_1(k10_funct_6(E),D)),C) ) ) ) ) ) )).

fof(t4_equation,axiom,(
    $true )).

fof(t5_equation,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ( r1_pzfmisc1(A,B,C)
           => ! [D] :
                ( ( v1_funcop_1(D)
                  & m1_pboole(D,A) )
               => ( ( r6_pboole(A,k1_extens_1(A,D),B)
                    & r2_pboole(A,k2_extens_1(A,D),C) )
                 => m3_pboole(D,A,B,C) ) ) ) ) ) )).

fof(t6_equation,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ! [D] :
              ( m3_pboole(D,A,B,C)
             => ! [E] :
                  ( m4_pboole(E,A,B)
                 => ! [F] :
                      ( m4_pboole(F,A,B)
                     => ! [G] :
                          ( m4_pboole(G,A,E)
                         => ( r6_pboole(A,F,G)
                           => r6_pboole(A,k1_msafree(A,E,C,G,k1_msafree(A,B,C,E,D)),k1_msafree(A,B,C,F,D)) ) ) ) ) ) ) ) )).

fof(t7_equation,axiom,(
    ! [A,B] :
      ( ( v2_relat_1(B)
        & m1_pboole(B,A) )
     => ! [C] :
          ( m1_pboole(C,A)
         => ! [D] :
              ( m4_pboole(D,A,C)
             => ! [E] :
                  ( m3_pboole(E,A,D,B)
                 => ? [F] :
                      ( m3_pboole(F,A,C,B)
                      & r6_pboole(A,k1_msafree(A,C,B,D,F),E) ) ) ) ) ) )).

fof(d1_equation,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( ( v1_funcop_1(C)
            & m1_pboole(C,A) )
         => ! [D] :
              ( m1_pboole(D,A)
             => ( D = k1_equation(A,B,C)
              <=> ! [E] :
                    ( r2_hidden(E,A)
                   => k1_funct_1(D,E) = k10_relat_1(k1_funct_1(C,E),k1_funct_1(B,E)) ) ) ) ) ) )).

fof(t8_equation,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ! [D] :
              ( m1_pboole(D,A)
             => ! [E] :
                  ( m3_pboole(E,A,B,C)
                 => m4_pboole(k14_pboole(A,D,E),A,C) ) ) ) ) )).

fof(t9_equation,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ! [D] :
              ( m1_pboole(D,A)
             => ! [E] :
                  ( m3_pboole(E,A,B,C)
                 => m4_pboole(k1_equation(A,D,E),A,B) ) ) ) ) )).

fof(t10_equation,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ! [D] :
              ( m3_pboole(D,A,B,C)
             => ( v2_msualg_3(D,A,B,C)
               => r6_pboole(A,k14_pboole(A,B,D),C) ) ) ) ) )).

fof(t11_equation,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ! [D] :
              ( m3_pboole(D,A,B,C)
             => ( r1_pzfmisc1(A,B,C)
               => r6_pboole(A,k1_equation(A,C,D),B) ) ) ) ) )).

fof(t12_equation,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( ( v1_funcop_1(C)
            & m1_pboole(C,A) )
         => ( r2_pboole(A,B,k2_extens_1(A,C))
           => r6_pboole(A,k14_pboole(A,k1_equation(A,B,C),C),B) ) ) ) )).

fof(t13_equation,axiom,(
    ! [A,B] :
      ( ( v1_funcop_1(B)
        & m1_pboole(B,A) )
     => ! [C] :
          ( m1_pboole(C,A)
         => r2_pboole(A,k14_pboole(A,C,B),k2_extens_1(A,B)) ) ) )).

fof(t14_equation,axiom,(
    ! [A,B] :
      ( ( v1_funcop_1(B)
        & m1_pboole(B,A) )
     => r6_pboole(A,k14_pboole(A,k1_extens_1(A,B),B),k2_extens_1(A,B)) ) )).

fof(t15_equation,axiom,(
    ! [A,B] :
      ( ( v1_funcop_1(B)
        & m1_pboole(B,A) )
     => r6_pboole(A,k1_equation(A,k2_extens_1(A,B),B),k1_extens_1(A,B)) ) )).

fof(t16_equation,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => r6_pboole(A,k14_pboole(A,B,k2_msualg_3(A,B)),B) ) )).

