SET007 Axioms: SET007+48.ax
%------------------------------------------------------------------------------
% File : SET007+48 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Integers
% 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 : int_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 121 ( 29 unt; 0 def)
% Number of atoms : 430 ( 53 equ)
% Maximal formula atoms : 10 ( 3 avg)
% Number of connectives : 351 ( 42 ~; 7 |; 97 &)
% ( 18 <=>; 187 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 23 ( 21 usr; 1 prp; 0-3 aty)
% Number of functors : 23 ( 23 usr; 10 con; 0-2 aty)
% Number of variables : 173 ( 164 !; 9 ?)
% 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_int_1,axiom,
? [A] :
( m1_subset_1(A,k1_numbers)
& v1_xcmplx_0(A)
& v1_xreal_0(A)
& v1_int_1(A) ) ).
fof(rc2_int_1,axiom,
? [A] : v1_int_1(A) ).
fof(cc1_int_1,axiom,
! [A] :
( m1_subset_1(A,k4_numbers)
=> v1_int_1(A) ) ).
fof(cc2_int_1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& v4_ordinal2(A)
& v1_xcmplx_0(A)
& v1_xreal_0(A)
& ~ v3_xreal_0(A)
& v1_int_1(A) ) ) ).
fof(cc3_int_1,axiom,
! [A] :
( v4_ordinal2(A)
=> v1_int_1(A) ) ).
fof(cc4_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ( v1_xcmplx_0(A)
& v1_xreal_0(A) ) ) ).
fof(fc1_int_1,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_int_1(B) )
=> ( v1_xcmplx_0(k2_xcmplx_0(A,B))
& v1_xreal_0(k2_xcmplx_0(A,B))
& v1_int_1(k2_xcmplx_0(A,B)) ) ) ).
fof(fc2_int_1,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_int_1(B) )
=> ( v1_xcmplx_0(k3_xcmplx_0(A,B))
& v1_xreal_0(k3_xcmplx_0(A,B))
& v1_int_1(k3_xcmplx_0(A,B)) ) ) ).
fof(fc3_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ( v1_xcmplx_0(k4_xcmplx_0(A))
& v1_xreal_0(k4_xcmplx_0(A))
& v1_int_1(k4_xcmplx_0(A)) ) ) ).
fof(fc4_int_1,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_int_1(B) )
=> ( v1_xcmplx_0(k6_xcmplx_0(A,B))
& v1_xreal_0(k6_xcmplx_0(A,B))
& v1_int_1(k6_xcmplx_0(A,B)) ) ) ).
fof(fc5_int_1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_xcmplx_0(k4_xcmplx_0(A))
& v1_xreal_0(k4_xcmplx_0(A))
& ~ v2_xreal_0(k4_xcmplx_0(A))
& v1_int_1(k4_xcmplx_0(A)) ) ) ).
fof(fc6_int_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& v1_int_1(B) )
=> ( v1_xcmplx_0(k2_xcmplx_0(B,A))
& v1_xreal_0(k2_xcmplx_0(B,A))
& v1_int_1(k2_xcmplx_0(B,A)) ) ) ).
fof(fc7_int_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& v1_int_1(B) )
=> ( v1_xcmplx_0(k3_xcmplx_0(B,A))
& v1_xreal_0(k3_xcmplx_0(B,A))
& v1_int_1(k3_xcmplx_0(B,A)) ) ) ).
fof(fc8_int_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& v1_int_1(B) )
=> ( v1_xcmplx_0(k6_xcmplx_0(B,A))
& v1_xreal_0(k6_xcmplx_0(B,A))
& v1_int_1(k6_xcmplx_0(B,A)) ) ) ).
fof(fc9_int_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> ( v1_xcmplx_0(k6_xcmplx_0(A,B))
& v1_xreal_0(k6_xcmplx_0(A,B))
& v1_int_1(k6_xcmplx_0(A,B)) ) ) ).
fof(fc10_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v1_xcmplx_0(k3_int_1(A))
& v1_xreal_0(k3_int_1(A)) ) ) ).
