SET007 Axioms: SET007+176.ax


%------------------------------------------------------------------------------
% File     : SET007+176 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Basic Properties of Extended Real Numbers
% 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    : extreal1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   61 (   7 unit)
%            Number of atoms       :  400 ( 155 equality)
%            Maximal formula depth :   21 (   9 average)
%            Number of connectives :  511 ( 172 ~  ;  26  |; 165  &)
%                                         (   3 <=>; 145 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    6 (   1 propositional; 0-2 arity)
%            Number of functors    :   18 (   8 constant; 0-2 arity)
%            Number of variables   :  127 (   0 singleton; 125 !;   2 ?)
%            Maximal term depth    :    3 (   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_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ~ ( A != k5_measure6
          & A != k4_measure6
          & ~ m1_subset_1(A,k1_numbers) ) ) )).

fof(t2_extreal1,axiom,(
    ~ r1_supinf_1(k5_measure6,k4_measure6) )).

fof(t3_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( ~ r1_supinf_1(B,A)
           => ( A != k5_measure6
              & B != k4_measure6 ) ) ) ) )).

fof(t4_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ( ( A = k5_measure6
         => k3_supinf_2(A) = k4_measure6 )
        & ( k3_supinf_2(A) = k4_measure6
         => A = k5_measure6 )
        & ( A = k4_measure6
         => k3_supinf_2(A) = k5_measure6 )
        & ( k3_supinf_2(A) = k5_measure6
         => A = k4_measure6 ) ) ) )).

fof(t5_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => k4_supinf_2(A,k3_supinf_2(B)) = k2_supinf_2(A,B) ) ) )).

fof(t6_extreal1,axiom,(
    $true )).

fof(t7_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( r1_supinf_1(A,B)
           => ( A = k4_measure6
              | B = k5_measure6
              | ( A != k5_measure6
                & B != k4_measure6 ) ) ) ) ) )).

fof(t8_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ! [C] :
              ( m1_subset_1(C,k3_supinf_1)
             => ~ ( ~ ( A = k5_measure6
                      & B = k4_measure6 )
                  & ~ ( A = k4_measure6
                      & B = k5_measure6 )
                  & ~ ( B = k5_measure6
                      & C = k4_measure6 )
                  & ~ ( B = k4_measure6
                      & C = k5_measure6 )
                  & ~ ( A = k5_measure6
                      & C = k4_measure6 )
                  & ~ ( A = k4_measure6
                      & C = k5_measure6 )
                  & k2_supinf_2(k2_supinf_2(A,B),C) != k2_supinf_2(A,k2_supinf_2(B,C)) ) ) ) ) )).

fof(t9_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => k2_supinf_2(A,k3_supinf_2(A)) = k1_supinf_2 ) )).

fof(t10_extreal1,axiom,(
    $true )).

fof(t11_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ! [C] :
              ( m1_subset_1(C,k3_supinf_1)
             => ~ ( ~ ( A = k5_measure6
                      & B = k4_measure6 )
                  & ~ ( A = k4_measure6
                      & B = k5_measure6 )
                  & ~ ( B = k5_measure6
                      & C = k5_measure6 )
                  & ~ ( B = k4_measure6
                      & C = k4_measure6 )
                  & ~ ( A = k5_measure6
                      & C = k5_measure6 )
                  & ~ ( A = k4_measure6
                      & C = k4_measure6 )
                  & k4_supinf_2(k2_supinf_2(A,B),C) != k2_supinf_2(A,k4_supinf_2(B,C)) ) ) ) ) )).

