SET007 Axioms: SET007+156.ax


%------------------------------------------------------------------------------
% File     : SET007+156 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Several Properties of the sigma-additive Measure
% 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    : measure2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   38 (   4 unit)
%            Number of atoms       :  390 (  42 equality)
%            Maximal formula depth :   17 (  10 average)
%            Number of connectives :  390 (  38 ~  ;   0  |; 214  &)
%                                         (   4 <=>; 134 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   24 (   1 propositional; 0-4 arity)
%            Number of functors    :   26 (   6 constant; 0-5 arity)
%            Number of variables   :  161 (   0 singleton; 149 !;  12 ?)
%            Maximal term depth    :    4 (   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_measure2,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_measure2(C,A,B)
          & ~ v1_xboole_0(C)
          & v1_card_4(C)
          & v1_measure2(C,A,B) ) ) )).

fof(rc2_measure2,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_measure2(C,A,B)
          & ~ v1_xboole_0(C)
          & v1_card_4(C)
          & v2_measure2(C,A,B) ) ) )).

fof(t1_measure2,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] :
          ( m3_measure1(C,A,B)
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,k5_numbers,B)
                & m2_relset_1(D,k5_numbers,B) )
             => v6_supinf_2(k7_funct_2(k5_numbers,B,k3_supinf_1,D,C),k5_numbers) ) ) ) )).

fof(d1_measure2,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] :
          ( ( ~ v1_xboole_0(C)
            & v1_card_4(C)
            & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => ( m1_measure2(C,A,B)
          <=> r1_tarski(C,B) ) ) ) )).

fof(t2_measure2,axiom,(
    $true )).

fof(t3_measure2,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_measure2(C,A,B)
         => ( r2_hidden(k6_setfam_1(A,C),B)
            & r2_hidden(k5_setfam_1(A,C),B) ) ) ) )).

fof(t4_measure2,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] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k5_numbers,B)
            & m2_relset_1(C,k5_numbers,B) )
         => ? [D] :
              ( v1_funct_1(D)
              & v1_funct_2(D,k5_numbers,B)
              & m2_relset_1(D,k5_numbers,B)
              & k8_funct_2(k5_numbers,B,D,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
              & ! [E] :
                  ( m2_subset_1(E,k1_numbers,k5_numbers)
                 => k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1)),k8_funct_2(k5_numbers,B,C,E)) ) ) ) ) )).

fof(t5_measure2,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] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k5_numbers,B)
            & m2_relset_1(C,k5_numbers,B) )
         => ? [D] :
              ( v1_funct_1(D)
              & v1_funct_2(D,k5_numbers,B)
              & m2_relset_1(D,k5_numbers,B)
              & k8_funct_2(k5_numbers,B,D,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
              & ! [E] :
                  ( m2_subset_1(E,k1_numbers,k5_numbers)
                 => k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)) = k2_measure1(A,B,k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1)),k8_funct_2(k5_numbers,B,D,E)) ) ) ) ) )).

fof(t6_measure2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k5_numbers,B)
            & m2_relset_1(C,k5_numbers,B) )
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,k5_numbers,B)
                & m2_relset_1(D,k5_numbers,B) )
             => ( ( k8_funct_2(k5_numbers,B,D,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
                  & ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)) = k2_xboole_0(k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1)),k8_funct_2(k5_numbers,B,D,E)) ) )
               => ! [E,F] :
                    ( m2_subset_1(F,k1_numbers,k5_numbers)
                   => ( r2_hidden(E,k8_funct_2(k5_numbers,B,D,F))
                    <=> ? [G] :
                          ( m2_subset_1(G,k1_numbers,k5_numbers)
                          & r1_xreal_0(G,F)
                          & r2_hidden(E,k8_funct_2(k5_numbers,B,C,G)) ) ) ) ) ) ) ) )).

fof(t7_measure2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k5_numbers,B)
            & m2_relset_1(C,k5_numbers,B) )
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,k5_numbers,B)
                & m2_relset_1(D,k5_numbers,B) )
             => ( ( k8_funct_2(k5_numbers,B,D,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
                  & ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)) = k2_xboole_0(k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1)),k8_funct_2(k5_numbers,B,D,E)) ) )
               => ! [E] :
                    ( m2_subset_1(E,k1_numbers,k5_numbers)
                   => ! [F] :
                        ( m2_subset_1(F,k1_numbers,k5_numbers)
                       => ( ~ r1_xreal_0(F,E)
                         => r1_tarski(k8_funct_2(k5_numbers,B,D,E),k8_funct_2(k5_numbers,B,D,F)) ) ) ) ) ) ) ) )).

