SET007 Axioms: SET007+49.ax
%------------------------------------------------------------------------------
% File : SET007+49 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Basic Properties of Rational 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 : rat_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 117 ( 45 unt; 0 def)
% Number of atoms : 379 ( 75 equ)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 326 ( 64 ~; 7 |; 132 &)
% ( 14 <=>; 109 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 4 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 10 ( 8 usr; 1 prp; 0-3 aty)
% Number of functors : 16 ( 16 usr; 5 con; 0-2 aty)
% Number of variables : 132 ( 117 !; 15 ?)
% 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_rat_1,axiom,
? [A] :
( m1_subset_1(A,k1_numbers)
& v1_xcmplx_0(A)
& v1_xreal_0(A)
& v1_rat_1(A) ) ).
fof(rc2_rat_1,axiom,
? [A] : v1_rat_1(A) ).
fof(cc1_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ( v1_xcmplx_0(A)
& v1_xreal_0(A) ) ) ).
fof(cc2_rat_1,axiom,
! [A] :
( v1_int_1(A)
=> ( v1_xcmplx_0(A)
& v1_xreal_0(A)
& v1_rat_1(A) ) ) ).
fof(fc1_rat_1,axiom,
! [A,B] :
( ( v1_rat_1(A)
& v1_rat_1(B) )
=> ( v1_xcmplx_0(k3_xcmplx_0(A,B))
& v1_xreal_0(k3_xcmplx_0(A,B))
& v1_rat_1(k3_xcmplx_0(A,B)) ) ) ).
fof(fc2_rat_1,axiom,
! [A,B] :
( ( v1_rat_1(A)
& v1_rat_1(B) )
=> ( v1_xcmplx_0(k2_xcmplx_0(A,B))
& v1_xreal_0(k2_xcmplx_0(A,B))
& v1_rat_1(k2_xcmplx_0(A,B)) ) ) ).
fof(fc3_rat_1,axiom,
! [A,B] :
( ( v1_rat_1(A)
& v1_rat_1(B) )
=> ( v1_xcmplx_0(k6_xcmplx_0(A,B))
& v1_xreal_0(k6_xcmplx_0(A,B))
& v1_rat_1(k6_xcmplx_0(A,B)) ) ) ).
fof(fc4_rat_1,axiom,
! [A,B] :
( ( v1_rat_1(A)
& v1_int_1(B) )
=> ( v1_xcmplx_0(k2_xcmplx_0(A,B))
& v1_xreal_0(k2_xcmplx_0(A,B))
& v1_rat_1(k2_xcmplx_0(A,B)) ) ) ).
fof(fc5_rat_1,axiom,
! [A,B] :
( ( v1_rat_1(A)
& v1_int_1(B) )
=> ( v1_xcmplx_0(k6_xcmplx_0(A,B))
& v1_xreal_0(k6_xcmplx_0(A,B))
& v1_rat_1(k6_xcmplx_0(A,B)) ) ) ).
fof(fc6_rat_1,axiom,
! [A,B] :
( ( v1_rat_1(A)
& v1_int_1(B) )
=> ( v1_xcmplx_0(k3_xcmplx_0(A,B))
& v1_xreal_0(k3_xcmplx_0(A,B))
& v1_rat_1(k3_xcmplx_0(A,B)) ) ) ).
fof(fc7_rat_1,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_rat_1(B) )
=> ( v1_xcmplx_0(k2_xcmplx_0(A,B))
& v1_xreal_0(k2_xcmplx_0(A,B))
& v1_rat_1(k2_xcmplx_0(A,B)) ) ) ).
fof(fc8_rat_1,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_rat_1(B) )
=> ( v1_xcmplx_0(k6_xcmplx_0(A,B))
& v1_xreal_0(k6_xcmplx_0(A,B))
& v1_rat_1(k6_xcmplx_0(A,B)) ) ) ).
