SET007 Axioms: SET007+851.ax


%------------------------------------------------------------------------------
% File     : SET007+851 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Logical Correctness of Vector Calculation Programs
% 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    : prgcor_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   34 (   1 unit)
%            Number of atoms       :  348 (  82 equality)
%            Maximal formula depth :   24 (  12 average)
%            Number of connectives :  365 (  51 ~  ;   9  |; 167  &)
%                                         (  11 <=>; 127 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   21 (   1 propositional; 0-4 arity)
%            Number of functors    :   29 (   5 constant; 0-5 arity)
%            Number of variables   :  120 (   0 singleton; 108 !;  12 ?)
%            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(rc1_prgcor_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ? [B] :
          ( m1_ordinal1(B,A)
          & ~ v1_xboole_0(B)
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v5_ordinal1(B)
          & v1_finset_1(B) ) ) )).

fof(t1_prgcor_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_hidden(A,B)
          <=> ~ r1_xreal_0(B,A) ) ) ) )).

fof(t2_prgcor_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_finset_1(B)
            & m1_ordinal1(B,A) )
         => ~ r1_xreal_0(k1_afinsq_1(B),np__0) ) ) )).

fof(d1_prgcor_2,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,A)
     => ! [C] :
          ( ( v1_finset_1(C)
            & m1_ordinal1(C,A) )
         => ( C = k1_prgcor_2(A,B)
          <=> ( k1_afinsq_1(C) = k3_finseq_1(B)
              & ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( ~ r1_xreal_0(k3_finseq_1(B),D)
                   => k1_funct_1(B,k1_nat_1(D,np__1)) = k1_funct_1(C,D) ) ) ) ) ) ) )).

fof(d2_prgcor_2,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(B)
        & m1_ordinal1(B,A) )
     => ! [C] :
          ( m2_finseq_1(C,A)
         => ( C = k2_prgcor_2(A,B)
          <=> ( k3_finseq_1(C) = k1_afinsq_1(B)
              & ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( ( r1_xreal_0(np__1,D)
                      & r1_xreal_0(D,k1_afinsq_1(B)) )
                   => k1_funct_1(B,k5_binarith(D,np__1)) = k1_funct_1(C,D) ) ) ) ) ) ) )).

fof(t3_prgcor_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_relat_1(k2_funcop_1(A,B))
          & v1_funct_1(k2_funcop_1(A,B))
          & v5_ordinal1(k2_funcop_1(A,B))
          & v1_finset_1(k2_funcop_1(A,B)) ) ) )).

fof(t4_prgcor_2,axiom,(
    ! [A,B] :
      ( m2_subset_1(B,k1_numbers,k5_numbers)
     => ! [C] :
          ( r2_hidden(C,A)
         => ( v1_finset_1(k2_funcop_1(B,C))
            & m1_ordinal1(k2_funcop_1(B,C),A)
            & ! [D] :
                ( ( v1_relat_1(D)
                  & v1_funct_1(D)
                  & v5_ordinal1(D)
                  & v1_finset_1(D) )
               => ( D = k2_funcop_1(B,C)
                 => k1_afinsq_1(D) = B ) ) ) ) ) )).

fof(d3_prgcor_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r1_tarski(k5_numbers,A)
               => ( r1_xreal_0(C,k3_finseq_1(B))
                  | ! [D] :
                      ( ( ~ v1_xboole_0(D)
                        & v1_finset_1(D)
                        & m1_ordinal1(D,A) )
                     => ( D = k3_prgcor_2(A,B,C)
                      <=> ( k3_finseq_1(B) = k1_funct_1(D,np__0)
                          & k1_afinsq_1(D) = C
                          & ! [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ( ( r1_xreal_0(np__1,E)
                                  & r1_xreal_0(E,k3_finseq_1(B)) )
                               => k1_funct_1(D,E) = k1_funct_1(B,E) ) )
                          & ! [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ~ ( ~ r1_xreal_0(E,k3_finseq_1(B))
                                  & ~ r1_xreal_0(C,E)
                                  & k1_funct_1(D,E) != np__0 ) ) ) ) ) ) ) ) ) ) )).