fof(d1_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ! [C] :
              ( m1_subset_1(C,k3_supinf_1)
             => ( C = k1_extreal1(A,B)
              <=> ~ ( ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ! [E] :
                            ( m1_subset_1(E,k1_numbers)
                           => ~ ( A = D
                                & B = E
                                & C = k4_real_1(D,E) ) ) )
                    & ~ ( ~ ( ~ ( ~ r1_supinf_1(A,k1_supinf_2)
                                & B = k5_measure6 )
                            & ~ ( ~ r1_supinf_1(B,k1_supinf_2)
                                & A = k5_measure6 )
                            & ~ ( ~ r1_supinf_1(k1_supinf_2,A)
                                & B = k4_measure6 )
                            & ~ ( ~ r1_supinf_1(k1_supinf_2,B)
                                & A = k4_measure6 ) )
                        & C = k5_measure6 )
                    & ~ ( ~ ( ~ ( ~ r1_supinf_1(k1_supinf_2,A)
                                & B = k5_measure6 )
                            & ~ ( ~ r1_supinf_1(k1_supinf_2,B)
                                & A = k5_measure6 )
                            & ~ ( ~ r1_supinf_1(A,k1_supinf_2)
                                & B = k4_measure6 )
                            & ~ ( ~ r1_supinf_1(B,k1_supinf_2)
                                & A = k4_measure6 ) )
                        & C = k4_measure6 )
                    & ~ ( ( A = k1_supinf_2
                          | B = k1_supinf_2 )
                        & C = k1_supinf_2 ) ) ) ) ) ) )).

fof(t12_extreal1,axiom,(
    $true )).

fof(t13_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( ( A = C
                      & B = D )
                   => k1_extreal1(A,B) = k4_real_1(C,D) ) ) ) ) ) )).

fof(t14_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ~ ( ( ( r1_supinf_1(k1_supinf_2,A)
                  & ~ r1_supinf_1(B,k1_supinf_2) )
                | ( ~ r1_supinf_1(A,k1_supinf_2)
                  & r1_supinf_1(k1_supinf_2,B) ) )
              & r1_supinf_1(k2_supinf_2(A,B),k1_supinf_2) ) ) ) )).

fof(t15_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ~ ( ( ( r1_supinf_1(A,k1_supinf_2)
                  & ~ r1_supinf_1(k1_supinf_2,B) )
                | ( ~ r1_supinf_1(k1_supinf_2,A)
                  & r1_supinf_1(B,k1_supinf_2) ) )
              & r1_supinf_1(k1_supinf_2,k2_supinf_2(A,B)) ) ) ) )).

fof(t16_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ~ ( r2_hidden(A,k6_supinf_1)
          & ~ ( ~ r1_supinf_1(A,k4_measure6)
              & ~ r1_supinf_1(k1_supinf_2,A) )
          & A != k1_supinf_2
          & ~ ( ~ r1_supinf_1(A,k1_supinf_2)
              & ~ r1_supinf_1(k5_measure6,A) ) ) ) )).

fof(t17_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => k1_extreal1(A,B) = k1_extreal1(B,A) ) ) )).

fof(t18_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( A = B
           => ( ~ ( ~ r1_xreal_0(B,np__0)
                  & r1_supinf_1(A,k1_supinf_2) )
              & ~ ( ~ r1_supinf_1(A,k1_supinf_2)
                  & r1_xreal_0(B,np__0) ) ) ) ) ) )).

fof(t19_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( A = B
           => ( ~ ( ~ r1_xreal_0(np__0,B)
                  & r1_supinf_1(k1_supinf_2,A) )
              & ~ ( ~ r1_supinf_1(k1_supinf_2,A)
                  & r1_xreal_0(np__0,B) ) ) ) ) ) )).

fof(t20_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ~ ( ( ( ~ r1_supinf_1(A,k1_supinf_2)
                  & ~ r1_supinf_1(B,k1_supinf_2) )
                | ( ~ r1_supinf_1(k1_supinf_2,A)
                  & ~ r1_supinf_1(k1_supinf_2,B) ) )
              & r1_supinf_1(k2_extreal1(A,B),k1_supinf_2) ) ) ) )).

fof(t21_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ~ ( ( ( ~ r1_supinf_1(A,k1_supinf_2)
                  & ~ r1_supinf_1(k1_supinf_2,B) )
                | ( ~ r1_supinf_1(k1_supinf_2,A)
                  & ~ r1_supinf_1(B,k1_supinf_2) ) )
              & r1_supinf_1(k1_supinf_2,k2_extreal1(A,B)) ) ) ) )).

fof(t22_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( k2_extreal1(A,B) = k1_supinf_2
          <=> ( A = k1_supinf_2
              | B = k1_supinf_2 ) ) ) ) )).