fof(t8_measure2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k5_numbers,B)
            & m2_relset_1(C,k5_numbers,B) )
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,k5_numbers,B)
                & m2_relset_1(D,k5_numbers,B) )
             => ! [E] :
                  ( ( v1_funct_1(E)
                    & v1_funct_2(E,k5_numbers,B)
                    & m2_relset_1(E,k5_numbers,B) )
                 => ( ( k8_funct_2(k5_numbers,B,D,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
                      & ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => k8_funct_2(k5_numbers,B,D,k1_nat_1(F,np__1)) = k2_xboole_0(k8_funct_2(k5_numbers,B,C,k1_nat_1(F,np__1)),k8_funct_2(k5_numbers,B,D,F)) )
                      & k8_funct_2(k5_numbers,B,E,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
                      & ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => k8_funct_2(k5_numbers,B,E,k1_nat_1(F,np__1)) = k4_xboole_0(k8_funct_2(k5_numbers,B,C,k1_nat_1(F,np__1)),k8_funct_2(k5_numbers,B,D,F)) ) )
                   => ! [F] :
                        ( m2_subset_1(F,k1_numbers,k5_numbers)
                       => ! [G] :
                            ( m2_subset_1(G,k1_numbers,k5_numbers)
                           => ( r1_xreal_0(F,G)
                             => r1_tarski(k8_funct_2(k5_numbers,B,E,F),k8_funct_2(k5_numbers,B,D,G)) ) ) ) ) ) ) ) ) )).

fof(t9_measure2,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] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k5_numbers,B)
            & m2_relset_1(C,k5_numbers,B) )
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,k5_numbers,B)
                & m2_relset_1(D,k5_numbers,B) )
             => ? [E] :
                  ( v1_funct_1(E)
                  & v1_funct_2(E,k5_numbers,B)
                  & m2_relset_1(E,k5_numbers,B)
                  & k8_funct_2(k5_numbers,B,E,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
                  & ! [F] :
                      ( m2_subset_1(F,k1_numbers,k5_numbers)
                     => k8_funct_2(k5_numbers,B,E,k1_nat_1(F,np__1)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,C,k1_nat_1(F,np__1)),k8_funct_2(k5_numbers,B,D,F)) ) ) ) ) ) )).

fof(t10_measure2,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] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k5_numbers,B)
            & m2_relset_1(C,k5_numbers,B) )
         => ? [D] :
              ( v1_funct_1(D)
              & v1_funct_2(D,k5_numbers,B)
              & m2_relset_1(D,k5_numbers,B)
              & k8_funct_2(k5_numbers,B,D,np__0) = k1_xboole_0
              & ! [E] :
                  ( m2_subset_1(E,k1_numbers,k5_numbers)
                 => k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,C,np__0),k8_funct_2(k5_numbers,B,C,E)) ) ) ) ) )).

fof(t11_measure2,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] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k5_numbers,B)
            & m2_relset_1(C,k5_numbers,B) )
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,k5_numbers,B)
                & m2_relset_1(D,k5_numbers,B) )
             => ! [E] :
                  ( ( v1_funct_1(E)
                    & v1_funct_2(E,k5_numbers,B)
                    & m2_relset_1(E,k5_numbers,B) )
                 => ( ( k8_funct_2(k5_numbers,B,D,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
                      & ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => k8_funct_2(k5_numbers,B,D,k1_nat_1(F,np__1)) = k2_measure1(A,B,k8_funct_2(k5_numbers,B,C,k1_nat_1(F,np__1)),k8_funct_2(k5_numbers,B,D,F)) )
                      & k8_funct_2(k5_numbers,B,E,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
                      & ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => k8_funct_2(k5_numbers,B,E,k1_nat_1(F,np__1)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,C,k1_nat_1(F,np__1)),k8_funct_2(k5_numbers,B,D,F)) ) )
                   => ! [F] :
                        ( m2_subset_1(F,k1_numbers,k5_numbers)
                       => ! [G] :
                            ( m2_subset_1(G,k1_numbers,k5_numbers)
                           => ( ~ r1_xreal_0(G,F)
                             => r1_xboole_0(k8_funct_2(k5_numbers,B,E,F),k8_funct_2(k5_numbers,B,E,G)) ) ) ) ) ) ) ) ) )).

fof(t12_measure2,axiom,(
    $true )).

