SET007 Axioms: SET007+178.ax


%------------------------------------------------------------------------------
% File     : SET007+178 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Properties of Extended Real Numbers Operations: abs, min and max
% 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    : extreal2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  112 (  13 unit)
%            Number of atoms       :  643 ( 273 equality)
%            Maximal formula depth :   16 (   8 average)
%            Number of connectives :  784 ( 253 ~  ;  27  |; 264  &)
%                                         (   6 <=>; 234 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    6 (   1 propositional; 0-2 arity)
%            Number of functors    :   21 (   9 constant; 0-2 arity)
%            Number of variables   :  202 (   0 singleton; 202 !;   0 ?)
%            Maximal term depth    :    5 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(t1_extreal2,axiom,(
    $true )).

fof(t2_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ~ ( A != k5_measure6
          & A != k4_measure6
          & ! [B] :
              ( m1_subset_1(B,k3_supinf_1)
             => k2_supinf_2(A,B) != k1_supinf_2 ) ) ) )).

fof(t3_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ~ ( A != k5_measure6
          & A != k4_measure6
          & A != k1_supinf_2
          & ! [B] :
              ( m1_subset_1(B,k3_supinf_1)
             => k2_extreal1(A,B) != k10_mesfunc1 ) ) ) )).

fof(t4_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ( k2_extreal1(k10_mesfunc1,A) = A
        & k2_extreal1(A,k10_mesfunc1) = A
        & k2_extreal1(k1_measure6(np__1),A) = A
        & k2_extreal1(A,k1_measure6(np__1)) = A ) ) )).

fof(t5_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => k4_supinf_2(k1_supinf_2,A) = k3_supinf_2(A) ) )).

fof(t6_extreal2,axiom,(
    $true )).

fof(t7_extreal2,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(k1_supinf_2,B) )
           => r1_supinf_1(k1_supinf_2,k2_supinf_2(A,B)) ) ) ) )).

fof(t8_extreal2,axiom,(
    $true )).

fof(t9_extreal2,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(k2_supinf_2(A,B),k1_supinf_2) ) ) ) )).

fof(t10_extreal2,axiom,(
    $true )).

fof(t11_extreal2,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_supinf_2(B,A) = C
               => ( A = k5_measure6
                  | A = k4_measure6
                  | B = k4_supinf_2(C,A) ) ) ) ) ) )).

fof(t12_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ~ ( A != k5_measure6
          & A != k4_measure6
          & A != k1_supinf_2
          & ~ ( k2_extreal1(A,k3_extreal1(k10_mesfunc1,A)) = k10_mesfunc1
              & k2_extreal1(k3_extreal1(k10_mesfunc1,A),A) = k10_mesfunc1 ) ) ) )).

fof(t13_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ~ ( A != k5_measure6
          & A != k4_measure6
          & k4_supinf_2(A,A) != k1_supinf_2 ) ) )).

fof(t14_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ~ ( ~ ( A = k5_measure6
                  & B = k4_measure6 )
              & ~ ( A = k4_measure6
                  & B = k5_measure6 )
              & ~ ( k3_supinf_2(k2_supinf_2(A,B)) = k2_supinf_2(k3_supinf_2(A),k3_supinf_2(B))
                  & k3_supinf_2(k2_supinf_2(A,B)) = k4_supinf_2(k3_supinf_2(B),A)
                  & k3_supinf_2(k2_supinf_2(A,B)) = k4_supinf_2(k3_supinf_2(A),B) ) ) ) ) )).

fof(t15_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ~ ( ~ ( A = k5_measure6
                  & B = k5_measure6 )
              & ~ ( A = k4_measure6
                  & B = k4_measure6 )
              & ~ ( k3_supinf_2(k4_supinf_2(A,B)) = k2_supinf_2(k3_supinf_2(A),B)
                  & k3_supinf_2(k4_supinf_2(A,B)) = k4_supinf_2(B,A) ) ) ) ) )).

