SET007 Axioms: SET007+631.ax


%------------------------------------------------------------------------------
% File     : SET007+631 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Scalar Multiple of Riemann Definite Integral
% 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    : integra2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   52 (   0 unit)
%            Number of atoms       :  454 (  27 equality)
%            Maximal formula depth :   16 (  10 average)
%            Number of connectives :  434 (  32 ~  ;   1  |; 195  &)
%                                         (   8 <=>; 198 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   30 (   0 propositional; 1-4 arity)
%            Number of functors    :   40 (   4 constant; 0-4 arity)
%            Number of variables   :  173 (   0 singleton; 168 !;   5 ?)
%            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_integra2,axiom,(
    ? [A] :
      ( m1_finseq_1(A,k1_numbers)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v1_finset_1(A)
      & v1_finseq_1(A)
      & v1_seq_1(A)
      & v1_integra2(A) ) )).

fof(t1_integra2,axiom,(
    ! [A] :
      ( ( v1_integra1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r2_hidden(B,A)
          <=> ( r1_xreal_0(k4_pscomp_1(A),B)
              & r1_xreal_0(B,k3_pscomp_1(A)) ) ) ) ) )).

fof(d1_integra2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ( v1_integra2(A)
      <=> ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( ( r2_hidden(B,k4_finseq_1(A))
                & r2_hidden(k1_nat_1(B,np__1),k4_finseq_1(A)) )
             => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,B),k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(B,np__1))) ) ) ) ) )).

fof(t2_integra2,axiom,(
    ! [A] :
      ( ( v1_integra2(A)
        & m2_finseq_1(A,k1_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r2_hidden(B,k4_finseq_1(A))
                  & r2_hidden(C,k4_finseq_1(A))
                  & r1_xreal_0(B,C) )
               => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,B),k2_seq_1(k5_numbers,k1_numbers,A,C)) ) ) ) ) )).

fof(t3_integra2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ? [B] :
          ( v1_integra2(B)
          & m2_finseq_1(B,k1_numbers)
          & r1_rfinseq(A,B) ) ) )).

fof(t4_integra2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( ( r1_xreal_0(np__1,C)
                          & r1_xreal_0(E,k3_finseq_1(B))
                          & r1_xreal_0(C,D) )
                       => ( r1_xreal_0(E,D)
                          | k8_finseq_1(A,k1_jordan3(A,B,C,D),k1_jordan3(A,B,k1_nat_1(D,np__1),E)) = k1_jordan3(A,B,C,E) ) ) ) ) ) ) ) )).

fof(d2_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_numbers))
             => ( C = k1_integra2(A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ( r2_hidden(D,C)
                    <=> ? [E] :
                          ( m1_subset_1(E,k1_numbers)
                          & r2_hidden(E,A)
                          & D = k3_xcmplx_0(B,E) ) ) ) ) ) ) ) )).

fof(t5_integra2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,A,k1_numbers) )
             => ( ( r1_rfunct_1(C,A)
                  & r1_tarski(B,A) )
               => r1_rfunct_1(k2_partfun1(A,k1_numbers,C,B),B) ) ) ) ) )).

fof(t6_integra2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,A,k1_numbers) )
             => ( ( r2_rfunct_1(C,A)
                  & r1_tarski(B,A) )
               => r2_rfunct_1(k2_partfun1(A,k1_numbers,C,B),B) ) ) ) ) )).

fof(t7_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ~ v1_xboole_0(k1_integra2(B,A)) ) ) )).

fof(t9_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ( ( v1_seq_4(B)
              & r1_xreal_0(np__0,A) )
           => v1_seq_4(k1_integra2(B,A)) ) ) ) )).

fof(t10_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ( ( v1_seq_4(B)
              & r1_xreal_0(A,np__0) )
           => v2_seq_4(k1_integra2(B,A)) ) ) ) )).

fof(t11_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ( ( v2_seq_4(B)
              & r1_xreal_0(np__0,A) )
           => v2_seq_4(k1_integra2(B,A)) ) ) ) )).

fof(t12_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ( ( v2_seq_4(B)
              & r1_xreal_0(A,np__0) )
           => v1_seq_4(k1_integra2(B,A)) ) ) ) )).

fof(t13_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ( ( v1_seq_4(B)
              & r1_xreal_0(np__0,A) )
           => k3_pscomp_1(k1_integra2(B,A)) = k4_real_1(A,k3_pscomp_1(B)) ) ) ) )).