fof(t13_measure2,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] :
          ( m3_measure1(C,A,B)
         => ! [D] :
              ( m1_measure2(D,A,B)
             => ! [E] :
                  ( ( v1_funct_1(E)
                    & v1_funct_2(E,k5_numbers,B)
                    & m2_relset_1(E,k5_numbers,B) )
                 => ( D = k5_measure1(A,B,E)
                   => r1_supinf_1(k1_measure1(B,C,k2_measure2(A,B,D)),k19_supinf_2(k7_funct_2(k5_numbers,B,k3_supinf_1,E,C))) ) ) ) ) ) )).

fof(t14_measure2,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_measure2(C,A,B)
         => ? [D] :
              ( v1_funct_1(D)
              & v1_funct_2(D,k5_numbers,B)
              & m2_relset_1(D,k5_numbers,B)
              & C = k5_measure1(A,B,D) ) ) ) )).

fof(t15_measure2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( k1_funct_1(B,np__0) = k1_xboole_0
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( k1_funct_1(B,k1_nat_1(C,np__1)) = k4_xboole_0(k1_funct_1(A,np__0),k1_funct_1(A,C))
                    & r1_tarski(k1_funct_1(A,k1_nat_1(C,np__1)),k1_funct_1(A,C)) ) ) )
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => r1_tarski(k1_funct_1(B,C),k1_funct_1(B,k1_nat_1(C,np__1))) ) ) ) ) )).

fof(t16_measure2,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] :
          ( m3_measure1(C,A,B)
         => ! [D] :
              ( m1_measure2(D,A,B)
             => ( ! [E] :
                    ( r2_hidden(E,D)
                   => m4_measure1(E,A,B,C) )
               => m4_measure1(k2_measure2(A,B,D),A,B,C) ) ) ) ) )).

fof(t17_measure2,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] :
          ( m3_measure1(C,A,B)
         => ! [D] :
              ( m1_measure2(D,A,B)
             => ( ? [E] :
                    ( r2_hidden(E,D)
                    & m4_measure1(E,A,B,C) )
               => m4_measure1(k1_measure2(A,B,D),A,B,C) ) ) ) ) )).

fof(t18_measure2,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] :
          ( m3_measure1(C,A,B)
         => ! [D] :
              ( m1_measure2(D,A,B)
             => ( ! [E] :
                    ( r2_hidden(E,D)
                   => m4_measure1(E,A,B,C) )
               => m4_measure1(k1_measure2(A,B,D),A,B,C) ) ) ) ) )).

fof(d2_measure2,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_measure2(C,A,B)
         => ( v1_measure2(C,A,B)
          <=> ? [D] :
                ( v1_funct_1(D)
                & v1_funct_2(D,k5_numbers,B)
                & m2_relset_1(D,k5_numbers,B)
                & C = k5_measure1(A,B,D)
                & ! [E] :
                    ( m2_subset_1(E,k1_numbers,k5_numbers)
                   => r1_tarski(k8_funct_2(k5_numbers,B,D,E),k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1))) ) ) ) ) ) )).

fof(d3_measure2,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_measure2(C,A,B)
         => ( v2_measure2(C,A,B)
          <=> ? [D] :
                ( v1_funct_1(D)
                & v1_funct_2(D,k5_numbers,B)
                & m2_relset_1(D,k5_numbers,B)
                & C = k5_measure1(A,B,D)
                & ! [E] :
                    ( m2_subset_1(E,k1_numbers,k5_numbers)
                   => r1_tarski(k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)),k8_funct_2(k5_numbers,B,D,E)) ) ) ) ) ) )).

fof(t19_measure2,axiom,(
    $true )).

fof(t20_measure2,axiom,(
    $true )).

fof(t21_measure2,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] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k5_numbers,B)
            & m2_relset_1(C,k5_numbers,B) )
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,k5_numbers,B)
                & m2_relset_1(D,k5_numbers,B) )
             => ( ( k8_funct_2(k5_numbers,B,D,np__0) = k1_xboole_0
                  & ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,C,np__0),k8_funct_2(k5_numbers,B,C,E))
                        & r1_tarski(k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1)),k8_funct_2(k5_numbers,B,C,E)) ) ) )
               => ( v1_measure2(k5_measure1(A,B,D),A,B)
                  & m1_measure2(k5_measure1(A,B,D),A,B) ) ) ) ) ) )).

fof(t22_measure2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => r1_tarski(k1_funct_1(A,B),k1_funct_1(A,k1_nat_1(B,np__1))) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( r1_xreal_0(C,B)
                 => r1_tarski(k1_funct_1(A,C),k1_funct_1(A,B)) ) ) ) ) ) )).