fof(d4_prgcor_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_finset_1(B)
            & m1_ordinal1(B,A) )
         => ( ( r1_tarski(k5_numbers,A)
              & m2_subset_1(k1_funct_1(B,np__0),k1_numbers,k5_numbers)
              & r2_hidden(k1_funct_1(B,np__0),k1_afinsq_1(B)) )
           => ! [C] :
                ( m2_finseq_1(C,A)
               => ( C = k4_prgcor_2(A,B)
                <=> ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( D = k1_funct_1(B,np__0)
                       => ( k3_finseq_1(C) = D
                          & ! [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ( ( r1_xreal_0(np__1,E)
                                  & r1_xreal_0(E,D) )
                               => k1_funct_1(C,E) = k1_funct_1(B,E) ) ) ) ) ) ) ) ) ) ) )).

fof(t5_prgcor_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_finset_1(B)
            & m1_ordinal1(B,A) )
         => ( ( r1_tarski(k5_numbers,A)
              & k1_funct_1(B,np__0) = np__0 )
           => ( r1_xreal_0(k1_afinsq_1(B),np__0)
              | k4_prgcor_2(A,B) = k1_xboole_0 ) ) ) ) )).

fof(d5_prgcor_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_finset_1(B)
            & m1_ordinal1(B,A) )
         => ! [C] :
              ( m2_finseq_1(C,A)
             => ( r1_prgcor_2(A,B,C)
              <=> ( r1_tarski(k5_numbers,A)
                  & k1_funct_1(B,np__0) = k3_finseq_1(C)
                  & ~ r1_xreal_0(k1_afinsq_1(B),k3_finseq_1(C))
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( ( r1_xreal_0(np__1,D)
                          & r1_xreal_0(D,k3_finseq_1(C)) )
                       => k1_funct_1(B,D) = k1_funct_1(C,D) ) ) ) ) ) ) ) )).

fof(t6_prgcor_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_finset_1(B)
            & m1_ordinal1(B,A) )
         => ( ( r1_tarski(k5_numbers,A)
              & m2_subset_1(k1_funct_1(B,np__0),k1_numbers,k5_numbers)
              & r2_hidden(k1_funct_1(B,np__0),k1_afinsq_1(B)) )
           => r1_prgcor_2(A,B,k4_prgcor_2(A,B)) ) ) ) )).

fof(d6_prgcor_2,axiom,(
    ! [A,B,C,D,E] :
      ( ( r2_hidden(A,B)
       => k5_prgcor_2(A,B,C,D,E) = C )
      & ( A = B
       => k5_prgcor_2(A,B,C,D,E) = D )
      & ~ ( ~ r2_hidden(A,B)
          & A != B
          & k5_prgcor_2(A,B,C,D,E) != E ) ) )).

fof(t7_prgcor_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ~ ( r1_tarski(k5_numbers,A)
                  & ~ r1_xreal_0(C,k3_finseq_1(B))
                  & ! [D] :
                      ( ( v1_finset_1(D)
                        & m1_ordinal1(D,A) )
                     => ~ ( k1_afinsq_1(D) = C
                          & r1_prgcor_2(A,D,B) ) ) ) ) ) ) )).

fof(d7_prgcor_2,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & m1_ordinal1(A,k1_numbers) )
     => ! [B] :
          ( ( v1_finset_1(B)
            & m1_ordinal1(B,k1_numbers) )
         => ( ( m2_subset_1(k6_prgcor_2(B,np__0),k1_numbers,k5_numbers)
              & r1_xreal_0(np__0,k6_prgcor_2(B,np__0)) )
           => ( r1_xreal_0(k1_afinsq_1(A),k6_prgcor_2(B,np__0))
              | ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ( C = k7_prgcor_2(A,B)
                  <=> ? [D] :
                        ( v1_finset_1(D)
                        & m1_ordinal1(D,k1_numbers)
                        & ? [E] :
                            ( v1_int_1(E)
                            & k1_afinsq_1(D) = k1_afinsq_1(A)
                            & k6_prgcor_2(D,np__0) = np__0
                            & E = k6_prgcor_2(B,np__0)
                            & ( E != np__0
                             => ! [F] :
                                  ( m2_subset_1(F,k1_numbers,k5_numbers)
                                 => ( ~ r1_xreal_0(E,F)
                                   => k6_prgcor_2(D,k1_nat_1(F,np__1)) = k3_real_1(k6_prgcor_2(D,F),k4_real_1(k6_prgcor_2(A,k1_nat_1(F,np__1)),k6_prgcor_2(B,k1_nat_1(F,np__1)))) ) ) )
                            & C = k1_funct_1(D,E) ) ) ) ) ) ) ) ) )).