fof(t23_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ! [C] :
              ( m1_subset_1(C,k3_supinf_1)
             => k2_extreal1(k2_extreal1(A,B),C) = k2_extreal1(A,k2_extreal1(B,C)) ) ) ) )).

fof(t24_extreal1,axiom,(
    k3_supinf_2(k1_supinf_2) = k1_supinf_2 )).

fof(t25_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ( ~ ( ~ r1_supinf_1(A,k1_supinf_2)
            & r1_supinf_1(k1_supinf_2,k3_supinf_2(A)) )
        & ~ ( ~ r1_supinf_1(k1_supinf_2,k3_supinf_2(A))
            & r1_supinf_1(A,k1_supinf_2) )
        & ~ ( ~ r1_supinf_1(k1_supinf_2,A)
            & r1_supinf_1(k3_supinf_2(A),k1_supinf_2) )
        & ~ ( ~ r1_supinf_1(k3_supinf_2(A),k1_supinf_2)
            & r1_supinf_1(k1_supinf_2,A) ) ) ) )).

fof(t26_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( k3_supinf_2(k2_extreal1(A,B)) = k2_extreal1(A,k3_supinf_2(B))
            & k3_supinf_2(k2_extreal1(A,B)) = k2_extreal1(k3_supinf_2(A),B) ) ) ) )).

fof(t27_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ~ ( A != k5_measure6
              & A != k4_measure6
              & k2_extreal1(A,B) = k5_measure6
              & B != k5_measure6
              & B != k4_measure6 ) ) ) )).

fof(t28_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ~ ( A != k5_measure6
              & A != k4_measure6
              & k2_extreal1(A,B) = k4_measure6
              & B != k5_measure6
              & B != k4_measure6 ) ) ) )).

fof(t29_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ! [C] :
              ( m1_subset_1(C,k3_supinf_1)
             => ( ~ ( ~ ( A != k5_measure6
                        & A != k4_measure6 )
                    & ~ ( B = k4_measure6
                        & C = k5_measure6 )
                    & ~ ( ~ r1_supinf_1(k1_supinf_2,B)
                        & ~ r1_supinf_1(k1_supinf_2,C) )
                    & B != k1_supinf_2
                    & C != k1_supinf_2
                    & ~ ( ~ r1_supinf_1(B,k1_supinf_2)
                        & ~ r1_supinf_1(C,k1_supinf_2) )
                    & ~ ( B = k5_measure6
                        & C = k4_measure6 ) )
               => k2_extreal1(A,k2_supinf_2(B,C)) = k2_supinf_2(k2_extreal1(A,B),k2_extreal1(A,C)) ) ) ) ) )).

fof(t30_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ! [C] :
              ( m1_subset_1(C,k3_supinf_1)
             => ~ ( ~ ( A = k5_measure6
                      & B = k5_measure6 )
                  & ~ ( A = k4_measure6
                      & B = k4_measure6 )
                  & C != k5_measure6
                  & C != k4_measure6
                  & k2_extreal1(C,k4_supinf_2(A,B)) != k4_supinf_2(k2_extreal1(C,A),k2_extreal1(C,B)) ) ) ) ) )).

fof(d2_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ~ ( ~ ( ( A = k4_measure6
                    | A = k5_measure6 )
                  & ( B = k4_measure6
                    | B = k5_measure6 ) )
              & B != k1_supinf_2
              & ~ ! [C] :
                    ( m1_subset_1(C,k3_supinf_1)
                   => ( C = k3_extreal1(A,B)
                    <=> ~ ( ! [D] :
                              ( m1_subset_1(D,k1_numbers)
                             => ! [E] :
                                  ( m1_subset_1(E,k1_numbers)
                                 => ~ ( A = D
                                      & B = E
                                      & C = k6_real_1(D,E) ) ) )
                          & ~ ( ( ( A = k5_measure6
                                  & ~ r1_supinf_1(B,k1_supinf_2) )
                                | ( A = k4_measure6
                                  & ~ r1_supinf_1(k1_supinf_2,B) ) )
                              & C = k5_measure6 )
                          & ~ ( ( ( A = k4_measure6
                                  & ~ r1_supinf_1(B,k1_supinf_2) )
                                | ( A = k5_measure6
                                  & ~ r1_supinf_1(k1_supinf_2,B) ) )
                              & C = k4_measure6 )
                          & ~ ( ( B = k4_measure6
                                | B = k5_measure6 )
                              & C = k1_supinf_2 ) ) ) ) ) ) ) )).