fof(t16_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ~ ( ~ ( A = k5_measure6
                  & B = k5_measure6 )
              & ~ ( A = k4_measure6
                  & B = k4_measure6 )
              & ~ ( k3_supinf_2(k2_supinf_2(k3_supinf_2(A),B)) = k4_supinf_2(A,B)
                  & k3_supinf_2(k2_supinf_2(k3_supinf_2(A),B)) = k2_supinf_2(A,k3_supinf_2(B)) ) ) ) ) )).

fof(t17_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( ( ( A = k5_measure6
                & ~ r1_supinf_1(B,k1_supinf_2)
                & ~ r1_supinf_1(k5_measure6,B) )
              | ( A = k4_measure6
                & ~ r1_supinf_1(k1_supinf_2,B)
                & ~ r1_supinf_1(B,k4_measure6) ) )
           => k3_extreal1(A,B) = k5_measure6 ) ) ) )).

fof(t18_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( ( ( A = k5_measure6
                & ~ r1_supinf_1(k1_supinf_2,B)
                & ~ r1_supinf_1(B,k4_measure6) )
              | ( A = k4_measure6
                & ~ r1_supinf_1(B,k1_supinf_2)
                & ~ r1_supinf_1(k5_measure6,B) ) )
           => k3_extreal1(A,B) = k4_measure6 ) ) ) )).

fof(t19_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ~ ( ~ r1_supinf_1(A,k4_measure6)
              & ~ r1_supinf_1(k5_measure6,A)
              & A != k1_supinf_2
              & ~ ( k3_extreal1(k2_extreal1(B,A),A) = B
                  & k2_extreal1(B,k3_extreal1(A,A)) = B ) ) ) ) )).

fof(t20_extreal2,axiom,
    ( ~ r1_supinf_1(k5_measure6,k10_mesfunc1)
    & ~ r1_supinf_1(k10_mesfunc1,k4_measure6) )).

fof(t21_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ( ( A = k5_measure6
          | A = k4_measure6 )
       => ! [B] :
            ( m1_subset_1(B,k3_supinf_1)
           => ~ ( r2_hidden(B,k6_supinf_1)
                & k2_supinf_2(A,B) = k1_supinf_2 ) ) ) ) )).

fof(t22_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ( ( A = k5_measure6
          | A = k4_measure6 )
       => ! [B] :
            ( m1_subset_1(B,k3_supinf_1)
           => k2_extreal1(A,B) != k10_mesfunc1 ) ) ) )).

fof(t23_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ~ ( ~ ( A = k5_measure6
                  & B = k4_measure6 )
              & ~ ( A = k4_measure6
                  & B = k5_measure6 )
              & ~ r1_supinf_1(k5_measure6,k2_supinf_2(A,B))
              & ~ ( A != k5_measure6
                  & B != k5_measure6 ) ) ) ) )).

fof(t24_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ~ ( ~ ( A = k5_measure6
                  & B = k4_measure6 )
              & ~ ( A = k4_measure6
                  & B = k5_measure6 )
              & ~ r1_supinf_1(k2_supinf_2(A,B),k4_measure6)
              & ~ ( A != k4_measure6
                  & B != k4_measure6 ) ) ) ) )).

fof(t25_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ~ ( ~ ( A = k5_measure6
                  & B = k5_measure6 )
              & ~ ( A = k4_measure6
                  & B = k4_measure6 )
              & ~ r1_supinf_1(k5_measure6,k4_supinf_2(A,B))
              & ~ ( A != k5_measure6
                  & B != k4_measure6 ) ) ) ) )).

fof(t26_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ~ ( ~ ( A = k5_measure6
                  & B = k5_measure6 )
              & ~ ( A = k4_measure6
                  & B = k4_measure6 )
              & ~ r1_supinf_1(k4_supinf_2(A,B),k4_measure6)
              & ~ ( A != k4_measure6
                  & B != k5_measure6 ) ) ) ) )).