fof(t8_prgcor_2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_finset_1(B)
            & m1_ordinal1(B,k1_numbers) )
         => ( ( k6_prgcor_2(B,np__0) = np__0
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( ~ r1_xreal_0(k3_finseq_1(A),C)
                   => k6_prgcor_2(B,k1_nat_1(C,np__1)) = k3_real_1(k6_prgcor_2(B,C),k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(C,np__1))) ) ) )
           => ( r1_xreal_0(k1_afinsq_1(B),k3_finseq_1(A))
              | k15_rvsum_1(A) = k6_prgcor_2(B,k3_finseq_1(A)) ) ) ) ) )).

fof(t9_prgcor_2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ? [B] :
          ( v1_finset_1(B)
          & m1_ordinal1(B,k1_numbers)
          & k1_afinsq_1(B) = k1_nat_1(k3_finseq_1(A),np__1)
          & k6_prgcor_2(B,np__0) = np__0
          & ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ~ r1_xreal_0(k3_finseq_1(A),C)
               => k6_prgcor_2(B,k1_nat_1(C,np__1)) = k3_real_1(k6_prgcor_2(B,C),k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(C,np__1))) ) )
          & k15_rvsum_1(A) = k6_prgcor_2(B,k3_finseq_1(A)) ) ) )).

fof(t10_prgcor_2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( k3_finseq_1(A) = k3_finseq_1(B)
               => ( r1_xreal_0(C,k3_finseq_1(A))
                  | k1_euclid_2(A,B) = k7_prgcor_2(k3_prgcor_2(k1_numbers,A,C),k3_prgcor_2(k1_numbers,B,C)) ) ) ) ) ) )).

fof(d8_prgcor_2,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & m1_ordinal1(A,k1_numbers) )
     => ! [B] :
          ( ( v1_finset_1(B)
            & m1_ordinal1(B,k1_numbers) )
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( v1_int_1(D)
                 => ( r2_prgcor_2(A,B,C,D)
                  <=> ( k1_afinsq_1(B) = D
                      & k1_afinsq_1(A) = D
                      & ? [E] :
                          ( v1_int_1(E)
                          & k6_prgcor_2(B,np__0) = k6_prgcor_2(A,np__0)
                          & E = k6_prgcor_2(A,np__0)
                          & ( E != np__0
                           => ! [F] :
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                               => ( ( r1_xreal_0(np__1,F)
                                    & r1_xreal_0(F,E) )
                                 => k6_prgcor_2(B,F) = k4_real_1(C,k6_prgcor_2(A,F)) ) ) ) ) ) ) ) ) ) ) )).

fof(t11_prgcor_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & m1_ordinal1(A,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( m2_subset_1(k6_prgcor_2(A,np__0),k1_numbers,k5_numbers)
                  & k1_afinsq_1(A) = C
                  & r1_xreal_0(np__0,k6_prgcor_2(A,np__0)) )
               => ( r1_xreal_0(C,k6_prgcor_2(A,np__0))
                  | ( ? [D] :
                        ( v1_finset_1(D)
                        & m1_ordinal1(D,k1_numbers)
                        & r2_prgcor_2(A,D,B,C) )
                    & ! [D] :
                        ( ( ~ v1_xboole_0(D)
                          & v1_finset_1(D)
                          & m1_ordinal1(D,k1_numbers) )
                       => ( r2_prgcor_2(A,D,B,C)
                         => k4_prgcor_2(k1_numbers,D) = k9_rvsum_1(B,k4_prgcor_2(k1_numbers,A)) ) ) ) ) ) ) ) ) )).

