SET007 Axioms: SET007+385.ax


%------------------------------------------------------------------------------
% File     : SET007+385 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Homomorphisms of Algebras. Quotient Universal Algebra
% 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    : alg_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   52 (   2 unit)
%            Number of atoms       :  685 (  41 equality)
%            Maximal formula depth :   20 (  10 average)
%            Number of connectives :  719 (  86 ~  ;   0  |; 435  &)
%                                         (  18 <=>; 180 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   33 (   1 propositional; 0-4 arity)
%            Number of functors    :   36 (   2 constant; 0-5 arity)
%            Number of variables   :  167 (   0 singleton; 164 !;   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(t1_alg_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,A,B)
                    & m2_relset_1(D,A,B) )
                 => ( k4_finseq_1(k5_finseqop(A,B,C,D)) = k4_finseq_1(C)
                    & k3_finseq_1(k5_finseqop(A,B,C,D)) = k3_finseq_1(C)
                    & ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( r2_hidden(E,k4_finseq_1(k5_finseqop(A,B,C,D)))
                         => k1_funct_1(k5_finseqop(A,B,C,D),E) = k1_funct_1(D,k1_funct_1(C,E)) ) ) ) ) ) ) ) )).

fof(t2_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ( B = u1_struct_0(A)
           => k3_unialg_2(A,B) = u1_unialg_1(A) ) ) ) )).

fof(t3_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => k5_finseqop(u1_struct_0(A),u1_struct_0(B),k6_finseq_1(u1_struct_0(A)),C) = k6_finseq_1(u1_struct_0(B)) ) ) ) )).

fof(t4_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(A))
         => k5_finseqop(u1_struct_0(A),u1_struct_0(A),B,k6_partfun1(u1_struct_0(A))) = B ) ) )).

fof(t5_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v6_unialg_1(C)
                & v7_unialg_1(C)
                & v8_unialg_1(C)
                & l1_unialg_1(C) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
                    & m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(B),u1_struct_0(C))
                        & m2_relset_1(E,u1_struct_0(B),u1_struct_0(C)) )
                     => ! [F] :
                          ( m2_finseq_1(F,u1_struct_0(A))
                         => k5_finseqop(u1_struct_0(B),u1_struct_0(C),k5_finseqop(u1_struct_0(A),u1_struct_0(B),F,D),E) = k5_finseqop(u1_struct_0(A),u1_struct_0(C),F,k7_funct_2(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),D,E)) ) ) ) ) ) ) )).

fof(d1_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r1_alg_1(A,B,C)
              <=> ( r1_unialg_2(A,B)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k4_finseq_1(u1_unialg_1(A)))
                       => ! [E] :
                            ( m2_unialg_2(E,u1_struct_0(A),k1_unialg_2(A))
                           => ! [F] :
                                ( m2_unialg_2(F,u1_struct_0(B),k1_unialg_2(B))
                               => ( ( E = k1_funct_1(u1_unialg_1(A),D)
                                    & F = k1_funct_1(u1_unialg_1(B),D) )
                                 => ! [G] :
                                      ( m2_finseq_1(G,u1_struct_0(A))
                                     => ( r2_hidden(G,k4_relset_1(k3_finseq_2(u1_struct_0(A)),u1_struct_0(A),E))
                                       => k1_funct_1(C,k1_funct_1(E,G)) = k1_funct_1(F,k5_finseqop(u1_struct_0(A),u1_struct_0(B),G,C)) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d2_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r2_alg_1(A,B,C)
              <=> ( r1_alg_1(A,B,C)
                  & v2_funct_1(C) ) ) ) ) ) )).

fof(d3_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r3_alg_1(A,B,C)
              <=> ( r1_alg_1(A,B,C)
                  & k2_relat_1(C) = u1_struct_0(B) ) ) ) ) ) )).

fof(d4_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r4_alg_1(A,B,C)
              <=> ( r2_alg_1(A,B,C)
                  & r3_alg_1(A,B,C) ) ) ) ) ) )).

fof(d5_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ( r5_alg_1(A,B)
          <=> ? [C] :
                ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B))
                & r4_alg_1(A,B,C) ) ) ) ) )).

fof(t6_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => r1_alg_1(A,A,k6_partfun1(u1_struct_0(A))) ) )).

