SET007 Axioms: SET007+181.ax


%------------------------------------------------------------------------------
% File     : SET007+181 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Measurability of Extended Real Valued Functions
% 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    : mesfunc2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   47 (   4 unit)
%            Number of atoms       :  371 (  43 equality)
%            Maximal formula depth :   22 (   9 average)
%            Number of connectives :  388 (  64 ~  ;   5  |; 151  &)
%                                         (   4 <=>; 164 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   26 (   1 propositional; 0-4 arity)
%            Number of functors    :   41 (  10 constant; 0-4 arity)
%            Number of variables   :  150 (   0 singleton; 147 !;   3 ?)
%            Maximal term depth    :    6 (   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_mesfunc2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v1_prob_1(B,A)
        & v1_measure1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => ? [C] :
          ( m1_finseq_1(C,B)
          & v1_relat_1(C)
          & v1_funct_1(C)
          & v1_finset_1(C)
          & v1_finseq_1(C)
          & v1_prob_2(C) ) ) )).

fof(d1_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => ( r1_mesfunc2(A,B)
          <=> ! [C] :
                ( m1_subset_1(C,A)
               => ~ ( r2_hidden(C,k1_relat_1(B))
                    & r1_supinf_1(k5_measure6,k4_extreal1(k4_mesfunc1(A,B,C))) ) ) ) ) ) )).

fof(t1_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => B = k8_mesfunc1(A,B,np__1) ) ) )).

fof(t2_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,A,k3_supinf_1) )
             => ( ( r1_mesfunc2(A,B)
                  | r1_mesfunc2(A,C) )
               => ( k1_relat_1(k13_mesfunc1(A,B,C)) = k3_xboole_0(k1_relat_1(B),k1_relat_1(C))
                  & k1_relat_1(k6_mesfunc1(A,B,C)) = k3_xboole_0(k1_relat_1(B),k1_relat_1(C)) ) ) ) ) ) )).

fof(t3_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_prob_1(B,A)
            & v1_measure1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,A,k3_supinf_1) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,A,k3_supinf_1) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k3_numbers,B)
                        & m2_relset_1(E,k3_numbers,B) )
                     => ! [F] :
                          ( m1_subset_1(F,k1_numbers)
                         => ! [G] :
                              ( m1_subset_1(G,B)
                             => ( ( r1_mesfunc2(A,C)
                                  & r1_mesfunc2(A,D)
                                  & ! [H] :
                                      ( v1_rat_1(H)
                                     => k1_funct_1(E,H) = k3_xboole_0(k3_xboole_0(G,k15_mesfunc1(A,C,k1_measure6(H))),k3_xboole_0(G,k15_mesfunc1(A,D,k1_measure6(k6_xcmplx_0(F,H))))) ) )
                               => k3_xboole_0(G,k15_mesfunc1(A,k13_mesfunc1(A,C,D),k1_measure6(F))) = k3_tarski(k2_relat_1(E)) ) ) ) ) ) ) ) ) )).

fof(t4_mesfunc2,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k5_numbers,k3_numbers)
      & m2_relset_1(A,k5_numbers,k3_numbers)
      & v2_funct_1(A)
      & k1_relat_1(A) = k5_numbers
      & k2_relat_1(A) = k3_numbers ) )).

fof(t5_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,A,C)
                    & m2_relset_1(D,A,C) )
                 => ~ ( r2_wellord2(A,B)
                      & ! [E] :
                          ( ( v1_funct_1(E)
                            & v1_funct_2(E,B,C)
                            & m2_relset_1(E,B,C) )
                         => k2_relat_1(D) != k2_relat_1(E) ) ) ) ) ) ) )).

fof(t6_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v1_prob_1(C,A)
                & v1_measure1(C,A)
                & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,A,k3_supinf_1) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & m2_relset_1(E,A,k3_supinf_1) )
                     => ! [F] :
                          ( m1_subset_1(F,C)
                         => ~ ( r2_mesfunc1(A,C,D,F)
                              & r2_mesfunc1(A,C,E,F)
                              & ! [G] :
                                  ( ( v1_funct_1(G)
                                    & v1_funct_2(G,k3_numbers,C)
                                    & m2_relset_1(G,k3_numbers,C) )
                                 => ~ ! [H] :
                                        ( v1_rat_1(H)
                                       => k1_funct_1(G,H) = k3_xboole_0(k3_xboole_0(F,k15_mesfunc1(A,D,k1_measure6(H))),k3_xboole_0(F,k15_mesfunc1(A,E,k1_measure6(k6_xcmplx_0(B,H))))) ) ) ) ) ) ) ) ) ) )).