fof(fc9_rat_1,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_rat_1(B) )
=> ( v1_xcmplx_0(k3_xcmplx_0(A,B))
& v1_xreal_0(k3_xcmplx_0(A,B))
& v1_rat_1(k3_xcmplx_0(A,B)) ) ) ).
fof(fc10_rat_1,axiom,
! [A,B] :
( ( v1_rat_1(A)
& m1_subset_1(B,k5_numbers) )
=> ( v1_xcmplx_0(k2_xcmplx_0(A,B))
& v1_xreal_0(k2_xcmplx_0(A,B))
& v1_rat_1(k2_xcmplx_0(A,B)) ) ) ).
fof(fc11_rat_1,axiom,
! [A,B] :
( ( v1_rat_1(A)
& m1_subset_1(B,k5_numbers) )
=> ( v1_xcmplx_0(k6_xcmplx_0(A,B))
& v1_xreal_0(k6_xcmplx_0(A,B))
& v1_rat_1(k6_xcmplx_0(A,B)) ) ) ).
fof(fc12_rat_1,axiom,
! [A,B] :
( ( v1_rat_1(A)
& m1_subset_1(B,k5_numbers) )
=> ( v1_xcmplx_0(k3_xcmplx_0(A,B))
& v1_xreal_0(k3_xcmplx_0(A,B))
& v1_rat_1(k3_xcmplx_0(A,B)) ) ) ).
fof(fc13_rat_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& v1_rat_1(B) )
=> ( v1_xcmplx_0(k2_xcmplx_0(A,B))
& v1_xreal_0(k2_xcmplx_0(A,B))
& v1_rat_1(k2_xcmplx_0(A,B)) ) ) ).
fof(fc14_rat_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& v1_rat_1(B) )
=> ( v1_xcmplx_0(k6_xcmplx_0(A,B))
& v1_xreal_0(k6_xcmplx_0(A,B))
& v1_rat_1(k6_xcmplx_0(A,B)) ) ) ).
fof(fc15_rat_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& v1_rat_1(B) )
=> ( v1_xcmplx_0(k3_xcmplx_0(A,B))
& v1_xreal_0(k3_xcmplx_0(A,B))
& v1_rat_1(k3_xcmplx_0(A,B)) ) ) ).
fof(fc16_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ( v1_xcmplx_0(k4_xcmplx_0(A))
& v1_xreal_0(k4_xcmplx_0(A))
& v1_rat_1(k4_xcmplx_0(A)) ) ) ).
fof(fc17_rat_1,axiom,
! [A,B] :
( ( v1_rat_1(A)
& v1_rat_1(B) )
=> ( v1_xcmplx_0(k7_xcmplx_0(A,B))
& v1_xreal_0(k7_xcmplx_0(A,B))
& v1_rat_1(k7_xcmplx_0(A,B)) ) ) ).
fof(fc18_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ( v1_xcmplx_0(k5_xcmplx_0(A))
& v1_xreal_0(k5_xcmplx_0(A))
& v1_rat_1(k5_xcmplx_0(A)) ) ) ).
fof(d1_rat_1,axiom,
! [A] :
( A = k3_numbers
<=> ! [B] :
( r2_hidden(B,A)
<=> ? [C] :
( v1_int_1(C)
& ? [D] :
( v1_int_1(D)
& B = k7_xcmplx_0(C,D) ) ) ) ) ).
fof(d2_rat_1,axiom,
! [A] :
( v1_rat_1(A)
<=> r2_hidden(A,k3_numbers) ) ).
fof(t1_rat_1,axiom,
! [A] :
~ ( r2_hidden(A,k3_numbers)
& ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ~ ( C != np__0
& A = k7_xcmplx_0(B,C) ) ) ) ) ).
fof(t2_rat_1,axiom,
$true ).
fof(t3_rat_1,axiom,
! [A] :
~ ( v1_rat_1(A)
& ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ~ ( C != np__0
& A = k7_xcmplx_0(B,C) ) ) ) ) ).