fof(d1_int_1,axiom,
! [A] :
( A = k4_numbers
<=> ! [B] :
( r2_hidden(B,A)
<=> ~ ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( B != C
& B != k1_real_1(C) ) ) ) ) ).
fof(d2_int_1,axiom,
! [A] :
( v1_int_1(A)
<=> r2_hidden(A,k4_numbers) ) ).
fof(t1_int_1,axiom,
$true ).
fof(t2_int_1,axiom,
$true ).
fof(t3_int_1,axiom,
$true ).
fof(t4_int_1,axiom,
$true ).
fof(t5_int_1,axiom,
$true ).
fof(t6_int_1,axiom,
$true ).
fof(t7_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( ( A = B
| A = k4_xcmplx_0(B) )
=> v1_int_1(A) ) ) ) ).
fof(t8_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ~ ( v1_int_1(A)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( A != B
& A != k1_real_1(B) ) ) ) ) ).
fof(t9_int_1,axiom,
$true ).
fof(t10_int_1,axiom,
$true ).
fof(t11_int_1,axiom,
$true ).
fof(t12_int_1,axiom,
$true ).
fof(t13_int_1,axiom,
$true ).
fof(t14_int_1,axiom,
$true ).
fof(t15_int_1,axiom,
$true ).
fof(t16_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ( r1_xreal_0(np__0,A)
=> r2_hidden(A,k5_numbers) ) ) ).
fof(t17_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v1_int_1(A)
=> ( v1_int_1(k2_xcmplx_0(A,np__1))
& v1_int_1(k6_xcmplx_0(A,np__1)) ) ) ) ).
fof(t18_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( r1_xreal_0(A,B)
=> r2_hidden(k6_xcmplx_0(B,A),k5_numbers) ) ) ) ).
fof(t19_int_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( k2_xcmplx_0(B,A) = C
=> r1_xreal_0(B,C) ) ) ) ) ).
fof(t20_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( ~ r1_xreal_0(B,A)
=> r1_xreal_0(k2_xcmplx_0(A,np__1),B) ) ) ) ).
fof(t21_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ( ~ r1_xreal_0(np__0,A)
=> r1_xreal_0(A,k1_real_1(np__1)) ) ) ).
fof(t22_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( k3_xcmplx_0(A,B) = np__1
<=> ( ( A = np__1
& B = np__1 )
| ( A = k1_real_1(np__1)
& B = k1_real_1(np__1) ) ) ) ) ) ).
fof(t23_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( k3_xcmplx_0(A,B) = k1_real_1(np__1)
<=> ( ( A = k1_real_1(np__1)
& B = np__1 )
| ( A = np__1
& B = k1_real_1(np__1) ) ) ) ) ) ).
fof(d3_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( r1_int_1(A,B,C)
<=> ? [D] :
( v1_int_1(D)
& k3_xcmplx_0(C,D) = k6_xcmplx_0(A,B) ) ) ) ) ) ).
fof(t24_int_1,axiom,
$true ).
fof(t25_int_1,axiom,
$true ).
fof(t26_int_1,axiom,
$true ).
fof(t27_int_1,axiom,
$true ).
fof(t28_int_1,axiom,
$true ).
fof(t29_int_1,axiom,
$true ).
fof(t30_int_1,axiom,
$true ).
fof(t31_int_1,axiom,
$true ).
fof(t32_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> r1_int_1(A,A,B) ) ) ).
fof(t33_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ( r1_int_1(A,np__0,A)
& r1_int_1(np__0,A,A) ) ) ).
fof(t34_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> r1_int_1(A,B,np__1) ) ) ).
fof(t35_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( r1_int_1(A,B,C)
=> r1_int_1(B,A,C) ) ) ) ) ).
fof(t36_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ! [D] :
( v1_int_1(D)
=> ( ( r1_int_1(A,B,C)
& r1_int_1(B,D,C) )
=> r1_int_1(A,D,C) ) ) ) ) ) ).
fof(t37_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ! [D] :
( v1_int_1(D)
=> ! [E] :
( v1_int_1(E)
=> ( ( r1_int_1(A,B,C)
& r1_int_1(D,E,C) )
=> r1_int_1(k2_xcmplx_0(A,D),k2_xcmplx_0(B,E),C) ) ) ) ) ) ) ).