fof(t14_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ( ( v1_seq_4(B)
              & r1_xreal_0(A,np__0) )
           => k4_pscomp_1(k1_integra2(B,A)) = k4_real_1(A,k3_pscomp_1(B)) ) ) ) )).

fof(t15_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ( ( v2_seq_4(B)
              & r1_xreal_0(np__0,A) )
           => k4_pscomp_1(k1_integra2(B,A)) = k4_real_1(A,k4_pscomp_1(B)) ) ) ) )).

fof(t16_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ( ( v2_seq_4(B)
              & r1_xreal_0(A,np__0) )
           => k3_pscomp_1(k1_integra2(B,A)) = k4_real_1(A,k4_pscomp_1(B)) ) ) ) )).

fof(t17_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,k1_numbers)
                & m2_relset_1(C,B,k1_numbers) )
             => k5_relset_1(B,k1_numbers,k13_seq_1(B,k1_numbers,C,A)) = k1_integra2(k1_pscomp_1(B,k1_numbers,C),A) ) ) ) )).

fof(t18_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,B,k1_numbers) )
                 => k5_relset_1(B,k1_numbers,k13_seq_1(B,k1_numbers,k2_partfun1(B,k1_numbers,D,C),A)) = k1_integra2(k5_relset_1(B,k1_numbers,k2_partfun1(B,k1_numbers,D,C)),A) ) ) ) ) )).

fof(t19_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_integra1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,k1_numbers)
                & m2_relset_1(C,B,k1_numbers) )
             => ! [D] :
                  ( m3_integra1(D,B,k8_integra1(B))
                 => ( ( r3_rfunct_1(C,B)
                      & r1_xreal_0(np__0,A) )
                   => r1_xreal_0(k4_real_1(k4_real_1(A,k4_pscomp_1(k1_pscomp_1(B,k1_numbers,C))),k3_integra1(B)),k2_seq_1(k8_integra1(B),k1_numbers,k9_integra1(B,k13_seq_1(B,k1_numbers,C,A)),D)) ) ) ) ) ) )).

fof(t20_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_integra1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,k1_numbers)
                & m2_relset_1(C,B,k1_numbers) )
             => ! [D] :
                  ( m3_integra1(D,B,k8_integra1(B))
                 => ( ( r3_rfunct_1(C,B)
                      & r1_xreal_0(A,np__0) )
                   => r1_xreal_0(k4_real_1(k4_real_1(A,k3_pscomp_1(k1_pscomp_1(B,k1_numbers,C))),k3_integra1(B)),k2_seq_1(k8_integra1(B),k1_numbers,k9_integra1(B,k13_seq_1(B,k1_numbers,C,A)),D)) ) ) ) ) ) )).

fof(t21_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_integra1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,k1_numbers)
                & m2_relset_1(C,B,k1_numbers) )
             => ! [D] :
                  ( m3_integra1(D,B,k8_integra1(B))
                 => ( ( r3_rfunct_1(C,B)
                      & r1_xreal_0(np__0,A) )
                   => r1_xreal_0(k2_seq_1(k8_integra1(B),k1_numbers,k10_integra1(B,k13_seq_1(B,k1_numbers,C,A)),D),k4_real_1(k4_real_1(A,k3_pscomp_1(k1_pscomp_1(B,k1_numbers,C))),k3_integra1(B))) ) ) ) ) ) )).

fof(t22_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_integra1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,k1_numbers)
                & m2_relset_1(C,B,k1_numbers) )
             => ! [D] :
                  ( m3_integra1(D,B,k8_integra1(B))
                 => ( ( r3_rfunct_1(C,B)
                      & r1_xreal_0(A,np__0) )
                   => r1_xreal_0(k2_seq_1(k8_integra1(B),k1_numbers,k10_integra1(B,k13_seq_1(B,k1_numbers,C,A)),D),k4_real_1(k4_real_1(A,k4_pscomp_1(k1_pscomp_1(B,k1_numbers,C))),k3_integra1(B))) ) ) ) ) ) )).

fof(t23_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_integra1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,k1_numbers)
                & m2_relset_1(C,B,k1_numbers) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m2_integra1(D,B) )
                 => ! [E] :
                      ( m3_integra1(E,B,D)
                     => ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => ( ( r2_hidden(F,k2_finseq_1(k3_finseq_1(E)))
                              & r1_rfunct_1(C,B)
                              & r1_xreal_0(np__0,A) )
                           => k2_seq_1(k5_numbers,k1_numbers,k14_integra1(B,k13_seq_1(B,k1_numbers,C,A),D,E),F) = k4_real_1(A,k2_seq_1(k5_numbers,k1_numbers,k14_integra1(B,C,D,E),F)) ) ) ) ) ) ) ) )).