fof(t7_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v6_unialg_1(C)
                & v7_unialg_1(C)
                & v8_unialg_1(C)
                & l1_unialg_1(C) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
                    & m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(B),u1_struct_0(C))
                        & m2_relset_1(E,u1_struct_0(B),u1_struct_0(C)) )
                     => ( ( r1_alg_1(A,B,D)
                          & r1_alg_1(B,C,E) )
                       => r1_alg_1(A,C,k7_funct_2(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),D,E)) ) ) ) ) ) ) )).

fof(t8_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r4_alg_1(A,B,C)
              <=> ( r1_alg_1(A,B,C)
                  & k2_relat_1(C) = u1_struct_0(B)
                  & v2_funct_1(C) ) ) ) ) ) )).

fof(t9_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r4_alg_1(A,B,C)
               => ( k4_relset_1(u1_struct_0(A),u1_struct_0(B),C) = u1_struct_0(A)
                  & k2_relat_1(C) = u1_struct_0(B) ) ) ) ) ) )).

fof(t10_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(B),u1_struct_0(A))
                    & m2_relset_1(D,u1_struct_0(B),u1_struct_0(A)) )
                 => ( ( r4_alg_1(A,B,C)
                      & D = k2_funct_1(C) )
                   => r1_alg_1(B,A,D) ) ) ) ) ) )).

fof(t11_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(B),u1_struct_0(A))
                    & m2_relset_1(D,u1_struct_0(B),u1_struct_0(A)) )
                 => ( ( r4_alg_1(A,B,C)
                      & D = k2_funct_1(C) )
                   => r4_alg_1(B,A,D) ) ) ) ) ) )).

fof(t12_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v6_unialg_1(C)
                & v7_unialg_1(C)
                & v8_unialg_1(C)
                & l1_unialg_1(C) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
                    & m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(B),u1_struct_0(C))
                        & m2_relset_1(E,u1_struct_0(B),u1_struct_0(C)) )
                     => ( ( r4_alg_1(A,B,D)
                          & r4_alg_1(B,C,E) )
                       => r4_alg_1(A,C,k7_funct_2(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),D,E)) ) ) ) ) ) ) )).

fof(t13_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => r5_alg_1(A,A) ) )).

fof(t14_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ( r5_alg_1(A,B)
           => r5_alg_1(B,A) ) ) ) )).

fof(t15_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v6_unialg_1(C)
                & v7_unialg_1(C)
                & v8_unialg_1(C)
                & l1_unialg_1(C) )
             => ( ( r5_alg_1(A,B)
                  & r5_alg_1(B,C) )
               => r5_alg_1(A,C) ) ) ) ) )).

fof(d6_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r1_alg_1(A,B,C)
               => ! [D] :
                    ( ( v3_unialg_1(D)
                      & m3_unialg_2(D,B) )
                   => ( D = k1_alg_1(A,B,C)
                    <=> u1_struct_0(D) = k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,u1_struct_0(A)) ) ) ) ) ) ) )).

fof(t16_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r1_alg_1(A,B,C)
               => k2_relat_1(C) = u1_struct_0(k1_alg_1(A,B,C)) ) ) ) ) )).

fof(t17_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_unialg_1(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r1_alg_1(A,B,C)
               => ( r3_alg_1(A,B,C)
                <=> k1_alg_1(A,B,C) = B ) ) ) ) ) )).

fof(d7_alg_1,axiom,(
    $true )).

fof(d8_alg_1,axiom,(
    $true )).

fof(d9_alg_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_relset_1(B,A,A)
         => ! [C] :
              ( m2_relset_1(C,k3_finseq_2(A),k3_finseq_2(A))
             => ( C = k2_alg_1(A,B)
              <=> ! [D] :
                    ( m2_finseq_1(D,A)
                   => ! [E] :
                        ( m2_finseq_1(E,A)
                       => ( r2_hidden(k4_tarski(D,E),C)
                        <=> ( k3_finseq_1(D) = k3_finseq_1(E)
                            & ! [F] :
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                               => ( r2_hidden(F,k4_finseq_1(D))
                                 => r2_hidden(k4_tarski(k1_funct_1(D,F),k1_funct_1(E,F)),B) ) ) ) ) ) ) ) ) ) ) )).