fof(t4_rat_1,axiom,
$true ).
fof(t5_rat_1,axiom,
$true ).
fof(t6_rat_1,axiom,
! [A] :
( ? [B] :
( v1_int_1(B)
& ? [C] :
( v1_int_1(C)
& A = k7_xcmplx_0(B,C) ) )
=> v1_rat_1(A) ) ).
fof(t7_rat_1,axiom,
! [A] :
( v1_int_1(A)
=> v1_rat_1(A) ) ).
fof(t8_rat_1,axiom,
$true ).
fof(t9_rat_1,axiom,
$true ).
fof(t10_rat_1,axiom,
$true ).
fof(t11_rat_1,axiom,
$true ).
fof(t12_rat_1,axiom,
$true ).
fof(t13_rat_1,axiom,
$true ).
fof(t14_rat_1,axiom,
$true ).
fof(t15_rat_1,axiom,
$true ).
fof(t16_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ! [B] :
( v1_rat_1(B)
=> v1_rat_1(k7_xcmplx_0(A,B)) ) ) ).
fof(t17_rat_1,axiom,
$true ).
fof(t18_rat_1,axiom,
$true ).
fof(t19_rat_1,axiom,
$true ).
fof(t20_rat_1,axiom,
$true ).
fof(t21_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> v1_rat_1(k5_xcmplx_0(A)) ) ).
fof(t22_rat_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( ~ r1_xreal_0(B,A)
& ! [C] :
( v1_rat_1(C)
=> ~ ( ~ r1_xreal_0(C,A)
& ~ r1_xreal_0(B,C) ) ) ) ) ) ).
fof(t23_rat_1,axiom,
$true ).
fof(t24_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ? [B] :
( v1_int_1(B)
& ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& C != np__0
& A = k7_xcmplx_0(B,C) ) ) ) ).
fof(t25_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ? [B] :
( v1_int_1(B)
& ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& C != np__0
& A = k7_xcmplx_0(B,C)
& ! [D] :
( v1_int_1(D)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( A = k7_xcmplx_0(D,E)
=> ( E = np__0
| r1_xreal_0(C,E) ) ) ) ) ) ) ) ).
fof(d3_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( B = k1_rat_1(A)
<=> ( B != np__0
& ? [C] :
( v1_int_1(C)
& A = k7_xcmplx_0(C,B) )
& ! [C] :
( v1_int_1(C)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( A = k7_xcmplx_0(C,D)
=> ( D = np__0
| r1_xreal_0(B,D) ) ) ) ) ) ) ) ) ).
fof(d4_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> k2_rat_1(A) = k3_xcmplx_0(k1_rat_1(A),A) ) ).
fof(t26_rat_1,axiom,
$true ).
fof(t27_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ~ r1_xreal_0(k1_rat_1(A),np__0) ) ).
fof(t28_rat_1,axiom,
$true ).
fof(t29_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> r1_xreal_0(np__1,k1_rat_1(A)) ) ).
fof(t30_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ~ r1_xreal_0(k2_real_1(k1_rat_1(A)),np__0) ) ).
fof(t31_rat_1,axiom,
$true ).
fof(t32_rat_1,axiom,
$true ).
fof(t33_rat_1,axiom,
$true ).
fof(t34_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> r1_xreal_0(k2_real_1(k1_rat_1(A)),np__1) ) ).
fof(t35_rat_1,axiom,
$true ).
fof(t36_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ( k2_rat_1(A) = np__0
<=> A = np__0 ) ) ).
fof(t37_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ( A = k7_xcmplx_0(k2_rat_1(A),k1_rat_1(A))
& A = k3_xcmplx_0(k2_rat_1(A),k2_real_1(k1_rat_1(A)))
& A = k3_xcmplx_0(k2_real_1(k1_rat_1(A)),k2_rat_1(A)) ) ) ).