fof(t38_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ! [D] :
( v1_int_1(D)
=> ! [E] :
( v1_int_1(E)
=> ( ( r1_int_1(A,B,C)
& r1_int_1(D,E,C) )
=> r1_int_1(k6_xcmplx_0(A,D),k6_xcmplx_0(B,E),C) ) ) ) ) ) ) ).
fof(t39_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ! [D] :
( v1_int_1(D)
=> ! [E] :
( v1_int_1(E)
=> ( ( r1_int_1(A,B,C)
& r1_int_1(D,E,C) )
=> r1_int_1(k3_xcmplx_0(A,D),k3_xcmplx_0(B,E),C) ) ) ) ) ) ) ).
fof(t40_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ! [D] :
( v1_int_1(D)
=> ( r1_int_1(k2_xcmplx_0(A,B),C,D)
<=> r1_int_1(A,k6_xcmplx_0(C,B),D) ) ) ) ) ) ).
fof(t41_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ! [D] :
( v1_int_1(D)
=> ! [E] :
( v1_int_1(E)
=> ( ( k3_xcmplx_0(A,B) = C
& r1_int_1(D,E,C) )
=> r1_int_1(D,E,A) ) ) ) ) ) ) ).
fof(t42_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( r1_int_1(A,B,C)
<=> r1_int_1(k2_xcmplx_0(A,C),B,C) ) ) ) ) ).
fof(t43_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( r1_int_1(A,B,C)
<=> r1_int_1(k6_xcmplx_0(A,C),B,C) ) ) ) ) ).
fof(t44_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( ( r1_xreal_0(B,A)
& r1_xreal_0(C,A) )
=> ( r1_xreal_0(B,k6_xcmplx_0(A,np__1))
| r1_xreal_0(C,k6_xcmplx_0(A,np__1))
| B = C ) ) ) ) ) ).
fof(t45_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( ( r1_xreal_0(A,B)
& r1_xreal_0(A,C) )
=> ( r1_xreal_0(k2_xcmplx_0(A,np__1),B)
| r1_xreal_0(k2_xcmplx_0(A,np__1),C)
| B = C ) ) ) ) ) ).
fof(d4_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_int_1(B)
=> ( B = k1_int_1(A)
<=> ( r1_xreal_0(B,A)
& ~ r1_xreal_0(B,k6_xcmplx_0(A,np__1)) ) ) ) ) ).
fof(t46_int_1,axiom,
$true ).
fof(t47_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k1_int_1(A) = A
<=> v1_int_1(A) ) ) ).
fof(t48_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( ~ ( ~ r1_xreal_0(A,k1_int_1(A))
& v1_int_1(A) )
& ~ ( ~ v1_int_1(A)
& r1_xreal_0(A,k1_int_1(A)) ) ) ) ).
fof(t49_int_1,axiom,
$true ).
fof(t50_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( ~ r1_xreal_0(A,k6_xcmplx_0(k1_int_1(A),np__1))
& ~ r1_xreal_0(k2_xcmplx_0(A,np__1),k1_int_1(A)) ) ) ).
fof(t51_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_int_1(B)
=> k2_xcmplx_0(k1_int_1(A),B) = k1_int_1(k2_xcmplx_0(A,B)) ) ) ).
fof(t52_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ~ r1_xreal_0(k2_xcmplx_0(k1_int_1(A),np__1),A) ) ).
fof(d5_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_int_1(B)
=> ( B = k2_int_1(A)
<=> ( r1_xreal_0(A,B)
& ~ r1_xreal_0(k2_xcmplx_0(A,np__1),B) ) ) ) ) ).
fof(t53_int_1,axiom,
$true ).
fof(t54_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k2_int_1(A) = A
<=> v1_int_1(A) ) ) ).
fof(t55_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( ~ ( ~ r1_xreal_0(k2_int_1(A),A)
& v1_int_1(A) )
& ~ ( ~ v1_int_1(A)
& r1_xreal_0(k2_int_1(A),A) ) ) ) ).
fof(t56_int_1,axiom,
$true ).