fof(t7_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_prob_1(B,A)
            & v1_measure1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,A,k3_supinf_1) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,A,k3_supinf_1) )
                 => ! [E] :
                      ( m1_subset_1(E,B)
                     => ( ( r1_mesfunc2(A,C)
                          & r1_mesfunc2(A,D)
                          & r2_mesfunc1(A,B,C,E)
                          & r2_mesfunc1(A,B,D,E) )
                       => r2_mesfunc1(A,B,k13_mesfunc1(A,C,D),E) ) ) ) ) ) ) )).

fof(t8_mesfunc2,axiom,(
    $true )).

fof(t9_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,A,k3_supinf_1) )
             => k6_mesfunc1(A,B,C) = k13_mesfunc1(A,B,k9_mesfunc1(A,C)) ) ) ) )).

fof(t10_mesfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => k1_measure6(k1_real_1(A)) = k3_supinf_2(k1_measure6(A)) ) )).

fof(t11_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => k9_mesfunc1(A,B) = k8_mesfunc1(A,B,k1_real_1(np__1)) ) ) )).

fof(t12_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( r1_mesfunc2(A,B)
               => r1_mesfunc2(A,k8_mesfunc1(A,B,C)) ) ) ) ) )).

fof(t13_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_prob_1(B,A)
            & v1_measure1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,A,k3_supinf_1) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,A,k3_supinf_1) )
                 => ! [E] :
                      ( m1_subset_1(E,B)
                     => ( ( r1_mesfunc2(A,C)
                          & r1_mesfunc2(A,D)
                          & r2_mesfunc1(A,B,C,E)
                          & r2_mesfunc1(A,B,D,E)
                          & r1_tarski(E,k1_relat_1(D)) )
                       => r2_mesfunc1(A,B,k6_mesfunc1(A,C,D),E) ) ) ) ) ) ) )).

fof(d2_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,A,k3_supinf_1) )
             => ( C = k1_mesfunc2(A,B)
              <=> ( k1_relat_1(C) = k1_relat_1(B)
                  & ! [D] :
                      ( m1_subset_1(D,A)
                     => ( r2_hidden(D,k1_relat_1(C))
                       => k4_mesfunc1(A,C,D) = k2_extreal2(k4_mesfunc1(A,B,D),k1_supinf_2) ) ) ) ) ) ) ) )).

fof(d3_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,A,k3_supinf_1) )
             => ( C = k2_mesfunc2(A,B)
              <=> ( k1_relat_1(C) = k1_relat_1(B)
                  & ! [D] :
                      ( m1_subset_1(D,A)
                     => ( r2_hidden(D,k1_relat_1(C))
                       => k4_mesfunc1(A,C,D) = k2_extreal2(k3_supinf_2(k4_mesfunc1(A,B,D)),k1_supinf_2) ) ) ) ) ) ) ) )).

fof(t14_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ( r2_hidden(C,k1_relat_1(B))
               => r1_supinf_1(k1_supinf_2,k4_mesfunc1(A,k1_mesfunc2(A,B),C)) ) ) ) ) )).

fof(t15_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ( r2_hidden(C,k1_relat_1(B))
               => r1_supinf_1(k1_supinf_2,k4_mesfunc1(A,k2_mesfunc2(A,B),C)) ) ) ) ) )).

fof(t16_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => k2_mesfunc2(A,B) = k1_mesfunc2(A,k9_mesfunc1(A,B)) ) ) )).

fof(t17_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ( r2_hidden(C,k1_relat_1(B))
               => ( r1_supinf_1(k4_mesfunc1(A,k1_mesfunc2(A,B),C),k1_supinf_2)
                  | k4_mesfunc1(A,k2_mesfunc2(A,B),C) = k1_supinf_2 ) ) ) ) ) )).

fof(t18_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ( r2_hidden(C,k1_relat_1(B))
               => ( r1_supinf_1(k4_mesfunc1(A,k2_mesfunc2(A,B),C),k1_supinf_2)
                  | k4_mesfunc1(A,k1_mesfunc2(A,B),C) = k1_supinf_2 ) ) ) ) ) )).