fof(t24_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_integra1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,k1_numbers)
                & m2_relset_1(C,B,k1_numbers) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m2_integra1(D,B) )
                 => ! [E] :
                      ( m3_integra1(E,B,D)
                     => ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => ( ( r2_hidden(F,k2_finseq_1(k3_finseq_1(E)))
                              & r1_rfunct_1(C,B)
                              & r1_xreal_0(A,np__0) )
                           => k2_seq_1(k5_numbers,k1_numbers,k15_integra1(B,k13_seq_1(B,k1_numbers,C,A),D,E),F) = k4_real_1(A,k2_seq_1(k5_numbers,k1_numbers,k14_integra1(B,C,D,E),F)) ) ) ) ) ) ) ) )).

fof(t25_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_integra1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,k1_numbers)
                & m2_relset_1(C,B,k1_numbers) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m2_integra1(D,B) )
                 => ! [E] :
                      ( m3_integra1(E,B,D)
                     => ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => ( ( r2_hidden(F,k2_finseq_1(k3_finseq_1(E)))
                              & r2_rfunct_1(C,B)
                              & r1_xreal_0(np__0,A) )
                           => k2_seq_1(k5_numbers,k1_numbers,k15_integra1(B,k13_seq_1(B,k1_numbers,C,A),D,E),F) = k4_real_1(A,k2_seq_1(k5_numbers,k1_numbers,k15_integra1(B,C,D,E),F)) ) ) ) ) ) ) ) )).

fof(t26_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_integra1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,k1_numbers)
                & m2_relset_1(C,B,k1_numbers) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m2_integra1(D,B) )
                 => ! [E] :
                      ( m3_integra1(E,B,D)
                     => ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => ( ( r2_hidden(F,k2_finseq_1(k3_finseq_1(E)))
                              & r2_rfunct_1(C,B)
                              & r1_xreal_0(A,np__0) )
                           => k2_seq_1(k5_numbers,k1_numbers,k14_integra1(B,k13_seq_1(B,k1_numbers,C,A),D,E),F) = k4_real_1(A,k2_seq_1(k5_numbers,k1_numbers,k15_integra1(B,C,D,E),F)) ) ) ) ) ) ) ) )).

fof(t27_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_integra1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,k1_numbers)
                & m2_relset_1(C,B,k1_numbers) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m2_integra1(D,B) )
                 => ! [E] :
                      ( m3_integra1(E,B,D)
                     => ( ( r1_rfunct_1(C,B)
                          & r1_xreal_0(np__0,A) )
                       => k6_integra1(B,k13_seq_1(B,k1_numbers,C,A),D,E) = k4_real_1(A,k6_integra1(B,C,D,E)) ) ) ) ) ) ) )).

fof(t28_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_integra1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,k1_numbers)
                & m2_relset_1(C,B,k1_numbers) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m2_integra1(D,B) )
                 => ! [E] :
                      ( m3_integra1(E,B,D)
                     => ( ( r1_rfunct_1(C,B)
                          & r1_xreal_0(A,np__0) )
                       => k7_integra1(B,k13_seq_1(B,k1_numbers,C,A),D,E) = k4_real_1(A,k6_integra1(B,C,D,E)) ) ) ) ) ) ) )).

fof(t29_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_integra1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,k1_numbers)
                & m2_relset_1(C,B,k1_numbers) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m2_integra1(D,B) )
                 => ! [E] :
                      ( m3_integra1(E,B,D)
                     => ( ( r2_rfunct_1(C,B)
                          & r1_xreal_0(np__0,A) )
                       => k7_integra1(B,k13_seq_1(B,k1_numbers,C,A),D,E) = k4_real_1(A,k7_integra1(B,C,D,E)) ) ) ) ) ) ) )).

fof(t30_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_integra1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,k1_numbers)
                & m2_relset_1(C,B,k1_numbers) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m2_integra1(D,B) )
                 => ! [E] :
                      ( m3_integra1(E,B,D)
                     => ( ( r2_rfunct_1(C,B)
                          & r1_xreal_0(A,np__0) )
                       => k6_integra1(B,k13_seq_1(B,k1_numbers,C,A),D,E) = k4_real_1(A,k7_integra1(B,C,D,E)) ) ) ) ) ) ) )).