fof(t27_extreal2,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 )
                  & ~ r1_supinf_1(C,k2_supinf_2(A,B))
                  & ~ ( A != k5_measure6
                      & B != k5_measure6
                      & C != k4_measure6
                      & ~ r1_supinf_1(k4_supinf_2(C,B),A) ) ) ) ) ) )).

fof(t28_extreal2,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 )
                  & ~ r1_supinf_1(k4_supinf_2(A,B),C)
                  & ~ ( C != k5_measure6
                      & B != k5_measure6
                      & A != k4_measure6
                      & ~ r1_supinf_1(A,k2_supinf_2(C,B)) ) ) ) ) ) )).

fof(t29_extreal2,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 )
                  & ~ r1_supinf_1(C,k4_supinf_2(A,B))
                  & ~ ( A != k5_measure6
                      & B != k4_measure6
                      & C != k4_measure6
                      & ~ r1_supinf_1(k2_supinf_2(C,B),A) ) ) ) ) ) )).

fof(t30_extreal2,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 )
                  & ~ r1_supinf_1(k2_supinf_2(A,B),C)
                  & ~ ( C != k5_measure6
                      & B != k4_measure6
                      & A != k4_measure6
                      & ~ r1_supinf_1(A,k4_supinf_2(C,B)) ) ) ) ) ) )).

fof(t31_extreal2,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 )
                  & r1_supinf_1(k2_supinf_2(A,B),C)
                  & ~ ( B != k5_measure6
                      & r1_supinf_1(A,k4_supinf_2(C,B)) ) ) ) ) ) )).

fof(t32_extreal2,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,k4_supinf_2(C,B))
               => ( ( A = k5_measure6
                    & B = k4_measure6 )
                  | ( A = k4_measure6
                    & B = k5_measure6 )
                  | ( B = k5_measure6
                    & C = k5_measure6 )
                  | ( B != k5_measure6
                    & r1_supinf_1(k2_supinf_2(A,B),C) ) ) ) ) ) ) )).

fof(t33_extreal2,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 )
                  & ~ ( B = k5_measure6
                      & C = k4_measure6 )
                  & ~ ( B = k4_measure6
                      & C = k5_measure6 )
                  & r1_supinf_1(k4_supinf_2(A,B),C)
                  & ~ ( B != k4_measure6
                      & r1_supinf_1(A,k2_supinf_2(C,B)) ) ) ) ) ) )).

fof(t34_extreal2,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,k2_supinf_2(C,B))
               => ( ( A = k5_measure6
                    & B = k5_measure6 )
                  | ( A = k4_measure6
                    & B = k4_measure6 )
                  | ( B = k4_measure6
                    & C = k5_measure6 )
                  | ( B != k4_measure6
                    & r1_supinf_1(k4_supinf_2(A,B),C) ) ) ) ) ) ) )).

fof(t35_extreal2,axiom,(
    $true )).

fof(t36_extreal2,axiom,(
    $true )).

fof(t37_extreal2,axiom,(
    $true )).

fof(t38_extreal2,axiom,(
    $true )).

fof(t39_extreal2,axiom,(
    $true )).

fof(t40_extreal2,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 )
                  & ~ ( B = k5_measure6
                      & C = k4_measure6 )
                  & ~ ( B = k4_measure6
                      & C = k5_measure6 )
                  & ~ ( A = k5_measure6
                      & C = k5_measure6 )
                  & ~ ( A = k4_measure6
                      & C = k4_measure6 )
                  & k4_supinf_2(k4_supinf_2(A,B),C) != k4_supinf_2(A,k2_supinf_2(B,C)) ) ) ) ) )).

fof(t41_extreal2,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 )
                  & ~ ( B = k5_measure6
                      & C = k5_measure6 )
                  & ~ ( B = k4_measure6
                      & C = k4_measure6 )
                  & ~ ( A = k5_measure6
                      & C = k4_measure6 )
                  & ~ ( A = k4_measure6
                      & C = k5_measure6 )
                  & k2_supinf_2(k4_supinf_2(A,B),C) != k4_supinf_2(A,k4_supinf_2(B,C)) ) ) ) ) )).