fof(t18_alg_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k2_alg_1(A,k6_partfun1(A)) = k6_partfun1(k3_finseq_2(A)) ) )).

fof(d10_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( v3_relat_2(B)
            & v8_relat_2(B)
            & v1_partfun1(B,u1_struct_0(A),u1_struct_0(A))
            & m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
         => ( m1_alg_1(B,A)
          <=> ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ! [D] :
                    ( m2_unialg_2(D,u1_struct_0(A),k1_unialg_2(A))
                   => ( ( r2_hidden(C,k4_finseq_1(u1_unialg_1(A)))
                        & D = k1_funct_1(u1_unialg_1(A),C) )
                     => ! [E] :
                          ( m2_finseq_1(E,u1_struct_0(A))
                         => ! [F] :
                              ( m2_finseq_1(F,u1_struct_0(A))
                             => ( ( r2_hidden(E,k4_relset_1(k3_finseq_2(u1_struct_0(A)),u1_struct_0(A),D))
                                  & r2_hidden(F,k4_relset_1(k3_finseq_2(u1_struct_0(A)),u1_struct_0(A),D))
                                  & r2_hidden(k4_tarski(E,F),k2_alg_1(u1_struct_0(A),B)) )
                               => r2_hidden(k4_tarski(k1_funct_1(D,E),k1_funct_1(D,F)),B) ) ) ) ) ) ) ) ) ) )).

fof(d11_alg_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v3_relat_2(B)
            & v8_relat_2(B)
            & v1_partfun1(B,A,A)
            & m2_relset_1(B,A,A) )
         => ! [C] :
              ( m2_finseq_1(C,k8_eqrel_1(A,B))
             => ! [D] :
                  ( m2_finseq_1(D,A)
                 => ( r6_alg_1(A,B,C,D)
                  <=> ( k3_finseq_1(D) = k3_finseq_1(C)
                      & ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ( r2_hidden(E,k4_finseq_1(D))
                           => k6_eqrel_1(A,B,k1_funct_1(D,E)) = k1_funct_1(C,E) ) ) ) ) ) ) ) ) )).

fof(t19_alg_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v3_relat_2(B)
            & v8_relat_2(B)
            & v1_partfun1(B,A,A)
            & m2_relset_1(B,A,A) )
         => ! [C] :
              ( m2_finseq_1(C,k8_eqrel_1(A,B))
             => ? [D] :
                  ( m2_finseq_1(D,A)
                  & r6_alg_1(A,B,C,D) ) ) ) ) )).

fof(d12_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( m1_alg_1(B,A)
         => ! [C] :
              ( m2_unialg_2(C,u1_struct_0(A),k1_unialg_2(A))
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & v1_funct_1(D)
                    & v1_unialg_1(D,k8_eqrel_1(u1_struct_0(A),B))
                    & v2_unialg_1(D,k8_eqrel_1(u1_struct_0(A),B))
                    & m2_relset_1(D,k3_finseq_2(k8_eqrel_1(u1_struct_0(A),B)),k8_eqrel_1(u1_struct_0(A),B)) )
                 => ( D = k3_alg_1(A,B,C)
                  <=> ( k4_relset_1(k3_finseq_2(k8_eqrel_1(u1_struct_0(A),B)),k8_eqrel_1(u1_struct_0(A),B),D) = k4_finseq_2(k2_unialg_1(u1_struct_0(A),C),k8_eqrel_1(u1_struct_0(A),B))
                      & ! [E] :
                          ( m2_finseq_1(E,k8_eqrel_1(u1_struct_0(A),B))
                         => ( r2_hidden(E,k4_relset_1(k3_finseq_2(k8_eqrel_1(u1_struct_0(A),B)),k8_eqrel_1(u1_struct_0(A),B),D))
                           => ! [F] :
                                ( m2_finseq_1(F,u1_struct_0(A))
                               => ( r6_alg_1(u1_struct_0(A),B,E,F)
                                 => k1_funct_1(D,E) = k6_eqrel_1(u1_struct_0(A),B,k1_funct_1(C,F)) ) ) ) ) ) ) ) ) ) ) )).