fof(t19_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => ( k1_relat_1(B) = k1_relat_1(k6_mesfunc1(A,k1_mesfunc2(A,B),k2_mesfunc2(A,B)))
            & k1_relat_1(B) = k1_relat_1(k13_mesfunc1(A,k1_mesfunc2(A,B),k2_mesfunc2(A,B))) ) ) ) )).

fof(t20_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ( r2_hidden(C,k1_relat_1(B))
               => ( ( k4_mesfunc1(A,k1_mesfunc2(A,B),C) = k4_mesfunc1(A,B,C)
                    | k4_mesfunc1(A,k1_mesfunc2(A,B),C) = k1_supinf_2 )
                  & ( k4_mesfunc1(A,k2_mesfunc2(A,B),C) = k3_supinf_2(k4_mesfunc1(A,B,C))
                    | k4_mesfunc1(A,k2_mesfunc2(A,B),C) = k1_supinf_2 ) ) ) ) ) ) )).

fof(t21_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ( ( r2_hidden(C,k1_relat_1(B))
                  & k4_mesfunc1(A,k1_mesfunc2(A,B),C) = k4_mesfunc1(A,B,C) )
               => k4_mesfunc1(A,k2_mesfunc2(A,B),C) = k1_supinf_2 ) ) ) ) )).

fof(t22_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ( ( r2_hidden(C,k1_relat_1(B))
                  & k4_mesfunc1(A,k1_mesfunc2(A,B),C) = k1_supinf_2 )
               => k4_mesfunc1(A,k2_mesfunc2(A,B),C) = k3_supinf_2(k4_mesfunc1(A,B,C)) ) ) ) ) )).

fof(t23_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ( ( r2_hidden(C,k1_relat_1(B))
                  & k4_mesfunc1(A,k2_mesfunc2(A,B),C) = k3_supinf_2(k4_mesfunc1(A,B,C)) )
               => k4_mesfunc1(A,k1_mesfunc2(A,B),C) = k1_supinf_2 ) ) ) ) )).

fof(t24_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ( ( r2_hidden(C,k1_relat_1(B))
                  & k4_mesfunc1(A,k2_mesfunc2(A,B),C) = k1_supinf_2 )
               => k4_mesfunc1(A,k1_mesfunc2(A,B),C) = k4_mesfunc1(A,B,C) ) ) ) ) )).

fof(t25_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => B = k6_mesfunc1(A,k1_mesfunc2(A,B),k2_mesfunc2(A,B)) ) ) )).

fof(t26_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => k12_mesfunc1(A,B) = k13_mesfunc1(A,k1_mesfunc2(A,B),k2_mesfunc2(A,B)) ) ) )).

fof(t27_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v1_prob_1(C,A)
                & v1_measure1(C,A)
                & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
             => ! [D] :
                  ( m1_subset_1(D,C)
                 => ( r2_mesfunc1(A,C,B,D)
                   => r2_mesfunc1(A,C,k1_mesfunc2(A,B),D) ) ) ) ) ) )).

fof(t28_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v1_prob_1(C,A)
                & v1_measure1(C,A)
                & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
             => ! [D] :
                  ( m1_subset_1(D,C)
                 => ( ( r2_mesfunc1(A,C,B,D)
                      & r1_tarski(D,k1_relat_1(B)) )
                   => r2_mesfunc1(A,C,k2_mesfunc2(A,B),D) ) ) ) ) ) )).

fof(t29_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_prob_1(B,A)
            & v1_measure1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,A,k3_supinf_1) )
             => ! [D] :
                  ( m1_subset_1(D,B)
                 => ( ( r2_mesfunc1(A,B,C,D)
                      & r1_tarski(D,k1_relat_1(C)) )
                   => r2_mesfunc1(A,B,k12_mesfunc1(A,C),D) ) ) ) ) ) )).

fof(t30_mesfunc2,axiom,(
    ! [A,B] : r1_tarski(k2_relat_1(k5_funct_3(A,B)),k12_supinf_1(k1_supinf_2,k10_mesfunc1)) )).

fof(t31_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_prob_1(B,A)
            & v1_measure1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => ! [C] :
              ( m1_subset_1(C,B)
             => r1_mesfunc2(A,k3_mesfunc2(C,A)) ) ) ) )).

