SET007 Axioms: SET007+89.ax


%------------------------------------------------------------------------------
% File     : SET007+89 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Independence of Events and Conditional Probability
% 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    : prob_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   65 (  10 unit)
%            Number of atoms       :  434 (  55 equality)
%            Maximal formula depth :   22 (  11 average)
%            Number of connectives :  446 (  77 ~  ;  14  |;  70  &)
%                                         (  14 <=>; 271 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   23 (   1 propositional; 0-6 arity)
%            Number of functors    :   31 (   5 constant; 0-4 arity)
%            Number of variables   :  264 (   0 singleton; 263 !;   1 ?)
%            Maximal term depth    :    7 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(t1_prob_2,axiom,(
    $true )).

fof(t2_prob_2,axiom,(
    $true )).

fof(t3_prob_2,axiom,(
    $true )).

fof(t4_prob_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ~ ( A != np__0
                      & B != np__0
                      & ~ ( k6_real_1(D,B) = k6_real_1(C,A)
                        <=> k4_real_1(D,A) = k4_real_1(C,B) ) ) ) ) ) ) )).

fof(t5_prob_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( ( v4_seq_2(B)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => k2_seq_1(k5_numbers,k1_numbers,C,D) = k5_real_1(A,k2_seq_1(k5_numbers,k1_numbers,B,D)) ) )
               => ( v4_seq_2(C)
                  & k2_seq_2(C) = k5_real_1(A,k2_seq_2(B)) ) ) ) ) ) )).

fof(t6_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m3_prob_1(C,A,B)
             => ( k3_xboole_0(C,A) = C
                & k6_prob_1(A,B,C,k5_prob_1(A,B)) = C ) ) ) ) )).

fof(d1_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m2_prob_1(C,A,B)
             => k2_prob_2(A,B,C) = k4_prob_1(A,C) ) ) ) )).

fof(t7_prob_2,axiom,(
    $true )).

fof(t8_prob_2,axiom,(
    $true )).

fof(t9_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m2_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ? [E] :
                      ( m2_prob_1(E,A,B)
                      & ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => k1_prob_2(A,B,E,F) = k6_prob_1(A,B,k1_prob_2(A,B,C,F),D) ) ) ) ) ) ) )).

fof(t10_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m2_prob_1(C,A,B)
             => ! [D] :
                  ( m2_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => ( ( v2_prob_1(C,A)
                          & ! [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                             => k1_prob_2(A,B,D,F) = k6_prob_1(A,B,k1_prob_2(A,B,C,F),E) ) )
                       => v2_prob_1(D,A) ) ) ) ) ) ) )).

fof(t11_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_prob_1(C,A)
             => ! [D] :
                  ( m2_prob_1(D,A,C)
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,C,k1_numbers)
                        & m2_relset_1(E,C,k1_numbers) )
                     => k2_seq_1(k5_numbers,k1_numbers,k9_prob_1(A,C,D,E),B) = k10_prob_1(A,C,E,k1_prob_2(A,C,D,B)) ) ) ) ) ) )).

fof(t12_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m2_prob_1(C,A,B)
             => ! [D] :
                  ( m2_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => ( ! [F] :
                            ( m2_subset_1(F,k1_numbers,k5_numbers)
                           => k1_prob_2(A,B,C,F) = k6_prob_1(A,B,k1_prob_2(A,B,D,F),E) )
                       => k5_subset_1(A,k4_prob_1(A,D),E) = k4_prob_1(A,C) ) ) ) ) ) ) )).

fof(t13_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m4_prob_1(D,A,B)
                 => ( ! [E] :
                        ( m3_prob_1(E,A,B)
                       => k10_prob_1(A,B,C,E) = k10_prob_1(A,B,D,E) )
                   => C = D ) ) ) ) ) )).

fof(t14_prob_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
        & m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
     => ( v2_prob_1(B,A)
      <=> ! [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
           => r1_tarski(k1_prob_1(A,B,k1_nat_1(C,np__1)),k1_prob_1(A,B,C)) ) ) ) )).