fof(t17_equation,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => r6_pboole(A,k1_equation(A,B,k2_msualg_3(A,B)),B) ) )).

fof(t18_equation,axiom,(
    $true )).

fof(t19_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( l3_msualg_1(B,A)
         => m1_msualg_2(B,A,g3_msualg_1(A,u4_msualg_1(A,B),u5_msualg_1(A,B))) ) ) )).

fof(t20_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( l3_msualg_1(B,A)
         => ! [C] :
              ( m1_msualg_2(C,A,B)
             => ! [D] :
                  ( m1_subset_1(D,u1_msualg_1(A))
                 => ! [E] :
                      ( r2_hidden(E,k3_msualg_1(A,D,C))
                     => r2_hidden(E,k3_msualg_1(A,D,B)) ) ) ) ) ) )).

fof(t21_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( l3_msualg_1(B,A)
         => ! [C] :
              ( m1_msualg_2(C,A,B)
             => ! [D] :
                  ( m1_subset_1(D,u1_msualg_1(A))
                 => ! [E] :
                      ( r2_hidden(E,k3_msualg_1(A,D,C))
                     => k1_funct_1(k5_msualg_1(A,D,C),E) = k1_funct_1(k5_msualg_1(A,D,B),E) ) ) ) ) ) )).

fof(t22_equation,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & ~ v2_msualg_1(B)
        & l1_msualg_1(B) )
     => ! [C] :
          ( m2_pralg_2(C,A,B)
         => ! [D] :
              ( m1_msualg_2(D,B,k15_pralg_2(A,B,C))
             => ! [E] :
                  ( m1_subset_1(E,u1_msualg_1(B))
                 => ! [F] :
                      ( r2_hidden(F,k3_msualg_1(B,E,D))
                     => ( v1_relat_1(k1_funct_1(k5_msualg_1(B,E,D),F))
                        & v1_funct_1(k1_funct_1(k5_msualg_1(B,E,D),F))
                        & v1_relat_1(k1_funct_1(k5_msualg_1(B,E,k15_pralg_2(A,B,C)),F))
                        & v1_funct_1(k1_funct_1(k5_msualg_1(B,E,k15_pralg_2(A,B,C)),F)) ) ) ) ) ) ) )).

fof(d2_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( l3_msualg_1(B,A)
         => ! [C] :
              ( m1_msualg_2(C,A,B)
             => ! [D] :
                  ( D = k2_equation(A,B,C)
                <=> ! [E] :
                      ( r2_hidden(E,D)
                    <=> ? [F] :
                          ( v4_msualg_1(F,A)
                          & m1_msualg_2(F,A,B)
                          & E = F
                          & m1_msualg_2(C,A,F) ) ) ) ) ) ) )).

fof(t23_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( l3_msualg_1(C,A)
             => ! [D] :
                  ( m1_msualg_2(D,A,C)
                 => ! [E] :
                      ( m4_pboole(E,u1_struct_0(A),u4_msualg_1(A,C))
                     => ( r6_pboole(u1_struct_0(A),E,u4_msualg_1(A,D))
                       => ! [F] :
                            ( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,B))
                           => ! [G] :
                                ( m3_pboole(G,u1_struct_0(A),u4_msualg_1(A,D),u4_msualg_1(A,B))
                               => ( r6_pboole(u1_struct_0(A),G,k1_msafree(u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,B),E,F))
                                 => ! [H] :
                                      ( m1_subset_1(H,u1_msualg_1(A))
                                     => ! [I] :
                                          ( m1_subset_1(I,k3_msualg_1(A,H,C))
                                         => ! [J] :
                                              ( m1_subset_1(J,k3_msualg_1(A,H,D))
                                             => ( I = J
                                               => ( k3_msualg_1(A,H,D) = k1_xboole_0
                                                  | k5_msualg_3(A,C,B,H,F,I) = k5_msualg_3(A,D,B,H,G,J) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t24_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( ( v1_msualg_6(C,A)
                & l3_msualg_1(C,A) )
             => ! [D] :
                  ( ( v1_msualg_6(D,A)
                    & m1_msualg_2(D,A,C) )
                 => ! [E] :
                      ( m4_pboole(E,u1_struct_0(A),u4_msualg_1(A,C))
                     => ( r6_pboole(u1_struct_0(A),E,u4_msualg_1(A,D))
                       => ! [F] :
                            ( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,B))
                           => ( r1_msualg_3(A,C,B,F)
                             => ! [G] :
                                  ( m3_pboole(G,u1_struct_0(A),u4_msualg_1(A,D),u4_msualg_1(A,B))
                                 => ( r6_pboole(u1_struct_0(A),G,k1_msafree(u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,B),E,F))
                                   => r1_msualg_3(A,D,B,G) ) ) ) ) ) ) ) ) ) ) )).