fof(t31_extreal1,axiom,(
    $true )).

fof(t32_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( B != k1_supinf_2
           => ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ( ( A = C
                        & B = D )
                     => k3_extreal1(A,B) = k6_real_1(C,D) ) ) ) ) ) ) )).

fof(t33_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ~ ( A != k4_measure6
              & A != k5_measure6
              & ( B = k4_measure6
                | B = k5_measure6 )
              & k3_extreal1(A,B) != k1_supinf_2 ) ) ) )).

fof(t34_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ~ ( A != k4_measure6
          & A != k5_measure6
          & A != k1_supinf_2
          & k3_extreal1(A,A) != np__1 ) ) )).

fof(d3_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ( ( r1_supinf_1(k1_supinf_2,A)
         => k4_extreal1(A) = A )
        & ( ~ r1_supinf_1(k1_supinf_2,A)
         => k4_extreal1(A) = k3_supinf_2(A) ) ) ) )).

fof(t35_extreal1,axiom,(
    $true )).

fof(t36_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ( ~ r1_supinf_1(A,k1_supinf_2)
       => k4_extreal1(A) = A ) ) )).

fof(t37_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ( ~ r1_supinf_1(k1_supinf_2,A)
       => k4_extreal1(A) = k3_supinf_2(A) ) ) )).

fof(t38_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k1_measure6(k4_real_1(A,B)) = k2_extreal1(k1_measure6(A),k1_measure6(B)) ) ) )).

fof(t39_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( B != np__0
           => k1_measure6(k6_real_1(A,B)) = k3_extreal1(k1_measure6(A),k1_measure6(B)) ) ) ) )).

fof(t40_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( r1_supinf_1(A,B)
           => ( r1_supinf_1(k5_measure6,A)
              | r1_supinf_1(B,k4_measure6)
              | r1_supinf_1(k1_supinf_2,k4_supinf_2(B,A)) ) ) ) ) )).

fof(t41_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ~ ( ~ r1_supinf_1(B,A)
              & ~ r1_supinf_1(k5_measure6,A)
              & ~ r1_supinf_1(B,k4_measure6)
              & r1_supinf_1(k4_supinf_2(B,A),k1_supinf_2) ) ) ) )).

fof(t42_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ! [C] :
              ( m1_subset_1(C,k3_supinf_1)
             => ( ( r1_supinf_1(A,B)
                  & r1_supinf_1(k1_supinf_2,C) )
               => r1_supinf_1(k2_extreal1(A,C),k2_extreal1(B,C)) ) ) ) ) )).

fof(t43_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ! [C] :
              ( m1_subset_1(C,k3_supinf_1)
             => ( ( r1_supinf_1(A,B)
                  & r1_supinf_1(C,k1_supinf_2) )
               => r1_supinf_1(k2_extreal1(B,C),k2_extreal1(A,C)) ) ) ) ) )).

fof(t44_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ! [C] :
              ( m1_subset_1(C,k3_supinf_1)
             => ~ ( ~ r1_supinf_1(B,A)
                  & ~ r1_supinf_1(C,k1_supinf_2)
                  & C != k5_measure6
                  & r1_supinf_1(k2_extreal1(B,C),k2_extreal1(A,C)) ) ) ) ) )).

fof(t45_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ! [C] :
              ( m1_subset_1(C,k3_supinf_1)
             => ~ ( ~ r1_supinf_1(B,A)
                  & ~ r1_supinf_1(k1_supinf_2,C)
                  & C != k4_measure6
                  & r1_supinf_1(k2_extreal1(A,C),k2_extreal1(B,C)) ) ) ) ) )).