fof(d9_prgcor_2,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & m1_ordinal1(A,k1_numbers) )
     => ! [B] :
          ( ( v1_finset_1(B)
            & m1_ordinal1(B,k1_numbers) )
         => ! [C] :
              ( v1_int_1(C)
             => ( r3_prgcor_2(A,B,C)
              <=> ( k1_afinsq_1(B) = C
                  & k1_afinsq_1(A) = C
                  & ? [D] :
                      ( v1_int_1(D)
                      & k6_prgcor_2(B,np__0) = k6_prgcor_2(A,np__0)
                      & D = k6_prgcor_2(A,np__0)
                      & ( D != np__0
                       => ! [E] :
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                           => ( ( r1_xreal_0(np__1,E)
                                & r1_xreal_0(E,D) )
                             => k6_prgcor_2(B,E) = k1_real_1(k6_prgcor_2(A,E)) ) ) ) ) ) ) ) ) ) )).

fof(t12_prgcor_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & m1_ordinal1(A,k1_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( m2_subset_1(k6_prgcor_2(A,np__0),k1_numbers,k5_numbers)
              & k1_afinsq_1(A) = B
              & r1_xreal_0(np__0,k6_prgcor_2(A,np__0)) )
           => ( r1_xreal_0(B,k6_prgcor_2(A,np__0))
              | ( ? [C] :
                    ( v1_finset_1(C)
                    & m1_ordinal1(C,k1_numbers)
                    & r3_prgcor_2(A,C,B) )
                & ! [C] :
                    ( ( ~ v1_xboole_0(C)
                      & v1_finset_1(C)
                      & m1_ordinal1(C,k1_numbers) )
                   => ( r3_prgcor_2(A,C,B)
                     => k4_prgcor_2(k1_numbers,C) = k5_rvsum_1(k4_prgcor_2(k1_numbers,A)) ) ) ) ) ) ) ) )).

fof(d10_prgcor_2,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & m1_ordinal1(A,k1_numbers) )
     => ! [B] :
          ( ( v1_finset_1(B)
            & m1_ordinal1(B,k1_numbers) )
         => ! [C] :
              ( ( v1_finset_1(C)
                & m1_ordinal1(C,k1_numbers) )
             => ! [D] :
                  ( v1_int_1(D)
                 => ( r4_prgcor_2(A,B,C,D)
                  <=> ( k1_afinsq_1(C) = D
                      & k1_afinsq_1(A) = D
                      & k1_afinsq_1(B) = D
                      & ? [E] :
                          ( v1_int_1(E)
                          & k6_prgcor_2(C,np__0) = k6_prgcor_2(B,np__0)
                          & E = k6_prgcor_2(B,np__0)
                          & ( E != np__0
                           => ! [F] :
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                               => ( ( r1_xreal_0(np__1,F)
                                    & r1_xreal_0(F,E) )
                                 => k6_prgcor_2(C,F) = k3_real_1(k6_prgcor_2(A,F),k6_prgcor_2(B,F)) ) ) ) ) ) ) ) ) ) ) )).

fof(t13_prgcor_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & m1_ordinal1(A,k1_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_finset_1(B)
            & m1_ordinal1(B,k1_numbers) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( m2_subset_1(k6_prgcor_2(B,np__0),k1_numbers,k5_numbers)
                  & k1_afinsq_1(A) = C
                  & k1_afinsq_1(B) = C
                  & k6_prgcor_2(A,np__0) = k6_prgcor_2(B,np__0)
                  & r1_xreal_0(np__0,k6_prgcor_2(B,np__0)) )
               => ( r1_xreal_0(C,k6_prgcor_2(B,np__0))
                  | ( ? [D] :
                        ( v1_finset_1(D)
                        & m1_ordinal1(D,k1_numbers)
                        & r4_prgcor_2(A,B,D,C) )
                    & ! [D] :
                        ( ( ~ v1_xboole_0(D)
                          & v1_finset_1(D)
                          & m1_ordinal1(D,k1_numbers) )
                       => ( r4_prgcor_2(A,B,D,C)
                         => k4_prgcor_2(k1_numbers,D) = k3_rvsum_1(k4_prgcor_2(k1_numbers,A),k4_prgcor_2(k1_numbers,B)) ) ) ) ) ) ) ) ) )).