fof(t15_prob_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
        & m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
     => ( v3_prob_1(B,A)
      <=> ! [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
           => r1_tarski(k1_prob_1(A,B,C),k1_prob_1(A,B,k1_nat_1(C,np__1))) ) ) ) )).

fof(t16_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
            & m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
                & m2_relset_1(C,k5_numbers,k1_zfmisc_1(A)) )
             => ( ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k1_prob_1(A,B,D) = k1_prob_1(A,C,D) )
               => B = C ) ) ) ) )).

fof(t17_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
            & m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
         => ( v2_prob_1(B,A)
          <=> v3_prob_1(k3_prob_1(A,B),A) ) ) ) )).

fof(d2_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m2_prob_1(C,A,B)
             => k3_prob_2(A,B,C) = k3_prob_1(A,C) ) ) ) )).

fof(d3_prob_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_prob_2(A)
      <=> ! [B,C] :
            ( B != C
           => r1_xboole_0(k1_funct_1(A,B),k1_funct_1(A,C)) ) ) ) )).

fof(d4_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m2_prob_1(C,A,B)
             => ( v1_prob_2(C)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( D != E
                         => r1_xboole_0(k1_prob_2(A,B,C,D),k1_prob_2(A,B,C,E)) ) ) ) ) ) ) ) )).

fof(t18_prob_2,axiom,(
    $true )).

fof(t19_prob_2,axiom,(
    $true )).

fof(t20_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,k1_numbers)
                & m2_relset_1(C,B,k1_numbers) )
             => ( m4_prob_1(C,A,B)
              <=> ( ! [D] :
                      ( m3_prob_1(D,A,B)
                     => r1_xreal_0(np__0,k10_prob_1(A,B,C,D)) )
                  & k2_seq_1(B,k1_numbers,C,A) = np__1
                  & ! [D] :
                      ( m3_prob_1(D,A,B)
                     => ! [E] :
                          ( m3_prob_1(E,A,B)
                         => ( r1_xboole_0(D,E)
                           => k10_prob_1(A,B,C,k7_prob_1(A,B,D,E)) = k3_real_1(k10_prob_1(A,B,C,D),k10_prob_1(A,B,C,E)) ) ) )
                  & ! [D] :
                      ( m2_prob_1(D,A,B)
                     => ( v3_prob_1(D,A)
                       => ( v4_seq_2(k9_prob_1(A,B,D,C))
                          & k2_seq_2(k9_prob_1(A,B,D,C)) = k2_seq_1(B,k1_numbers,C,k2_prob_1(A,D)) ) ) ) ) ) ) ) ) )).

fof(t21_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => ! [F] :
                          ( m3_prob_1(F,A,B)
                         => k10_prob_1(A,B,C,k7_prob_1(A,B,k7_prob_1(A,B,D,E),F)) = k3_real_1(k5_real_1(k3_real_1(k3_real_1(k10_prob_1(A,B,C,D),k10_prob_1(A,B,C,E)),k10_prob_1(A,B,C,F)),k3_real_1(k3_real_1(k10_prob_1(A,B,C,k6_prob_1(A,B,D,E)),k10_prob_1(A,B,C,k6_prob_1(A,B,E,F))),k10_prob_1(A,B,C,k6_prob_1(A,B,D,F)))),k10_prob_1(A,B,C,k6_prob_1(A,B,k6_prob_1(A,B,D,E),F))) ) ) ) ) ) ) )).

fof(t22_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => k10_prob_1(A,B,C,k8_prob_1(A,B,D,k6_prob_1(A,B,D,E))) = k5_real_1(k10_prob_1(A,B,C,D),k10_prob_1(A,B,C,k6_prob_1(A,B,D,E))) ) ) ) ) ) )).

fof(t23_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => ( r1_xreal_0(k10_prob_1(A,B,C,k6_prob_1(A,B,D,E)),k10_prob_1(A,B,C,E))
                        & r1_xreal_0(k10_prob_1(A,B,C,k6_prob_1(A,B,D,E)),k10_prob_1(A,B,C,D)) ) ) ) ) ) ) )).