fof(t38_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ( A != np__0
=> k1_rat_1(A) = k7_xcmplx_0(k2_rat_1(A),A) ) ) ).
fof(t39_rat_1,axiom,
$true ).
fof(t40_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ( v1_int_1(A)
=> ( k1_rat_1(A) = np__1
& k2_rat_1(A) = A ) ) ) ).
fof(t41_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ( ( k2_rat_1(A) = A
| k1_rat_1(A) = np__1 )
=> v1_int_1(A) ) ) ).
fof(t42_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ( k2_rat_1(A) = A
<=> k1_rat_1(A) = np__1 ) ) ).
fof(t43_rat_1,axiom,
$true ).
fof(t44_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ( r1_xreal_0(np__0,A)
=> ( ( k2_rat_1(A) != A
& k1_rat_1(A) != np__1 )
| m2_subset_1(A,k1_numbers,k5_numbers) ) ) ) ).
fof(t45_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ( ~ ( ~ r1_xreal_0(k1_rat_1(A),np__1)
& v1_int_1(A) )
& ~ ( ~ v1_int_1(A)
& r1_xreal_0(k1_rat_1(A),np__1) ) ) ) ).
fof(t46_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ( ~ ( ~ r1_xreal_0(np__1,k2_real_1(k1_rat_1(A)))
& v1_int_1(A) )
& ~ ( ~ v1_int_1(A)
& r1_xreal_0(np__1,k2_real_1(k1_rat_1(A))) ) ) ) ).
fof(t47_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ( k2_rat_1(A) = k1_rat_1(A)
<=> A = np__1 ) ) ).
fof(t48_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ( k2_rat_1(A) = k1_real_1(k1_rat_1(A))
<=> A = k1_real_1(np__1) ) ) ).
fof(t49_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ( k4_xcmplx_0(k2_rat_1(A)) = k1_rat_1(A)
<=> A = k1_real_1(np__1) ) ) ).
fof(t50_rat_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_rat_1(B)
=> ( A != np__0
=> B = k7_xcmplx_0(k3_xcmplx_0(k2_rat_1(B),A),k3_xcmplx_0(k1_rat_1(B),A)) ) ) ) ).
fof(t51_rat_1,axiom,
$true ).
fof(t52_rat_1,axiom,
$true ).
fof(t53_rat_1,axiom,
$true ).
fof(t54_rat_1,axiom,
$true ).
fof(t55_rat_1,axiom,
$true ).
fof(t56_rat_1,axiom,
$true ).
fof(t57_rat_1,axiom,
$true ).
fof(t58_rat_1,axiom,
$true ).
fof(t59_rat_1,axiom,
$true ).
fof(t60_rat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_rat_1(C)
=> ~ ( A != np__0
& C = k7_xcmplx_0(B,A)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( B = k3_xcmplx_0(k2_rat_1(C),D)
& A = k2_nat_1(k1_rat_1(C),D) ) ) ) ) ) ) ).
fof(t61_rat_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_rat_1(C)
=> ~ ( C = k7_xcmplx_0(A,B)
& B != np__0
& ! [D] :
( v1_int_1(D)
=> ~ ( A = k3_xcmplx_0(k2_rat_1(C),D)
& B = k3_xcmplx_0(k1_rat_1(C),D) ) ) ) ) ) ) ).
fof(t62_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(B,np__1)
& ? [C] :
( v1_int_1(C)
& ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& k2_rat_1(A) = k3_xcmplx_0(C,B)
& k1_rat_1(A) = k2_nat_1(D,B) ) ) ) ) ) ).
fof(t63_rat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_rat_1(C)
=> ( ( C = k7_xcmplx_0(B,A)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(D,np__1)
& ? [E] :
( v1_int_1(E)
& ? [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
& B = k3_xcmplx_0(E,D)
& A = k2_nat_1(F,D) ) ) ) ) )
=> ( A = np__0
| ( A = k1_rat_1(C)
& B = k2_rat_1(C) ) ) ) ) ) ) ).