fof(t25_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( ( v4_msualg_1(C,A)
                & v5_msualg_1(C,A)
                & l3_msualg_1(C,A) )
             => ! [D] :
                  ( m1_msafree(D,A,B)
                 => ! [E] :
                      ( ( v2_relat_1(E)
                        & m1_msafree(E,A,C) )
                     => ! [F] :
                          ( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C))
                         => ( ( r2_pboole(u1_struct_0(A),E,k14_pboole(u1_struct_0(A),D,F))
                              & r1_msualg_3(A,B,C,F) )
                           => r2_msualg_3(A,B,C,F) ) ) ) ) ) ) ) )).

fof(t26_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( ( v5_msualg_1(C,A)
                & l3_msualg_1(C,A) )
             => ! [D] :
                  ( ( v4_msualg_1(D,A)
                    & v5_msualg_1(D,A)
                    & v2_msafree(D,A)
                    & l3_msualg_1(D,A) )
                 => ! [E] :
                      ( m3_pboole(E,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C))
                     => ( r2_msualg_3(A,B,C,E)
                       => ! [F] :
                            ( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,D),u4_msualg_1(A,C))
                           => ~ ( r1_msualg_3(A,D,C,F)
                                & ! [G] :
                                    ( m3_pboole(G,u1_struct_0(A),u4_msualg_1(A,D),u4_msualg_1(A,B))
                                   => ~ ( r1_msualg_3(A,D,B,G)
                                        & r6_pboole(u1_struct_0(A),F,k3_msualg_3(u1_struct_0(A),u4_msualg_1(A,D),u4_msualg_1(A,B),u4_msualg_1(A,C),G,E)) ) ) ) ) ) ) ) ) ) ) )).

fof(t27_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_finset_1(B) )
         => ! [C] :
              ( ( v5_msualg_1(C,A)
                & l3_msualg_1(C,A) )
             => ! [D] :
                  ( m2_pralg_2(D,B,A)
                 => ~ ( ! [E] :
                          ( m1_subset_1(E,B)
                         => ? [F] :
                              ( v4_msualg_1(F,A)
                              & v5_msualg_1(F,A)
                              & v3_msafree2(F,A)
                              & m1_msualg_2(F,A,C)
                              & F = k6_pralg_2(B,A,D,E) ) )
                      & ! [E] :
                          ( ( v4_msualg_1(E,A)
                            & v5_msualg_1(E,A)
                            & v3_msafree2(E,A)
                            & m1_msualg_2(E,A,C) )
                         => ~ ! [F] :
                                ( m1_subset_1(F,B)
                               => m1_msualg_2(k6_pralg_2(B,A,D,F),A,E) ) ) ) ) ) ) ) )).

fof(t28_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( ( v4_msualg_1(C,A)
                & v5_msualg_1(C,A)
                & v3_msafree2(C,A)
                & m1_msualg_2(C,A,B) )
             => ! [D] :
                  ( ( v4_msualg_1(D,A)
                    & v5_msualg_1(D,A)
                    & v3_msafree2(D,A)
                    & m1_msualg_2(D,A,B) )
                 => ? [E] :
                      ( v4_msualg_1(E,A)
                      & v5_msualg_1(E,A)
                      & v3_msafree2(E,A)
                      & m1_msualg_2(E,A,B)
                      & m1_msualg_2(C,A,E)
                      & m1_msualg_2(D,A,E) ) ) ) ) ) )).