fof(d13_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( m1_alg_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,k4_partfun1(k13_finseq_1(k8_eqrel_1(u1_struct_0(A),B)),k8_eqrel_1(u1_struct_0(A),B)))
             => ( C = k4_alg_1(A,B)
              <=> ( k3_finseq_1(C) = k3_finseq_1(u1_unialg_1(A))
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k4_finseq_1(C))
                       => ! [E] :
                            ( m2_unialg_2(E,u1_struct_0(A),k1_unialg_2(A))
                           => ( k1_funct_1(u1_unialg_1(A),D) = E
                             => k1_funct_1(C,D) = k3_alg_1(A,B,E) ) ) ) ) ) ) ) ) ) )).

fof(t20_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( m1_alg_1(B,A)
         => ( ~ v3_struct_0(g1_unialg_1(k8_eqrel_1(u1_struct_0(A),B),k4_alg_1(A,B)))
            & v3_unialg_1(g1_unialg_1(k8_eqrel_1(u1_struct_0(A),B),k4_alg_1(A,B)))
            & v6_unialg_1(g1_unialg_1(k8_eqrel_1(u1_struct_0(A),B),k4_alg_1(A,B)))
            & v7_unialg_1(g1_unialg_1(k8_eqrel_1(u1_struct_0(A),B),k4_alg_1(A,B)))
            & v8_unialg_1(g1_unialg_1(k8_eqrel_1(u1_struct_0(A),B),k4_alg_1(A,B)))
            & l1_unialg_1(g1_unialg_1(k8_eqrel_1(u1_struct_0(A),B),k4_alg_1(A,B))) ) ) ) )).

fof(d14_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( m1_alg_1(B,A)
         => k5_alg_1(A,B) = g1_unialg_1(k8_eqrel_1(u1_struct_0(A),B),k4_alg_1(A,B)) ) ) )).

fof(d15_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( m1_alg_1(B,A)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(k5_alg_1(A,B)))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(k5_alg_1(A,B))) )
             => ( C = k6_alg_1(A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => k8_funct_2(u1_struct_0(A),u1_struct_0(k5_alg_1(A,B)),C,D) = k6_eqrel_1(u1_struct_0(A),B,D) ) ) ) ) ) )).

fof(t21_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( m1_alg_1(B,A)
         => r1_alg_1(A,k5_alg_1(A,B),k6_alg_1(A,B)) ) ) )).

fof(t22_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( m1_alg_1(B,A)
         => r3_alg_1(A,k5_alg_1(A,B),k6_alg_1(A,B)) ) ) )).

fof(d16_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r1_alg_1(A,B,C)
               => ! [D] :
                    ( m1_alg_1(D,A)
                   => ( D = k7_alg_1(A,B,C)
                    <=> ! [E] :
                          ( m1_subset_1(E,u1_struct_0(A))
                         => ! [F] :
                              ( m1_subset_1(F,u1_struct_0(A))
                             => ( r2_hidden(k4_tarski(E,F),D)
                              <=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E) = k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,F) ) ) ) ) ) ) ) ) ) )).

fof(d17_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r1_alg_1(A,B,C)
               => ! [D] :
                    ( ( v1_funct_1(D)
                      & v1_funct_2(D,u1_struct_0(k5_alg_1(A,k7_alg_1(A,B,C))),u1_struct_0(B))
                      & m2_relset_1(D,u1_struct_0(k5_alg_1(A,k7_alg_1(A,B,C))),u1_struct_0(B)) )
                   => ( D = k8_alg_1(A,B,C)
                    <=> ! [E] :
                          ( m1_subset_1(E,u1_struct_0(A))
                         => k1_funct_1(D,k6_eqrel_1(u1_struct_0(A),k7_alg_1(A,B,C),E)) = k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E) ) ) ) ) ) ) ) )).

fof(t23_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r1_alg_1(A,B,C)
               => ( r1_alg_1(k5_alg_1(A,k7_alg_1(A,B,C)),B,k8_alg_1(A,B,C))
                  & r2_alg_1(k5_alg_1(A,k7_alg_1(A,B,C)),B,k8_alg_1(A,B,C)) ) ) ) ) ) )).

fof(t24_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r3_alg_1(A,B,C)
               => r4_alg_1(k5_alg_1(A,k7_alg_1(A,B,C)),B,k8_alg_1(A,B,C)) ) ) ) ) )).