fof(t42_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ~ ( k2_extreal1(A,B) != k5_measure6
              & k2_extreal1(A,B) != k4_measure6
              & ~ m1_subset_1(A,k1_numbers)
              & ~ m1_subset_1(B,k1_numbers) ) ) ) )).

fof(t43_extreal2,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) )
            & ~ ( ~ r1_supinf_1(k2_extreal1(A,B),k1_supinf_2)
                & ~ ( ~ 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) ) ) ) ) ) )).

fof(t44_extreal2,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)) )
            & ~ ( ~ r1_supinf_1(k1_supinf_2,k2_extreal1(A,B))
                & ~ ( ~ 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) ) ) ) ) ) )).

fof(t45_extreal2,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(A,k1_supinf_2) )
                & ~ ( ~ r1_supinf_1(k1_supinf_2,B)
                    & r1_supinf_1(B,k1_supinf_2) ) )
              | ( ~ ( ~ r1_supinf_1(A,k1_supinf_2)
                    & r1_supinf_1(k1_supinf_2,A) )
                & ~ ( ~ r1_supinf_1(B,k1_supinf_2)
                    & r1_supinf_1(k1_supinf_2,B) ) ) )
          <=> r1_supinf_1(k1_supinf_2,k2_extreal1(A,B)) ) ) ) )).

fof(t46_extreal2,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,A) )
                & ~ ( ~ r1_supinf_1(k1_supinf_2,B)
                    & r1_supinf_1(B,k1_supinf_2) ) )
              | ( ~ ( ~ r1_supinf_1(k1_supinf_2,A)
                    & r1_supinf_1(A,k1_supinf_2) )
                & ~ ( ~ r1_supinf_1(B,k1_supinf_2)
                    & r1_supinf_1(k1_supinf_2,B) ) ) )
          <=> r1_supinf_1(k2_extreal1(A,B),k1_supinf_2) ) ) ) )).

fof(t47_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( ( r1_supinf_1(A,k3_supinf_2(B))
             => r1_supinf_1(B,k3_supinf_2(A)) )
            & ( r1_supinf_1(B,k3_supinf_2(A))
             => r1_supinf_1(A,k3_supinf_2(B)) )
            & ( r1_supinf_1(k3_supinf_2(A),B)
             => r1_supinf_1(k3_supinf_2(B),A) )
            & ( r1_supinf_1(k3_supinf_2(B),A)
             => r1_supinf_1(k3_supinf_2(A),B) ) ) ) ) )).

fof(t48_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( ~ ( ~ r1_supinf_1(k3_supinf_2(B),A)
                & r1_supinf_1(k3_supinf_2(A),B) )
            & ~ ( ~ r1_supinf_1(k3_supinf_2(A),B)
                & r1_supinf_1(k3_supinf_2(B),A) )
            & ~ ( ~ r1_supinf_1(B,k3_supinf_2(A))
                & r1_supinf_1(A,k3_supinf_2(B)) )
            & ~ ( ~ r1_supinf_1(A,k3_supinf_2(B))
                & r1_supinf_1(B,k3_supinf_2(A)) ) ) ) ) )).

fof(t49_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( A = B
           => k4_extreal1(A) = k18_complex1(B) ) ) ) )).

fof(t50_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ( k4_extreal1(A) = A
        | k4_extreal1(A) = k3_supinf_2(A) ) ) )).

fof(t51_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => r1_supinf_1(k1_supinf_2,k4_extreal1(A)) ) )).

fof(t52_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ~ ( A != k1_supinf_2
          & r1_supinf_1(k4_extreal1(A),k1_supinf_2) ) ) )).

fof(t53_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ( A = k1_supinf_2
      <=> k4_extreal1(A) = k1_supinf_2 ) ) )).

fof(t54_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ~ ( k4_extreal1(A) = k3_supinf_2(A)
          & A != k1_supinf_2
          & r1_supinf_1(k1_supinf_2,A) ) ) )).