fof(t24_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => ! [F] :
                          ( m3_prob_1(F,A,B)
                         => ( D = k3_subset_1(A,E)
                           => k10_prob_1(A,B,C,F) = k3_real_1(k10_prob_1(A,B,C,k6_prob_1(A,B,F,E)),k10_prob_1(A,B,C,k6_prob_1(A,B,F,D))) ) ) ) ) ) ) ) )).

fof(t25_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => r1_xreal_0(k5_real_1(k3_real_1(k10_prob_1(A,B,C,D),k10_prob_1(A,B,C,E)),np__1),k10_prob_1(A,B,C,k6_prob_1(A,B,D,E))) ) ) ) ) ) )).

fof(t26_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => k10_prob_1(A,B,C,D) = k5_real_1(np__1,k10_prob_1(A,B,C,k8_prob_1(A,B,k5_prob_1(A,B),D))) ) ) ) ) )).

fof(t27_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ( ~ ( ~ r1_xreal_0(np__1,k10_prob_1(A,B,C,D))
                        & r1_xreal_0(k10_prob_1(A,B,C,k8_prob_1(A,B,k5_prob_1(A,B),D)),np__0) )
                    & ~ ( ~ r1_xreal_0(k10_prob_1(A,B,C,k8_prob_1(A,B,k5_prob_1(A,B),D)),np__0)
                        & r1_xreal_0(np__1,k10_prob_1(A,B,C,D)) ) ) ) ) ) ) )).

fof(t28_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ( ~ ( ~ r1_xreal_0(np__1,k10_prob_1(A,B,C,k8_prob_1(A,B,k5_prob_1(A,B),D)))
                        & r1_xreal_0(k10_prob_1(A,B,C,D),np__0) )
                    & ~ ( ~ r1_xreal_0(k10_prob_1(A,B,C,D),np__0)
                        & r1_xreal_0(np__1,k10_prob_1(A,B,C,k8_prob_1(A,B,k5_prob_1(A,B),D))) ) ) ) ) ) ) )).

fof(d5_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => ( r1_prob_2(A,B,C,D,E)
                      <=> k10_prob_1(A,B,C,k6_prob_1(A,B,D,E)) = k4_real_1(k10_prob_1(A,B,C,D),k10_prob_1(A,B,C,E)) ) ) ) ) ) ) )).

fof(d6_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => ! [F] :
                          ( m3_prob_1(F,A,B)
                         => ( r2_prob_2(A,B,C,D,E,F)
                          <=> ( k10_prob_1(A,B,C,k6_prob_1(A,B,k6_prob_1(A,B,D,E),F)) = k4_real_1(k4_real_1(k10_prob_1(A,B,C,D),k10_prob_1(A,B,C,E)),k10_prob_1(A,B,C,F))
                              & k10_prob_1(A,B,C,k6_prob_1(A,B,D,E)) = k4_real_1(k10_prob_1(A,B,C,D),k10_prob_1(A,B,C,E))
                              & k10_prob_1(A,B,C,k6_prob_1(A,B,D,F)) = k4_real_1(k10_prob_1(A,B,C,D),k10_prob_1(A,B,C,F))
                              & k10_prob_1(A,B,C,k6_prob_1(A,B,E,F)) = k4_real_1(k10_prob_1(A,B,C,E),k10_prob_1(A,B,C,F)) ) ) ) ) ) ) ) ) )).

fof(t29_prob_2,axiom,(
    $true )).

fof(t30_prob_2,axiom,(
    $true )).

fof(t31_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => ( r1_prob_2(A,B,C,D,E)
                      <=> r1_prob_2(A,B,C,E,D) ) ) ) ) ) ) )).

fof(t32_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => ! [F] :
                          ( m3_prob_1(F,A,B)
                         => ( r2_prob_2(A,B,C,D,E,F)
                          <=> ( k10_prob_1(A,B,C,k6_prob_1(A,B,k6_prob_1(A,B,D,E),F)) = k4_real_1(k4_real_1(k10_prob_1(A,B,C,D),k10_prob_1(A,B,C,E)),k10_prob_1(A,B,C,F))
                              & r1_prob_2(A,B,C,D,E)
                              & r1_prob_2(A,B,C,E,F)
                              & r1_prob_2(A,B,C,D,F) ) ) ) ) ) ) ) ) )).