fof(t46_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( ( m1_subset_1(A,k1_numbers)
              & m1_subset_1(B,k1_numbers) )
           => ( ~ ( ~ r1_supinf_1(B,A)
                  & ! [C] :
                      ( m1_subset_1(C,k1_numbers)
                     => ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ~ ( C = A
                              & D = B
                              & ~ r1_xreal_0(D,C) ) ) ) )
              & ~ ( ? [C] :
                      ( m1_subset_1(C,k1_numbers)
                      & ? [D] :
                          ( m1_subset_1(D,k1_numbers)
                          & C = A
                          & D = B
                          & ~ r1_xreal_0(D,C) ) )
                  & r1_supinf_1(B,A) ) ) ) ) ) )).

fof(t47_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ! [C] :
              ( m1_subset_1(C,k3_supinf_1)
             => ( r1_supinf_1(A,B)
               => ( A = k4_measure6
                  | B = k5_measure6
                  | r1_supinf_1(C,k1_supinf_2)
                  | r1_supinf_1(k3_extreal1(A,C),k3_extreal1(B,C)) ) ) ) ) ) )).

fof(t48_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ! [C] :
              ( m1_subset_1(C,k3_supinf_1)
             => ( r1_supinf_1(A,B)
               => ( r1_supinf_1(C,k1_supinf_2)
                  | C = k5_measure6
                  | r1_supinf_1(k3_extreal1(A,C),k3_extreal1(B,C)) ) ) ) ) ) )).

fof(t49_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ! [C] :
              ( m1_subset_1(C,k3_supinf_1)
             => ( r1_supinf_1(A,B)
               => ( A = k4_measure6
                  | B = k5_measure6
                  | r1_supinf_1(k1_supinf_2,C)
                  | r1_supinf_1(k3_extreal1(B,C),k3_extreal1(A,C)) ) ) ) ) ) )).

fof(t50_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ! [C] :
              ( m1_subset_1(C,k3_supinf_1)
             => ( r1_supinf_1(A,B)
               => ( r1_supinf_1(k1_supinf_2,C)
                  | C = k4_measure6
                  | r1_supinf_1(k3_extreal1(B,C),k3_extreal1(A,C)) ) ) ) ) ) )).

fof(t51_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ! [C] :
              ( m1_subset_1(C,k3_supinf_1)
             => ~ ( ~ r1_supinf_1(B,A)
                  & ~ r1_supinf_1(C,k1_supinf_2)
                  & C != k5_measure6
                  & r1_supinf_1(k3_extreal1(B,C),k3_extreal1(A,C)) ) ) ) ) )).

fof(t52_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ! [C] :
              ( m1_subset_1(C,k3_supinf_1)
             => ~ ( ~ r1_supinf_1(B,A)
                  & ~ r1_supinf_1(k1_supinf_2,C)
                  & C != k4_measure6
                  & r1_supinf_1(k3_extreal1(A,C),k3_extreal1(B,C)) ) ) ) ) )).

fof(dt_k1_extreal1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k3_supinf_1)
        & m1_subset_1(B,k3_supinf_1) )
     => m1_subset_1(k1_extreal1(A,B),k3_supinf_1) ) )).

fof(dt_k2_extreal1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k3_supinf_1)
        & m1_subset_1(B,k3_supinf_1) )
     => m1_subset_1(k2_extreal1(A,B),k3_supinf_1) ) )).

fof(commutativity_k2_extreal1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k3_supinf_1)
        & m1_subset_1(B,k3_supinf_1) )
     => k2_extreal1(A,B) = k2_extreal1(B,A) ) )).

fof(redefinition_k2_extreal1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k3_supinf_1)
        & m1_subset_1(B,k3_supinf_1) )
     => k2_extreal1(A,B) = k1_extreal1(A,B) ) )).

fof(dt_k3_extreal1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k3_supinf_1)
        & m1_subset_1(B,k3_supinf_1) )
     => m1_subset_1(k3_extreal1(A,B),k3_supinf_1) ) )).

fof(dt_k4_extreal1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => m1_subset_1(k4_extreal1(A),k3_supinf_1) ) )).
%------------------------------------------------------------------------------