fof(t30_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v2_msafree(B,A)
            & v1_msualg_6(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( ( v1_msafree(C,A,B)
                & m1_msafree(C,A,B) )
             => ! [D] :
                  ( m4_pboole(D,u1_struct_0(A),u4_msualg_1(A,B))
                 => ( ( r2_pboole(u1_struct_0(A),D,C)
                      & v1_msualg_6(k12_msualg_2(A,B,D),A) )
                   => v2_msafree(k12_msualg_2(A,B,D),A) ) ) ) ) ) )).

fof(d3_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => k3_equation(A) = k11_msafree(A,k2_pre_circ(u1_struct_0(A),k5_numbers)) ) )).

fof(d4_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => k4_equation(A) = k11_pboole(u1_struct_0(A),u4_msualg_1(A,k3_equation(A)),u4_msualg_1(A,k3_equation(A))) ) )).

fof(t31_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,k1_funct_1(k4_equation(A),B))
             => r2_hidden(k1_mcart_1(C),k1_funct_1(u4_msualg_1(A,k3_equation(A)),B)) ) ) ) )).

fof(t32_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,k1_funct_1(k4_equation(A),B))
             => r2_hidden(k2_mcart_1(C),k1_funct_1(u4_msualg_1(A,k3_equation(A)),B)) ) ) ) )).

fof(d5_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( l3_msualg_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_funct_1(k4_equation(A),C))
                 => ( r1_equation(A,B,C,D)
                  <=> ! [E] :
                        ( m3_pboole(E,u1_struct_0(A),u4_msualg_1(A,k3_equation(A)),u4_msualg_1(A,B))
                       => ( r1_msualg_3(A,k3_equation(A),B,E)
                         => k1_funct_1(k1_msualg_3(u1_struct_0(A),u4_msualg_1(A,k3_equation(A)),u4_msualg_1(A,B),E,C),k1_mcart_1(D)) = k1_funct_1(k1_msualg_3(u1_struct_0(A),u4_msualg_1(A,k3_equation(A)),u4_msualg_1(A,B),E,C),k2_mcart_1(D)) ) ) ) ) ) ) ) )).

fof(d6_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( l3_msualg_1(B,A)
         => ! [C] :
              ( m4_pboole(C,u1_struct_0(A),k4_equation(A))
             => ( r2_equation(A,B,C)
              <=> ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ! [E] :
                        ( m1_subset_1(E,k1_funct_1(k4_equation(A),D))
                       => ( r2_hidden(E,k1_funct_1(C,D))
                         => r1_equation(A,B,D,E) ) ) ) ) ) ) ) )).

fof(t33_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_funct_1(k4_equation(A),C))
                 => ! [E] :
                      ( ( v4_msualg_1(E,A)
                        & v5_msualg_1(E,A)
                        & m1_msualg_2(E,A,B) )
                     => ( r1_equation(A,B,C,D)
                       => r1_equation(A,E,C,D) ) ) ) ) ) ) )).

fof(t34_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( m4_pboole(C,u1_struct_0(A),k4_equation(A))
             => ! [D] :
                  ( ( v4_msualg_1(D,A)
                    & v5_msualg_1(D,A)
                    & m1_msualg_2(D,A,B) )
                 => ( r2_equation(A,B,C)
                   => r2_equation(A,D,C) ) ) ) ) ) )).

fof(t35_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( ( v5_msualg_1(C,A)
                & l3_msualg_1(C,A) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_subset_1(E,k1_funct_1(k4_equation(A),D))
                     => ( ( r6_msualg_3(A,B,C)
                          & r1_equation(A,B,D,E) )
                       => r1_equation(A,C,D,E) ) ) ) ) ) ) )).

fof(t36_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( ( v5_msualg_1(C,A)
                & l3_msualg_1(C,A) )
             => ! [D] :
                  ( m4_pboole(D,u1_struct_0(A),k4_equation(A))
                 => ( ( r6_msualg_3(A,B,C)
                      & r2_equation(A,B,D) )
                   => r2_equation(A,C,D) ) ) ) ) ) )).

fof(t37_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_funct_1(k4_equation(A),C))
                 => ! [E] :
                      ( ( v3_msualg_4(E,A,B)
                        & v4_msualg_4(E,A,B)
                        & m1_msualg_4(E,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) )
                     => ( r1_equation(A,B,C,D)
                       => r1_equation(A,k14_msualg_4(A,B,E),C,D) ) ) ) ) ) ) )).