fof(t33_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => ! [F] :
                          ( m3_prob_1(F,A,B)
                         => ( r2_prob_2(A,B,C,D,E,F)
                           => r2_prob_2(A,B,C,E,D,F) ) ) ) ) ) ) ) )).

fof(t34_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => ! [F] :
                          ( m3_prob_1(F,A,B)
                         => ( r2_prob_2(A,B,C,D,E,F)
                           => r2_prob_2(A,B,C,D,F,E) ) ) ) ) ) ) ) )).

fof(t35_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => ( E = k1_xboole_0
                       => r1_prob_2(A,B,C,D,E) ) ) ) ) ) ) )).

fof(t36_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => r1_prob_2(A,B,C,D,k5_prob_1(A,B)) ) ) ) ) )).

fof(t37_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m3_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m4_prob_1(E,A,B)
                     => ( r1_prob_2(A,B,E,C,D)
                       => r1_prob_2(A,B,E,C,k8_prob_1(A,B,k5_prob_1(A,B),D)) ) ) ) ) ) ) )).

fof(t38_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => ( r1_prob_2(A,B,C,D,E)
                       => r1_prob_2(A,B,C,k8_prob_1(A,B,k5_prob_1(A,B),D),k8_prob_1(A,B,k5_prob_1(A,B),E)) ) ) ) ) ) ) )).

fof(t39_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m3_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => ! [F] :
                          ( m4_prob_1(F,A,B)
                         => ( ( r1_prob_2(A,B,F,C,D)
                              & r1_prob_2(A,B,F,C,E)
                              & r1_xboole_0(D,E) )
                           => r1_prob_2(A,B,F,C,k7_prob_1(A,B,D,E)) ) ) ) ) ) ) ) )).

fof(t40_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => ~ ( r1_prob_2(A,B,C,D,E)
                          & ~ r1_xreal_0(np__1,k10_prob_1(A,B,C,D))
                          & ~ r1_xreal_0(np__1,k10_prob_1(A,B,C,E))
                          & r1_xreal_0(np__1,k10_prob_1(A,B,C,k7_prob_1(A,B,D,E))) ) ) ) ) ) ) )).

fof(d7_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ( ~ r1_xreal_0(k10_prob_1(A,B,C,D),np__0)
                   => ! [E] :
                        ( m4_prob_1(E,A,B)
                       => ( E = k4_prob_2(A,B,C,D)
                        <=> ! [F] :
                              ( m3_prob_1(F,A,B)
                             => k10_prob_1(A,B,E,F) = k6_real_1(k10_prob_1(A,B,C,k6_prob_1(A,B,F,D)),k10_prob_1(A,B,C,D)) ) ) ) ) ) ) ) ) )).

fof(t41_prob_2,axiom,(
    $true )).

fof(t42_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => ( ~ r1_xreal_0(k10_prob_1(A,B,C,D),np__0)
                       => k10_prob_1(A,B,C,k6_prob_1(A,B,E,D)) = k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,C,D),E),k10_prob_1(A,B,C,D)) ) ) ) ) ) ) )).

fof(t43_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => ! [F] :
                          ( m3_prob_1(F,A,B)
                         => ( ~ r1_xreal_0(k10_prob_1(A,B,C,k6_prob_1(A,B,D,E)),np__0)
                           => k10_prob_1(A,B,C,k6_prob_1(A,B,k6_prob_1(A,B,D,E),F)) = k4_real_1(k4_real_1(k10_prob_1(A,B,C,D),k10_prob_1(A,B,k4_prob_2(A,B,C,D),E)),k10_prob_1(A,B,k4_prob_2(A,B,C,k6_prob_1(A,B,D,E)),F)) ) ) ) ) ) ) ) )).