fof(t55_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ( r1_supinf_1(A,k1_supinf_2)
       => k4_extreal1(A) = k3_supinf_2(A) ) ) )).

fof(t56_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => k4_extreal1(k2_extreal1(A,B)) = k2_extreal1(k4_extreal1(A),k4_extreal1(B)) ) ) )).

fof(t57_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ( r1_supinf_1(k3_supinf_2(k4_extreal1(A)),A)
        & r1_supinf_1(A,k4_extreal1(A)) ) ) )).

fof(t58_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( ~ r1_supinf_1(B,k4_extreal1(A))
           => ( ~ r1_supinf_1(A,k3_supinf_2(B))
              & ~ r1_supinf_1(B,A) ) ) ) ) )).

fof(t59_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ~ ( ~ r1_supinf_1(B,k3_supinf_2(A))
              & ~ r1_supinf_1(A,B)
              & ~ ( ~ r1_supinf_1(A,k1_supinf_2)
                  & ~ r1_supinf_1(A,k4_extreal1(B)) ) ) ) ) )).

fof(t60_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( ( r1_supinf_1(k3_supinf_2(A),B)
              & r1_supinf_1(B,A) )
          <=> r1_supinf_1(k4_extreal1(B),A) ) ) ) )).

fof(t61_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ~ ( ~ ( A = k5_measure6
                  & B = k4_measure6 )
              & ~ ( A = k4_measure6
                  & B = k5_measure6 )
              & ~ r1_supinf_1(k4_extreal1(k2_supinf_2(A,B)),k2_supinf_2(k4_extreal1(A),k4_extreal1(B))) ) ) ) )).

fof(t62_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ~ ( ~ r1_supinf_1(A,k4_measure6)
          & ~ r1_supinf_1(k5_measure6,A)
          & A != k1_supinf_2
          & k2_extreal1(k4_extreal1(A),k4_extreal1(k3_extreal1(k10_mesfunc1,A))) != k10_mesfunc1 ) ) )).

fof(t63_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ( ( A = k5_measure6
          | A = k4_measure6 )
       => k2_extreal1(k4_extreal1(A),k4_extreal1(k3_extreal1(k10_mesfunc1,A))) = k1_supinf_2 ) ) )).

fof(t64_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ( A != k1_supinf_2
       => k4_extreal1(k3_extreal1(k10_mesfunc1,A)) = k3_extreal1(k10_mesfunc1,k4_extreal1(A)) ) ) )).

fof(t65_extreal2,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
              & k4_extreal1(k3_extreal1(A,B)) != k3_extreal1(k4_extreal1(A),k4_extreal1(B)) ) ) ) )).

fof(t66_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => k4_extreal1(A) = k4_extreal1(k3_supinf_2(A)) ) )).

fof(t67_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ( ( A = k5_measure6
          | A = k4_measure6 )
       => k4_extreal1(A) = k5_measure6 ) ) )).

fof(t68_extreal2,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(k4_supinf_2(k4_extreal1(A),k4_extreal1(B)),k4_extreal1(k4_supinf_2(A,B))) ) ) ) )).

fof(t69_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ~ ( ~ ( A = k5_measure6
                  & B = k5_measure6 )
              & ~ ( A = k4_measure6
                  & B = k4_measure6 )
              & ~ r1_supinf_1(k4_extreal1(k4_supinf_2(A,B)),k2_supinf_2(k4_extreal1(A),k4_extreal1(B))) ) ) ) )).

fof(t70_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => k4_extreal1(k4_extreal1(A)) = k4_extreal1(A) ) )).

fof(t71_extreal2,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)
             => ! [D] :
                  ( m1_subset_1(D,k3_supinf_1)
                 => ~ ( ~ ( A = k5_measure6
                          & B = k4_measure6 )
                      & ~ ( A = k4_measure6
                          & B = k5_measure6 )
                      & r1_supinf_1(k4_extreal1(A),C)
                      & r1_supinf_1(k4_extreal1(B),D)
                      & ~ r1_supinf_1(k4_extreal1(k2_supinf_2(A,B)),k2_supinf_2(C,D)) ) ) ) ) ) )).