fof(t25_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_unialg_1(B)
            & v7_unialg_1(B)
            & v8_unialg_1(B)
            & l1_unialg_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( r3_alg_1(A,B,C)
               => r5_alg_1(k5_alg_1(A,k7_alg_1(A,B,C)),B) ) ) ) ) )).

fof(dt_m1_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( m1_alg_1(B,A)
         => ( v3_relat_2(B)
            & v8_relat_2(B)
            & v1_partfun1(B,u1_struct_0(A),u1_struct_0(A))
            & m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) ) ) ) )).

fof(existence_m1_alg_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ? [B] : m1_alg_1(B,A) ) )).

fof(dt_k1_alg_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A)
        & ~ v3_struct_0(B)
        & v6_unialg_1(B)
        & v7_unialg_1(B)
        & v8_unialg_1(B)
        & l1_unialg_1(B)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
        & m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
     => ( v3_unialg_1(k1_alg_1(A,B,C))
        & m3_unialg_2(k1_alg_1(A,B,C),B) ) ) )).

fof(dt_k2_alg_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_relset_1(B,A,A) )
     => m2_relset_1(k2_alg_1(A,B),k3_finseq_2(A),k3_finseq_2(A)) ) )).

fof(dt_k3_alg_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A)
        & m1_alg_1(B,A)
        & m1_subset_1(C,k1_unialg_2(A)) )
     => ( ~ v1_xboole_0(k3_alg_1(A,B,C))
        & v1_funct_1(k3_alg_1(A,B,C))
        & v1_unialg_1(k3_alg_1(A,B,C),k8_eqrel_1(u1_struct_0(A),B))
        & v2_unialg_1(k3_alg_1(A,B,C),k8_eqrel_1(u1_struct_0(A),B))
        & m2_relset_1(k3_alg_1(A,B,C),k3_finseq_2(k8_eqrel_1(u1_struct_0(A),B)),k8_eqrel_1(u1_struct_0(A),B)) ) ) )).

fof(dt_k4_alg_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A)
        & m1_alg_1(B,A) )
     => m2_finseq_1(k4_alg_1(A,B),k4_partfun1(k13_finseq_1(k8_eqrel_1(u1_struct_0(A),B)),k8_eqrel_1(u1_struct_0(A),B))) ) )).

fof(dt_k5_alg_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A)
        & m1_alg_1(B,A) )
     => ( ~ v3_struct_0(k5_alg_1(A,B))
        & v3_unialg_1(k5_alg_1(A,B))
        & v6_unialg_1(k5_alg_1(A,B))
        & v7_unialg_1(k5_alg_1(A,B))
        & v8_unialg_1(k5_alg_1(A,B))
        & l1_unialg_1(k5_alg_1(A,B)) ) ) )).

fof(dt_k6_alg_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A)
        & m1_alg_1(B,A) )
     => ( v1_funct_1(k6_alg_1(A,B))
        & v1_funct_2(k6_alg_1(A,B),u1_struct_0(A),u1_struct_0(k5_alg_1(A,B)))
        & m2_relset_1(k6_alg_1(A,B),u1_struct_0(A),u1_struct_0(k5_alg_1(A,B))) ) ) )).

fof(dt_k7_alg_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A)
        & ~ v3_struct_0(B)
        & v6_unialg_1(B)
        & v7_unialg_1(B)
        & v8_unialg_1(B)
        & l1_unialg_1(B)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
        & m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
     => m1_alg_1(k7_alg_1(A,B,C),A) ) )).

fof(dt_k8_alg_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A)
        & ~ v3_struct_0(B)
        & v6_unialg_1(B)
        & v7_unialg_1(B)
        & v8_unialg_1(B)
        & l1_unialg_1(B)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
        & m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
     => ( v1_funct_1(k8_alg_1(A,B,C))
        & v1_funct_2(k8_alg_1(A,B,C),u1_struct_0(k5_alg_1(A,k7_alg_1(A,B,C))),u1_struct_0(B))
        & m2_relset_1(k8_alg_1(A,B,C),u1_struct_0(k5_alg_1(A,k7_alg_1(A,B,C))),u1_struct_0(B)) ) ) )).
%------------------------------------------------------------------------------