fof(t44_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => ! [F] :
                          ( m3_prob_1(F,A,B)
                         => ( F = k3_subset_1(A,E)
                           => ( r1_xreal_0(k10_prob_1(A,B,C,E),np__0)
                              | r1_xreal_0(k10_prob_1(A,B,C,F),np__0)
                              | k10_prob_1(A,B,C,D) = k3_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,C,E),D),k10_prob_1(A,B,C,E)),k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,C,F),D),k10_prob_1(A,B,C,F))) ) ) ) ) ) ) ) ) )).

fof(t45_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => ! [F] :
                          ( m3_prob_1(F,A,B)
                         => ! [G] :
                              ( m3_prob_1(G,A,B)
                             => ( ( r1_xboole_0(E,F)
                                  & G = k3_subset_1(A,k7_prob_1(A,B,E,F)) )
                               => ( r1_xreal_0(k10_prob_1(A,B,C,E),np__0)
                                  | r1_xreal_0(k10_prob_1(A,B,C,F),np__0)
                                  | r1_xreal_0(k10_prob_1(A,B,C,G),np__0)
                                  | k10_prob_1(A,B,C,D) = k3_real_1(k3_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,C,E),D),k10_prob_1(A,B,C,E)),k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,C,F),D),k10_prob_1(A,B,C,F))),k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,C,G),D),k10_prob_1(A,B,C,G))) ) ) ) ) ) ) ) ) ) )).

fof(t46_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => ( ~ r1_xreal_0(k10_prob_1(A,B,C,E),np__0)
                       => ( k10_prob_1(A,B,k4_prob_2(A,B,C,E),D) = k10_prob_1(A,B,C,D)
                        <=> r1_prob_2(A,B,C,D,E) ) ) ) ) ) ) ) )).

fof(t47_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => ( k10_prob_1(A,B,k4_prob_2(A,B,C,E),D) = k10_prob_1(A,B,k4_prob_2(A,B,C,k8_prob_1(A,B,k5_prob_1(A,B),E)),D)
                       => ( r1_xreal_0(k10_prob_1(A,B,C,E),np__0)
                          | r1_xreal_0(np__1,k10_prob_1(A,B,C,E))
                          | r1_prob_2(A,B,C,D,E) ) ) ) ) ) ) ) )).

fof(t48_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m4_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => ( ~ r1_xreal_0(k10_prob_1(A,B,C,E),np__0)
                       => r1_xreal_0(k6_real_1(k5_real_1(k3_real_1(k10_prob_1(A,B,C,D),k10_prob_1(A,B,C,E)),np__1),k10_prob_1(A,B,C,E)),k10_prob_1(A,B,k4_prob_2(A,B,C,E),D)) ) ) ) ) ) ) )).

fof(t49_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m3_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m4_prob_1(E,A,B)
                     => ~ ( ~ r1_xreal_0(k10_prob_1(A,B,E,C),np__0)
                          & ~ r1_xreal_0(k10_prob_1(A,B,E,D),np__0)
                          & k10_prob_1(A,B,k4_prob_2(A,B,E,D),C) != k6_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,E,C),D),k10_prob_1(A,B,E,C)),k10_prob_1(A,B,E,D)) ) ) ) ) ) ) )).

fof(t50_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m3_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => ! [F] :
                          ( m4_prob_1(F,A,B)
                         => ( E = k3_subset_1(A,D)
                           => ( r1_xreal_0(k10_prob_1(A,B,F,C),np__0)
                              | r1_xreal_0(k10_prob_1(A,B,F,D),np__0)
                              | r1_xreal_0(k10_prob_1(A,B,F,E),np__0)
                              | ( k10_prob_1(A,B,k4_prob_2(A,B,F,C),D) = k6_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,F,D),C),k10_prob_1(A,B,F,D)),k3_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,F,D),C),k10_prob_1(A,B,F,D)),k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,F,E),C),k10_prob_1(A,B,F,E))))
                                & k10_prob_1(A,B,k4_prob_2(A,B,F,C),E) = k6_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,F,E),C),k10_prob_1(A,B,F,E)),k3_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,F,D),C),k10_prob_1(A,B,F,D)),k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,F,E),C),k10_prob_1(A,B,F,E)))) ) ) ) ) ) ) ) ) ) )).