fof(t57_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( ~ r1_xreal_0(k2_int_1(A),k6_xcmplx_0(A,np__1))
& ~ r1_xreal_0(k2_xcmplx_0(k2_int_1(A),np__1),A) ) ) ).
fof(t58_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_int_1(B)
=> k2_xcmplx_0(k2_int_1(A),B) = k2_int_1(k2_xcmplx_0(A,B)) ) ) ).
fof(t59_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k1_int_1(A) = k2_int_1(A)
<=> v1_int_1(A) ) ) ).
fof(t60_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( ~ ( ~ r1_xreal_0(k2_int_1(A),k1_int_1(A))
& v1_int_1(A) )
& ~ ( ~ v1_int_1(A)
& r1_xreal_0(k2_int_1(A),k1_int_1(A)) ) ) ) ).
fof(t61_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> r1_xreal_0(k1_int_1(A),k2_int_1(A)) ) ).
fof(t62_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> k1_int_1(k2_int_1(A)) = k2_int_1(A) ) ).
fof(t63_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> k1_int_1(k1_int_1(A)) = k1_int_1(A) ) ).
fof(t64_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> k2_int_1(k2_int_1(A)) = k2_int_1(A) ) ).
fof(t65_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> k2_int_1(k1_int_1(A)) = k1_int_1(A) ) ).
fof(t66_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k1_int_1(A) = k2_int_1(A)
<=> k2_xcmplx_0(k1_int_1(A),np__1) != k2_int_1(A) ) ) ).
fof(d6_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> k3_int_1(A) = k6_xcmplx_0(A,k1_int_1(A)) ) ).
fof(t67_int_1,axiom,
$true ).
fof(t68_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> A = k2_xcmplx_0(k1_int_1(A),k4_int_1(A)) ) ).
fof(t69_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( ~ r1_xreal_0(np__1,k4_int_1(A))
& r1_xreal_0(np__0,k4_int_1(A)) ) ) ).
fof(t70_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> k1_int_1(k4_int_1(A)) = np__0 ) ).
fof(t71_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k4_int_1(A) = np__0
<=> v1_int_1(A) ) ) ).
fof(t72_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( ~ ( ~ r1_xreal_0(k4_int_1(A),np__0)
& v1_int_1(A) )
& ~ ( ~ v1_int_1(A)
& r1_xreal_0(k4_int_1(A),np__0) ) ) ) ).
fof(d7_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> k5_int_1(A,B) = k1_int_1(k7_xcmplx_0(A,B)) ) ) ).
fof(d8_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( ( B != np__0
=> k6_int_1(A,B) = k6_xcmplx_0(A,k3_xcmplx_0(k5_int_1(A,B),B)) )
& ( B = np__0
=> k6_int_1(A,B) = np__0 ) ) ) ) ).
fof(d9_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( r2_int_1(A,B)
<=> ? [C] :
( v1_int_1(C)
& B = k3_xcmplx_0(A,C) ) ) ) ) ).
fof(t73_int_1,axiom,
$true ).
fof(t74_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( A != np__0
=> k1_int_1(k7_xcmplx_0(A,A)) = np__1 ) ) ).
fof(t75_int_1,axiom,
! [A] :
( v1_int_1(A)
=> k5_int_1(A,np__0) = np__0 ) ).
fof(t76_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ( A != np__0
=> k5_int_1(A,A) = np__1 ) ) ).
fof(t77_int_1,axiom,
! [A] :
( v1_int_1(A)
=> k6_int_1(A,A) = np__0 ) ).
fof(t78_int_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_int_1(B)
=> ~ ( ~ r1_xreal_0(B,A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( C = k6_xcmplx_0(B,A)
& r1_xreal_0(np__1,C) ) ) ) ) ) ).
fof(t79_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( ~ r1_xreal_0(B,A)
=> r1_xreal_0(A,k6_xcmplx_0(B,np__1)) ) ) ) ).
fof(t80_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r1_xreal_0(np__0,A)
=> ( r1_xreal_0(np__0,k2_int_1(A))
& r1_xreal_0(np__0,k1_int_1(A))
& m2_subset_1(k2_int_1(A),k1_numbers,k5_numbers)
& m2_subset_1(k1_int_1(A),k1_numbers,k5_numbers) ) ) ) ).