fof(d11_prgcor_2,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & m1_ordinal1(A,k1_numbers) )
     => ! [B] :
          ( ( v1_finset_1(B)
            & m1_ordinal1(B,k1_numbers) )
         => ! [C] :
              ( ( v1_finset_1(C)
                & m1_ordinal1(C,k1_numbers) )
             => ! [D] :
                  ( v1_int_1(D)
                 => ( r5_prgcor_2(A,B,C,D)
                  <=> ( k1_afinsq_1(C) = D
                      & k1_afinsq_1(A) = D
                      & k1_afinsq_1(B) = D
                      & ? [E] :
                          ( v1_int_1(E)
                          & k6_prgcor_2(C,np__0) = k6_prgcor_2(B,np__0)
                          & E = k6_prgcor_2(B,np__0)
                          & ( E != np__0
                           => ! [F] :
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                               => ( ( r1_xreal_0(np__1,F)
                                    & r1_xreal_0(F,E) )
                                 => k6_prgcor_2(C,F) = k5_real_1(k6_prgcor_2(A,F),k6_prgcor_2(B,F)) ) ) ) ) ) ) ) ) ) ) )).

fof(t14_prgcor_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & m1_ordinal1(A,k1_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_finset_1(B)
            & m1_ordinal1(B,k1_numbers) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( m2_subset_1(k6_prgcor_2(B,np__0),k1_numbers,k5_numbers)
                  & k1_afinsq_1(A) = C
                  & k1_afinsq_1(B) = C
                  & k6_prgcor_2(A,np__0) = k6_prgcor_2(B,np__0)
                  & r1_xreal_0(np__0,k6_prgcor_2(B,np__0)) )
               => ( r1_xreal_0(C,k6_prgcor_2(B,np__0))
                  | ( ? [D] :
                        ( v1_finset_1(D)
                        & m1_ordinal1(D,k1_numbers)
                        & r5_prgcor_2(A,B,D,C) )
                    & ! [D] :
                        ( ( ~ v1_xboole_0(D)
                          & v1_finset_1(D)
                          & m1_ordinal1(D,k1_numbers) )
                       => ( r5_prgcor_2(A,B,D,C)
                         => k4_prgcor_2(k1_numbers,D) = k7_rvsum_1(k4_prgcor_2(k1_numbers,A),k4_prgcor_2(k1_numbers,B)) ) ) ) ) ) ) ) ) )).

fof(dt_k1_prgcor_2,axiom,(
    ! [A,B] :
      ( m1_finseq_1(B,A)
     => ( v1_finset_1(k1_prgcor_2(A,B))
        & m1_ordinal1(k1_prgcor_2(A,B),A) ) ) )).

fof(dt_k2_prgcor_2,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(B)
        & m1_ordinal1(B,A) )
     => m2_finseq_1(k2_prgcor_2(A,B),A) ) )).

fof(dt_k3_prgcor_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(B,A)
        & m1_subset_1(C,k5_numbers) )
     => ( ~ v1_xboole_0(k3_prgcor_2(A,B,C))
        & v1_finset_1(k3_prgcor_2(A,B,C))
        & m1_ordinal1(k3_prgcor_2(A,B,C),A) ) ) )).

fof(dt_k4_prgcor_2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & v1_finset_1(B)
        & m1_ordinal1(B,A) )
     => m2_finseq_1(k4_prgcor_2(A,B),A) ) )).

fof(dt_k5_prgcor_2,axiom,(
    $true )).

fof(dt_k6_prgcor_2,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(A)
        & m1_ordinal1(A,k1_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m1_subset_1(k6_prgcor_2(A,B),k1_numbers) ) )).

fof(redefinition_k6_prgcor_2,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(A)
        & m1_ordinal1(A,k1_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k6_prgcor_2(A,B) = k1_funct_1(A,B) ) )).

fof(dt_k7_prgcor_2,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(A)
        & m1_ordinal1(A,k1_numbers)
        & v1_finset_1(B)
        & m1_ordinal1(B,k1_numbers) )
     => m1_subset_1(k7_prgcor_2(A,B),k1_numbers) ) )).
%------------------------------------------------------------------------------