fof(t72_extreal2,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(k4_extreal1(k4_supinf_2(k4_extreal1(A),k4_extreal1(B))),k4_extreal1(k4_supinf_2(A,B))) ) ) ) )).

fof(t73_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( r1_supinf_1(k1_supinf_2,k2_extreal1(A,B))
           => k4_extreal1(k2_supinf_2(A,B)) = k2_supinf_2(k4_extreal1(A),k4_extreal1(B)) ) ) ) )).

fof(t74_extreal2,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 )
                   => ( ~ ( ~ r1_xreal_0(C,D)
                          & r1_supinf_1(A,B) )
                      & ~ ( ~ r1_supinf_1(A,B)
                          & r1_xreal_0(C,D) )
                      & ( r1_xreal_0(D,C)
                       => r1_supinf_1(B,A) )
                      & ( r1_supinf_1(B,A)
                       => r1_xreal_0(D,C) ) ) ) ) ) ) ) )).

fof(d1_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( ( r1_supinf_1(A,B)
             => k1_extreal2(A,B) = A )
            & ( ~ r1_supinf_1(A,B)
             => k1_extreal2(A,B) = B ) ) ) ) )).

fof(d2_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( ( r1_supinf_1(B,A)
             => k2_extreal2(A,B) = A )
            & ( ~ r1_supinf_1(B,A)
             => k2_extreal2(A,B) = B ) ) ) ) )).

fof(t75_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( ( A = k4_measure6
              | B = k4_measure6 )
           => k1_extreal2(A,B) = k4_measure6 ) ) ) )).

fof(t76_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( ( A = k5_measure6
              | B = k5_measure6 )
           => k2_extreal2(A,B) = k5_measure6 ) ) ) )).

fof(t77_extreal2,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_extreal2(A,B) = k3_square_1(C,D)
                      & k2_extreal2(A,B) = k4_square_1(C,D) ) ) ) ) ) ) )).

fof(t78_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( r1_supinf_1(A,B)
           => k1_extreal2(B,A) = A ) ) ) )).

fof(t79_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( ~ r1_supinf_1(A,B)
           => k1_extreal2(B,A) = B ) ) ) )).

fof(t80_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ~ ( A != k5_measure6
              & B != k5_measure6
              & ~ ( A = k5_measure6
                  & B = k5_measure6 )
              & ~ ( A = k4_measure6
                  & B = k4_measure6 )
              & k1_extreal2(A,B) != k3_extreal1(k4_supinf_2(k2_supinf_2(A,B),k4_extreal1(k4_supinf_2(A,B))),k1_measure6(np__2)) ) ) ) )).

fof(t81_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( r1_supinf_1(k1_extreal2(A,B),A)
            & r1_supinf_1(k1_extreal2(B,A),A) ) ) ) )).

fof(t82_extreal2,axiom,(
    $true )).

fof(t83_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => k1_extreal2(A,B) = k1_extreal2(B,A) ) ) )).

fof(t84_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( k1_extreal2(A,B) = A
            | k1_extreal2(A,B) = B ) ) ) )).

fof(t85_extreal2,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(A,C) )
              <=> r1_supinf_1(A,k1_extreal2(B,C)) ) ) ) ) )).

fof(t86_extreal2,axiom,(
    $true )).

fof(t87_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( k1_extreal2(A,B) = B
           => r1_supinf_1(B,A) ) ) ) )).

fof(t88_extreal2,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)
             => k1_extreal2(A,k1_extreal2(B,C)) = k1_extreal2(k1_extreal2(A,B),C) ) ) ) )).

fof(t89_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( r1_supinf_1(A,B)
           => k2_extreal2(A,B) = B ) ) ) )).

fof(t90_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( ~ r1_supinf_1(A,B)
           => k2_extreal2(A,B) = A ) ) ) )).