fof(t32_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_prob_1(B,A)
            & v1_measure1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => ! [C] :
              ( m1_subset_1(C,B)
             => ! [D] :
                  ( m1_subset_1(D,B)
                 => r2_mesfunc1(A,B,k3_mesfunc2(C,A),D) ) ) ) ) )).

fof(t33_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_prob_1(B,A)
            & v1_measure1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ~ ( v1_prob_2(C)
                  & m2_finseq_1(C,B)
                  & ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,k5_numbers,B)
                        & v1_prob_2(D)
                        & m2_relset_1(D,k5_numbers,B) )
                     => ~ ( k3_tarski(k2_relat_1(C)) = k2_measure2(A,B,k1_measure3(A,B,D))
                          & ! [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ( r2_hidden(E,k1_relat_1(C))
                               => k1_funct_1(C,E) = k8_funct_2(k5_numbers,B,D,E) ) )
                          & ! [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ( ~ r2_hidden(E,k1_relat_1(C))
                               => k8_funct_2(k5_numbers,B,D,E) = k1_xboole_0 ) ) ) ) ) ) ) ) )).

fof(t34_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_prob_1(B,A)
            & v1_measure1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ( ( v1_prob_2(C)
                  & m2_finseq_1(C,B) )
               => r2_hidden(k3_tarski(k2_relat_1(C)),B) ) ) ) ) )).

fof(d4_mesfunc2,axiom,(
    $true )).

fof(d5_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_prob_1(B,A)
            & v1_measure1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,A,k3_supinf_1) )
             => ( r2_mesfunc2(A,B,C)
              <=> ( r1_mesfunc2(A,C)
                  & ? [D] :
                      ( v1_prob_2(D)
                      & m2_finseq_1(D,B)
                      & k1_relat_1(C) = k3_tarski(k2_relat_1(D))
                      & ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ! [F] :
                              ( m1_subset_1(F,A)
                             => ! [G] :
                                  ( m1_subset_1(G,A)
                                 => ( ( r2_hidden(E,k4_finseq_1(D))
                                      & r2_hidden(F,k1_funct_1(D,E))
                                      & r2_hidden(G,k1_funct_1(D,E)) )
                                   => k4_mesfunc1(A,C,F) = k4_mesfunc1(A,C,G) ) ) ) ) ) ) ) ) ) ) )).

fof(t35_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => ( r1_mesfunc2(A,B)
           => m1_subset_1(k2_relat_1(B),k1_zfmisc_1(k6_supinf_1)) ) ) ) )).

fof(t36_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_prob_1(B,A)
            & v1_measure1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( v1_relat_1(D)
                 => ( ( v1_prob_2(D)
                      & m2_finseq_1(D,B) )
                   => ( v1_prob_2(k7_relat_1(D,k2_finseq_1(C)))
                      & m2_finseq_1(k7_relat_1(D,k2_finseq_1(C)),B) ) ) ) ) ) ) )).

fof(t37_mesfunc2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k3_supinf_1) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v1_prob_1(C,A)
                & v1_measure1(C,A)
                & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
             => ! [D] :
                  ( m1_subset_1(D,C)
                 => ( r2_mesfunc2(A,C,B)
                   => r2_mesfunc1(A,C,B,D) ) ) ) ) ) )).

fof(dt_k1_mesfunc2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & m1_relset_1(B,A,k3_supinf_1) )
     => ( v1_funct_1(k1_mesfunc2(A,B))
        & m2_relset_1(k1_mesfunc2(A,B),A,k3_supinf_1) ) ) )).

fof(dt_k2_mesfunc2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & m1_relset_1(B,A,k3_supinf_1) )
     => ( v1_funct_1(k2_mesfunc2(A,B))
        & m2_relset_1(k2_mesfunc2(A,B),A,k3_supinf_1) ) ) )).

fof(dt_k3_mesfunc2,axiom,(
    ! [A,B] :
      ( v1_funct_1(k3_mesfunc2(A,B))
      & m2_relset_1(k3_mesfunc2(A,B),B,k3_supinf_1) ) )).

fof(redefinition_k3_mesfunc2,axiom,(
    ! [A,B] : k3_mesfunc2(A,B) = k4_funct_3(A,B) )).
%------------------------------------------------------------------------------