fof(t31_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_integra1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,k1_numbers)
                & m2_relset_1(C,B,k1_numbers) )
             => ( ( r3_rfunct_1(C,B)
                  & r3_integra1(B,C) )
               => ( r3_integra1(B,k13_seq_1(B,k1_numbers,C,A))
                  & k13_integra1(B,k13_seq_1(B,k1_numbers,C,A)) = k4_real_1(A,k13_integra1(B,C)) ) ) ) ) ) )).

fof(t32_integra2,axiom,(
    ! [A] :
      ( ( v1_integra1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,A,k1_numbers)
            & m2_relset_1(B,A,k1_numbers) )
         => ( ( r3_rfunct_1(B,A)
              & r3_integra1(A,B)
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ( r2_hidden(C,A)
                   => r1_xreal_0(np__0,k2_seq_1(A,k1_numbers,B,C)) ) ) )
           => r1_xreal_0(np__0,k13_integra1(A,B)) ) ) ) )).

fof(t33_integra2,axiom,(
    ! [A] :
      ( ( v1_integra1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,A,k1_numbers)
            & m2_relset_1(B,A,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,A,k1_numbers)
                & m2_relset_1(C,A,k1_numbers) )
             => ( ( r3_rfunct_1(B,A)
                  & r3_integra1(A,B)
                  & r3_rfunct_1(C,A)
                  & r3_integra1(A,C) )
               => ( r3_integra1(A,k7_seq_1(A,k1_numbers,B,C))
                  & k13_integra1(A,k7_seq_1(A,k1_numbers,B,C)) = k5_real_1(k13_integra1(A,B),k13_integra1(A,C)) ) ) ) ) ) )).

fof(t34_integra2,axiom,(
    ! [A] :
      ( ( v1_integra1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,A,k1_numbers)
            & m2_relset_1(B,A,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,A,k1_numbers)
                & m2_relset_1(C,A,k1_numbers) )
             => ( ( r3_rfunct_1(B,A)
                  & r3_integra1(A,B)
                  & r3_rfunct_1(C,A)
                  & r3_integra1(A,C)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ( r2_hidden(D,A)
                       => r1_xreal_0(k2_seq_1(A,k1_numbers,C,D),k2_seq_1(A,k1_numbers,B,D)) ) ) )
               => r1_xreal_0(k13_integra1(A,C),k13_integra1(A,B)) ) ) ) ) )).

fof(t35_integra2,axiom,(
    ! [A] :
      ( ( v1_integra1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,A,k1_numbers)
            & m2_relset_1(B,A,k1_numbers) )
         => ( r3_rfunct_1(B,A)
           => v2_seq_4(k5_relset_1(k8_integra1(A),k1_numbers,k9_integra1(A,B))) ) ) ) )).

fof(t36_integra2,axiom,(
    ! [A] :
      ( ( v1_integra1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,A,k1_numbers)
            & m2_relset_1(B,A,k1_numbers) )
         => ( r3_rfunct_1(B,A)
           => v1_seq_4(k5_relset_1(k8_integra1(A),k1_numbers,k10_integra1(A,B))) ) ) ) )).

fof(d3_integra2,axiom,(
    ! [A] :
      ( ( v1_integra1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k8_integra1(A))
            & m2_relset_1(B,k5_numbers,k8_integra1(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( C = k2_integra2(A,B)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_seq_1(k5_numbers,k1_numbers,C,D) = k17_integra1(A,k8_funct_2(k5_numbers,k8_integra1(A),B,D)) ) ) ) ) ) )).

fof(d4_integra2,axiom,(
    ! [A] :
      ( ( v1_integra1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k8_integra1(A))
                & m2_relset_1(C,k5_numbers,k8_integra1(A)) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,k1_numbers)
                    & m2_relset_1(D,k5_numbers,k1_numbers) )
                 => ( D = k3_integra2(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_seq_1(k5_numbers,k1_numbers,D,E) = k6_integra1(A,B,k8_integra1(A),k8_funct_2(k5_numbers,k8_integra1(A),C,E)) ) ) ) ) ) ) )).