fof(t51_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m3_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m3_prob_1(E,A,B)
                     => ! [F] :
                          ( m3_prob_1(F,A,B)
                         => ! [G] :
                              ( m4_prob_1(G,A,B)
                             => ( ( r1_xboole_0(D,E)
                                  & F = k3_subset_1(A,k7_prob_1(A,B,D,E)) )
                               => ( r1_xreal_0(k10_prob_1(A,B,G,C),np__0)
                                  | r1_xreal_0(k10_prob_1(A,B,G,D),np__0)
                                  | r1_xreal_0(k10_prob_1(A,B,G,E),np__0)
                                  | r1_xreal_0(k10_prob_1(A,B,G,F),np__0)
                                  | ( k10_prob_1(A,B,k4_prob_2(A,B,G,C),D) = k6_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,G,D),C),k10_prob_1(A,B,G,D)),k3_real_1(k3_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,G,D),C),k10_prob_1(A,B,G,D)),k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,G,E),C),k10_prob_1(A,B,G,E))),k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,G,F),C),k10_prob_1(A,B,G,F))))
                                    & k10_prob_1(A,B,k4_prob_2(A,B,G,C),E) = k6_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,G,E),C),k10_prob_1(A,B,G,E)),k3_real_1(k3_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,G,D),C),k10_prob_1(A,B,G,D)),k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,G,E),C),k10_prob_1(A,B,G,E))),k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,G,F),C),k10_prob_1(A,B,G,F))))
                                    & k10_prob_1(A,B,k4_prob_2(A,B,G,C),F) = k6_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,G,F),C),k10_prob_1(A,B,G,F)),k3_real_1(k3_real_1(k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,G,D),C),k10_prob_1(A,B,G,D)),k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,G,E),C),k10_prob_1(A,B,G,E))),k4_real_1(k10_prob_1(A,B,k4_prob_2(A,B,G,F),C),k10_prob_1(A,B,G,F)))) ) ) ) ) ) ) ) ) ) ) )).

fof(t52_prob_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_prob_1(B,A)
         => ! [C] :
              ( m3_prob_1(C,A,B)
             => ! [D] :
                  ( m3_prob_1(D,A,B)
                 => ! [E] :
                      ( m4_prob_1(E,A,B)
                     => ( ~ r1_xreal_0(k10_prob_1(A,B,E,D),np__0)
                       => r1_xreal_0(k5_real_1(np__1,k6_real_1(k10_prob_1(A,B,E,k8_prob_1(A,B,k5_prob_1(A,B),C)),k10_prob_1(A,B,E,D))),k10_prob_1(A,B,k4_prob_2(A,B,E,D),C)) ) ) ) ) ) ) )).

fof(redefinition_v2_prob_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_prob_1(B,A)
        & m2_prob_1(C,A,B) )
     => ( v2_prob_2(C,A,B)
      <=> v1_prob_2(C) ) ) )).

fof(dt_k1_prob_2,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & m1_prob_1(B,A)
        & m2_prob_1(C,A,B)
        & m1_subset_1(D,k5_numbers) )
     => m3_prob_1(k1_prob_2(A,B,C,D),A,B) ) )).

fof(redefinition_k1_prob_2,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & m1_prob_1(B,A)
        & m2_prob_1(C,A,B)
        & m1_subset_1(D,k5_numbers) )
     => k1_prob_2(A,B,C,D) = k1_funct_1(C,D) ) )).

fof(dt_k2_prob_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_prob_1(B,A)
        & m2_prob_1(C,A,B) )
     => m3_prob_1(k2_prob_2(A,B,C),A,B) ) )).

fof(dt_k3_prob_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_prob_1(B,A)
        & m2_prob_1(C,A,B) )
     => m2_prob_1(k3_prob_2(A,B,C),A,B) ) )).

fof(dt_k4_prob_2,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & m1_prob_1(B,A)
        & m4_prob_1(C,A,B)
        & m3_prob_1(D,A,B) )
     => m4_prob_1(k4_prob_2(A,B,C,D),A,B) ) )).
%------------------------------------------------------------------------------