fof(t81_int_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> r1_xreal_0(A,k1_int_1(B)) ) ) ) ).
fof(t82_int_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> r1_xreal_0(np__0,k5_int_1(A,B)) ) ) ).
fof(s1_int_1,axiom,
? [A] :
( m1_subset_1(A,k1_zfmisc_1(k4_numbers))
& ! [B] :
( v1_int_1(B)
=> ( r2_hidden(B,A)
<=> p1_s1_int_1(B) ) ) ) ).
fof(s2_int_1,axiom,
( ( p1_s2_int_1(f1_s2_int_1)
& ! [A] :
( v1_int_1(A)
=> ( ( r1_xreal_0(f1_s2_int_1,A)
& p1_s2_int_1(A) )
=> p1_s2_int_1(k2_xcmplx_0(A,np__1)) ) ) )
=> ! [A] :
( v1_int_1(A)
=> ( r1_xreal_0(f1_s2_int_1,A)
=> p1_s2_int_1(A) ) ) ) ).
fof(s3_int_1,axiom,
( ( p1_s3_int_1(f1_s3_int_1)
& ! [A] :
( v1_int_1(A)
=> ( ( r1_xreal_0(A,f1_s3_int_1)
& p1_s3_int_1(A) )
=> p1_s3_int_1(k6_xcmplx_0(A,np__1)) ) ) )
=> ! [A] :
( v1_int_1(A)
=> ( r1_xreal_0(A,f1_s3_int_1)
=> p1_s3_int_1(A) ) ) ) ).
fof(s4_int_1,axiom,
( ( p1_s4_int_1(f1_s4_int_1)
& ! [A] :
( v1_int_1(A)
=> ( p1_s4_int_1(A)
=> ( p1_s4_int_1(k6_xcmplx_0(A,np__1))
& p1_s4_int_1(k2_xcmplx_0(A,np__1)) ) ) ) )
=> ! [A] :
( v1_int_1(A)
=> p1_s4_int_1(A) ) ) ).
fof(s5_int_1,axiom,
( ( ! [A] :
( v1_int_1(A)
=> ( p1_s5_int_1(A)
=> r1_xreal_0(f1_s5_int_1,A) ) )
& ? [A] :
( v1_int_1(A)
& p1_s5_int_1(A) ) )
=> ? [A] :
( v1_int_1(A)
& p1_s5_int_1(A)
& ! [B] :
( v1_int_1(B)
=> ( p1_s5_int_1(B)
=> r1_xreal_0(A,B) ) ) ) ) ).
fof(s6_int_1,axiom,
( ( ! [A] :
( v1_int_1(A)
=> ( p1_s6_int_1(A)
=> r1_xreal_0(A,f1_s6_int_1) ) )
& ? [A] :
( v1_int_1(A)
& p1_s6_int_1(A) ) )
=> ? [A] :
( v1_int_1(A)
& p1_s6_int_1(A)
& ! [B] :
( v1_int_1(B)
=> ( p1_s6_int_1(B)
=> r1_xreal_0(B,A) ) ) ) ) ).
fof(reflexivity_r2_int_1,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_int_1(B) )
=> r2_int_1(A,A) ) ).
fof(dt_k1_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> v1_int_1(k1_int_1(A)) ) ).
fof(dt_k2_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> v1_int_1(k2_int_1(A)) ) ).
fof(dt_k3_int_1,axiom,
$true ).
fof(dt_k4_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> m1_subset_1(k4_int_1(A),k1_numbers) ) ).
fof(redefinition_k4_int_1,axiom,
! [A] :
( v1_xreal_0(A)
=> k4_int_1(A) = k3_int_1(A) ) ).
fof(dt_k5_int_1,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_int_1(B) )
=> v1_int_1(k5_int_1(A,B)) ) ).
fof(dt_k6_int_1,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_int_1(B) )
=> v1_int_1(k6_int_1(A,B)) ) ).
%------------------------------------------------------------------------------