fof(t23_measure2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( k1_funct_1(B,np__0) = k1_funct_1(A,np__0)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( k1_funct_1(B,k1_nat_1(C,np__1)) = k4_xboole_0(k1_funct_1(A,k1_nat_1(C,np__1)),k1_funct_1(A,C))
                    & r1_tarski(k1_funct_1(A,C),k1_funct_1(A,k1_nat_1(C,np__1))) ) ) )
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( ~ r1_xreal_0(D,C)
                     => r1_xboole_0(k1_funct_1(B,C),k1_funct_1(B,D)) ) ) ) ) ) ) )).

fof(t24_measure2,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] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k5_numbers,B)
            & m2_relset_1(C,k5_numbers,B) )
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,k5_numbers,B)
                & m2_relset_1(D,k5_numbers,B) )
             => ( ( k8_funct_2(k5_numbers,B,D,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
                  & ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1)),k8_funct_2(k5_numbers,B,C,E))
                        & r1_tarski(k8_funct_2(k5_numbers,B,C,E),k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1))) ) ) )
               => k5_setfam_1(A,k5_measure1(A,B,D)) = k5_setfam_1(A,k5_measure1(A,B,C)) ) ) ) ) )).

fof(t25_measure2,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] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k5_numbers,B)
            & m2_relset_1(C,k5_numbers,B) )
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,k5_numbers,B)
                & m2_relset_1(D,k5_numbers,B) )
             => ( ( k8_funct_2(k5_numbers,B,D,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
                  & ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1)),k8_funct_2(k5_numbers,B,C,E))
                        & r1_tarski(k8_funct_2(k5_numbers,B,C,E),k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1))) ) ) )
               => ( v1_funct_1(D)
                  & v1_funct_2(D,k5_numbers,B)
                  & v1_prob_2(D)
                  & m2_relset_1(D,k5_numbers,B) ) ) ) ) ) )).

fof(t26_measure2,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] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k5_numbers,B)
            & m2_relset_1(C,k5_numbers,B) )
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,k5_numbers,B)
                & m2_relset_1(D,k5_numbers,B) )
             => ( ( k8_funct_2(k5_numbers,B,D,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
                  & ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)) = k4_measure1(A,B,k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1)),k8_funct_2(k5_numbers,B,C,E))
                        & r1_tarski(k8_funct_2(k5_numbers,B,C,E),k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1))) ) ) )
               => ( k8_funct_2(k5_numbers,B,C,np__0) = k8_funct_2(k5_numbers,B,D,np__0)
                  & ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1)) = k2_measure1(A,B,k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)),k8_funct_2(k5_numbers,B,C,E)) ) ) ) ) ) ) )).

fof(t27_measure2,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] :
          ( m3_measure1(C,A,B)
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,k5_numbers,B)
                & m2_relset_1(D,k5_numbers,B) )
             => ( ! [E] :
                    ( m2_subset_1(E,k1_numbers,k5_numbers)
                   => r1_tarski(k8_funct_2(k5_numbers,B,D,E),k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1))) )
               => k1_funct_1(C,k5_setfam_1(A,k5_measure1(A,B,D))) = k9_supinf_1(k17_supinf_2(k7_funct_2(k5_numbers,B,k3_supinf_1,D,C))) ) ) ) ) )).

fof(dt_m1_measure2,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_measure2(C,A,B)
         => ( ~ v1_xboole_0(C)
            & v1_card_4(C)
            & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) ) ) ) )).

fof(existence_m1_measure2,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_measure2(C,A,B) ) )).

fof(dt_k1_measure2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & v1_prob_1(B,A)
        & v1_measure1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
        & m1_measure2(C,A,B) )
     => m2_subset_1(k1_measure2(A,B,C),k1_zfmisc_1(A),B) ) )).

fof(redefinition_k1_measure2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & v1_prob_1(B,A)
        & v1_measure1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
        & m1_measure2(C,A,B) )
     => k1_measure2(A,B,C) = k1_setfam_1(C) ) )).

fof(dt_k2_measure2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & v1_prob_1(B,A)
        & v1_measure1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
        & m1_measure2(C,A,B) )
     => m2_subset_1(k2_measure2(A,B,C),k1_zfmisc_1(A),B) ) )).

fof(redefinition_k2_measure2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & v1_prob_1(B,A)
        & v1_measure1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
        & m1_measure2(C,A,B) )
     => k2_measure2(A,B,C) = k3_tarski(C) ) )).
%------------------------------------------------------------------------------