fof(t91_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ~ ( A != k4_measure6
              & B != k4_measure6
              & ~ ( A = k5_measure6
                  & B = k5_measure6 )
              & ~ ( A = k4_measure6
                  & B = k4_measure6 )
              & k2_extreal2(A,B) != k3_extreal1(k2_supinf_2(k2_supinf_2(A,B),k4_extreal1(k4_supinf_2(A,B))),k1_measure6(np__2)) ) ) ) )).

fof(t92_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( r1_supinf_1(A,k2_extreal2(A,B))
            & r1_supinf_1(A,k2_extreal2(B,A)) ) ) ) )).

fof(t93_extreal2,axiom,(
    $true )).

fof(t94_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => k2_extreal2(A,B) = k2_extreal2(B,A) ) ) )).

fof(t95_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( k2_extreal2(A,B) = A
            | k2_extreal2(A,B) = B ) ) ) )).

fof(t96_extreal2,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,B) )
              <=> r1_supinf_1(k2_extreal2(A,C),B) ) ) ) ) )).

fof(t97_extreal2,axiom,(
    $true )).

fof(t98_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( k2_extreal2(A,B) = B
           => r1_supinf_1(A,B) ) ) ) )).

fof(t99_extreal2,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_extreal2(A,k2_extreal2(B,C)) = k2_extreal2(k2_extreal2(A,B),C) ) ) ) )).

fof(t100_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => k2_supinf_2(k1_extreal2(A,B),k2_extreal2(A,B)) = k2_supinf_2(A,B) ) ) )).

fof(t101_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( k2_extreal2(A,k1_extreal2(A,B)) = A
            & k2_extreal2(k1_extreal2(A,B),A) = A
            & k2_extreal2(k1_extreal2(B,A),A) = A
            & k2_extreal2(A,k1_extreal2(B,A)) = A ) ) ) )).

fof(t102_extreal2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( k1_extreal2(A,k2_extreal2(A,B)) = A
            & k1_extreal2(k2_extreal2(A,B),A) = A
            & k1_extreal2(k2_extreal2(B,A),A) = A
            & k1_extreal2(A,k2_extreal2(B,A)) = A ) ) ) )).

fof(t103_extreal2,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)
             => ( k1_extreal2(A,k2_extreal2(B,C)) = k2_extreal2(k1_extreal2(A,B),k1_extreal2(A,C))
                & k1_extreal2(k2_extreal2(B,C),A) = k2_extreal2(k1_extreal2(B,A),k1_extreal2(C,A)) ) ) ) ) )).

fof(t104_extreal2,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_extreal2(A,k1_extreal2(B,C)) = k1_extreal2(k2_extreal2(A,B),k2_extreal2(A,C))
                & k2_extreal2(k1_extreal2(B,C),A) = k1_extreal2(k2_extreal2(B,A),k2_extreal2(C,A)) ) ) ) ) )).

fof(dt_k1_extreal2,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k3_supinf_1)
        & m1_subset_1(B,k3_supinf_1) )
     => m1_subset_1(k1_extreal2(A,B),k3_supinf_1) ) )).

fof(commutativity_k1_extreal2,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k3_supinf_1)
        & m1_subset_1(B,k3_supinf_1) )
     => k1_extreal2(A,B) = k1_extreal2(B,A) ) )).

fof(idempotence_k1_extreal2,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k3_supinf_1)
        & m1_subset_1(B,k3_supinf_1) )
     => k1_extreal2(A,A) = A ) )).

fof(dt_k2_extreal2,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k3_supinf_1)
        & m1_subset_1(B,k3_supinf_1) )
     => m1_subset_1(k2_extreal2(A,B),k3_supinf_1) ) )).

fof(commutativity_k2_extreal2,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k3_supinf_1)
        & m1_subset_1(B,k3_supinf_1) )
     => k2_extreal2(A,B) = k2_extreal2(B,A) ) )).

fof(idempotence_k2_extreal2,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k3_supinf_1)
        & m1_subset_1(B,k3_supinf_1) )
     => k2_extreal2(A,A) = A ) )).
%------------------------------------------------------------------------------