fof(d5_integra2,axiom,(
    ! [A] :
      ( ( v1_integra1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k8_integra1(A))
                & m2_relset_1(C,k5_numbers,k8_integra1(A)) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,k1_numbers)
                    & m2_relset_1(D,k5_numbers,k1_numbers) )
                 => ( D = k4_integra2(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_seq_1(k5_numbers,k1_numbers,D,E) = k7_integra1(A,B,k8_integra1(A),k8_funct_2(k5_numbers,k8_integra1(A),C,E)) ) ) ) ) ) ) )).

fof(t37_integra2,axiom,(
    ! [A] :
      ( ( v1_integra1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( m3_integra1(B,A,k8_integra1(A))
         => ! [C] :
              ( m3_integra1(C,A,k8_integra1(A))
             => ( r4_integra1(A,k8_integra1(A),B,C)
               => ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ~ ( r2_hidden(D,k4_finseq_1(C))
                        & ! [E] :
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                           => ~ ( r2_hidden(E,k4_finseq_1(B))
                                & r1_tarski(k2_integra1(A,k8_integra1(A),C,D),k2_integra1(A,k8_integra1(A),B,E)) ) ) ) ) ) ) ) ) )).

fof(t38_integra2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_finset_1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ( r1_tarski(A,B)
           => r1_xreal_0(k1_pre_circ(A),k1_pre_circ(B)) ) ) ) )).

fof(t39_integra2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_finset_1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ( ? [C] :
                ( m1_subset_1(C,k1_numbers)
                & r2_hidden(C,B)
                & r1_xreal_0(k1_pre_circ(A),C) )
           => r1_xreal_0(k1_pre_circ(A),k1_pre_circ(B)) ) ) ) )).

fof(t40_integra2,axiom,(
    ! [A] :
      ( ( v1_integra1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( v1_integra1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ( r1_tarski(A,B)
           => r1_xreal_0(k3_integra1(A),k3_integra1(B)) ) ) ) )).

fof(t41_integra2,axiom,(
    ! [A] :
      ( ( v1_integra1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( m3_integra1(B,A,k8_integra1(A))
         => ! [C] :
              ( m3_integra1(C,A,k8_integra1(A))
             => ( r4_integra1(A,k8_integra1(A),B,C)
               => r1_xreal_0(k17_integra1(A,C),k17_integra1(A,B)) ) ) ) ) )).

fof(dt_k1_integra2,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
        & v1_xreal_0(B) )
     => m1_subset_1(k1_integra2(A,B),k1_zfmisc_1(k1_numbers)) ) )).

fof(dt_k2_integra2,axiom,(
    ! [A,B] :
      ( ( v1_integra1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers))
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k8_integra1(A))
        & m1_relset_1(B,k5_numbers,k8_integra1(A)) )
     => ( v1_funct_1(k2_integra2(A,B))
        & v1_funct_2(k2_integra2(A,B),k5_numbers,k1_numbers)
        & m2_relset_1(k2_integra2(A,B),k5_numbers,k1_numbers) ) ) )).

fof(dt_k3_integra2,axiom,(
    ! [A,B,C] :
      ( ( v1_integra1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers))
        & v1_funct_1(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,k8_integra1(A))
        & m1_relset_1(C,k5_numbers,k8_integra1(A)) )
     => ( v1_funct_1(k3_integra2(A,B,C))
        & v1_funct_2(k3_integra2(A,B,C),k5_numbers,k1_numbers)
        & m2_relset_1(k3_integra2(A,B,C),k5_numbers,k1_numbers) ) ) )).

fof(dt_k4_integra2,axiom,(
    ! [A,B,C] :
      ( ( v1_integra1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers))
        & v1_funct_1(B)
        & m1_relset_1(B,A,k1_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,k8_integra1(A))
        & m1_relset_1(C,k5_numbers,k8_integra1(A)) )
     => ( v1_funct_1(k4_integra2(A,B,C))
        & v1_funct_2(k4_integra2(A,B,C),k5_numbers,k1_numbers)
        & m2_relset_1(k4_integra2(A,B,C),k5_numbers,k1_numbers) ) ) )).

fof(t8_integra2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
         => k1_integra2(B,A) = a_2_0_integra2(A,B) ) ) )).

fof(fraenkel_a_2_0_integra2,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(B,k1_numbers)
        & m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
     => ( r2_hidden(A,a_2_0_integra2(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,k1_numbers)
            & A = k4_real_1(B,D)
            & r2_hidden(D,C) ) ) ) )).
%------------------------------------------------------------------------------