fof(t38_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( m4_pboole(C,u1_struct_0(A),k4_equation(A))
             => ! [D] :
                  ( ( v3_msualg_4(D,A,B)
                    & v4_msualg_4(D,A,B)
                    & m1_msualg_4(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) )
                 => ( r2_equation(A,B,C)
                   => r2_equation(A,k14_msualg_4(A,B,D),C) ) ) ) ) ) )).

fof(t39_equation,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & ~ v2_msualg_1(B)
        & l1_msualg_1(B) )
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(B))
         => ! [D] :
              ( m1_subset_1(D,k1_funct_1(k4_equation(B),C))
             => ! [E] :
                  ( m2_pralg_2(E,A,B)
                 => ( ! [F] :
                        ~ ( r2_hidden(F,A)
                          & ! [G] :
                              ( l3_msualg_1(G,B)
                             => ~ ( G = k1_funct_1(E,F)
                                  & r1_equation(B,G,C,D) ) ) )
                   => r1_equation(B,k15_pralg_2(A,B,E),C,D) ) ) ) ) ) )).

fof(t40_equation,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & ~ v2_msualg_1(B)
        & l1_msualg_1(B) )
     => ! [C] :
          ( m4_pboole(C,u1_struct_0(B),k4_equation(B))
         => ! [D] :
              ( m2_pralg_2(D,A,B)
             => ( ! [E] :
                    ~ ( r2_hidden(E,A)
                      & ! [F] :
                          ( l3_msualg_1(F,B)
                         => ~ ( F = k1_funct_1(D,E)
                              & r2_equation(B,F,C) ) ) )
               => r2_equation(B,k15_pralg_2(A,B,D),C) ) ) ) ) )).

fof(dt_k1_equation,axiom,(
    ! [A,B,C] :
      ( ( m1_pboole(B,A)
        & v1_funcop_1(C)
        & m1_pboole(C,A) )
     => m1_pboole(k1_equation(A,B,C),A) ) )).

fof(dt_k2_equation,axiom,(
    $true )).

fof(dt_k3_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => l3_msualg_1(k3_equation(A),A) ) )).

fof(dt_k4_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => m1_pboole(k4_equation(A),u1_struct_0(A)) ) )).

fof(dt_k5_equation,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,k1_funct_1(u4_msualg_1(A,k3_equation(A)),B))
        & m1_subset_1(D,k1_funct_1(u4_msualg_1(A,k3_equation(A)),B)) )
     => m1_subset_1(k5_equation(A,B,C,D),k1_funct_1(k4_equation(A),B)) ) )).

fof(redefinition_k5_equation,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,k1_funct_1(u4_msualg_1(A,k3_equation(A)),B))
        & m1_subset_1(D,k1_funct_1(u4_msualg_1(A,k3_equation(A)),B)) )
     => k5_equation(A,B,C,D) = k4_tarski(C,D) ) )).

fof(t29_equation,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( C = a_2_0_equation(A,B)
             => ! [D] :
                  ( m2_pralg_2(D,C,A)
                 => ~ ( ! [E] :
                          ( r2_hidden(E,C)
                         => E = k1_funct_1(D,E) )
                      & ! [E] :
                          ( ( v4_msualg_1(E,A)
                            & v5_msualg_1(E,A)
                            & m1_msualg_2(E,A,k15_pralg_2(C,A,D)) )
                         => ! [F] :
                              ( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,E),u4_msualg_1(A,B))
                             => ~ r2_msualg_3(A,E,B,F) ) ) ) ) ) ) ) )).

fof(fraenkel_a_2_0_equation,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & ~ v2_msualg_1(B)
        & l1_msualg_1(B)
        & v5_msualg_1(C,B)
        & l3_msualg_1(C,B) )
     => ( r2_hidden(A,a_2_0_equation(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,k14_msualg_2(B,C))
            & A = D
            & ? [E] :
                ( v4_msualg_1(E,B)
                & v5_msualg_1(E,B)
                & v3_msafree2(E,B)
                & m1_msualg_2(E,B,C)
                & E = D ) ) ) ) )).
%------------------------------------------------------------------------------