fof(t64_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ( ~ ( ~ r1_xreal_0(k1_real_1(np__1),A)
& r1_xreal_0(k1_real_1(k1_rat_1(A)),k2_rat_1(A)) )
& ~ ( ~ r1_xreal_0(k1_real_1(k1_rat_1(A)),k2_rat_1(A))
& r1_xreal_0(k1_real_1(np__1),A) ) ) ) ).
fof(t65_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ( r1_xreal_0(A,k1_real_1(np__1))
<=> r1_xreal_0(k2_rat_1(A),k1_real_1(k1_rat_1(A))) ) ) ).
fof(t66_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ( ~ ( ~ r1_xreal_0(k1_real_1(np__1),A)
& r1_xreal_0(k4_xcmplx_0(k2_rat_1(A)),k1_rat_1(A)) )
& ~ ( ~ r1_xreal_0(k4_xcmplx_0(k2_rat_1(A)),k1_rat_1(A))
& r1_xreal_0(k1_real_1(np__1),A) ) ) ) ).
fof(t67_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ( r1_xreal_0(A,k1_real_1(np__1))
<=> r1_xreal_0(k1_rat_1(A),k4_xcmplx_0(k2_rat_1(A))) ) ) ).
fof(t68_rat_1,axiom,
$true ).
fof(t69_rat_1,axiom,
$true ).
fof(t70_rat_1,axiom,
$true ).
fof(t71_rat_1,axiom,
$true ).
fof(t72_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ( ~ ( ~ r1_xreal_0(np__1,A)
& r1_xreal_0(k1_rat_1(A),k2_rat_1(A)) )
& ~ ( ~ r1_xreal_0(k1_rat_1(A),k2_rat_1(A))
& r1_xreal_0(np__1,A) ) ) ) ).
fof(t73_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ( r1_xreal_0(A,np__1)
<=> r1_xreal_0(k2_rat_1(A),k1_rat_1(A)) ) ) ).
fof(t74_rat_1,axiom,
$true ).
fof(t75_rat_1,axiom,
$true ).
fof(t76_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ( ~ ( ~ r1_xreal_0(np__0,A)
& r1_xreal_0(np__0,k2_rat_1(A)) )
& ~ ( ~ r1_xreal_0(np__0,k2_rat_1(A))
& r1_xreal_0(np__0,A) ) ) ) ).
fof(t77_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ( r1_xreal_0(A,np__0)
<=> r1_xreal_0(k2_rat_1(A),np__0) ) ) ).
fof(t78_rat_1,axiom,
$true ).
fof(t79_rat_1,axiom,
$true ).
fof(t80_rat_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_rat_1(B)
=> ( ~ ( ~ r1_xreal_0(B,A)
& r1_xreal_0(k2_rat_1(B),k3_xcmplx_0(A,k1_rat_1(B))) )
& ~ ( ~ r1_xreal_0(k2_rat_1(B),k3_xcmplx_0(A,k1_rat_1(B)))
& r1_xreal_0(B,A) ) ) ) ) ).
fof(t81_rat_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_rat_1(B)
=> ( r1_xreal_0(A,B)
<=> r1_xreal_0(k3_xcmplx_0(A,k1_rat_1(B)),k2_rat_1(B)) ) ) ) ).
fof(t82_rat_1,axiom,
$true ).
fof(t83_rat_1,axiom,
$true ).
fof(t84_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ! [B] :
( v1_rat_1(B)
=> ( ( k1_rat_1(A) = k1_rat_1(B)
& k2_rat_1(A) = k2_rat_1(B) )
=> A = B ) ) ) ).
fof(t85_rat_1,axiom,
$true ).
fof(t86_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ! [B] :
( v1_rat_1(B)
=> ( ~ ( ~ r1_xreal_0(B,A)
& r1_xreal_0(k3_xcmplx_0(k2_rat_1(B),k1_rat_1(A)),k3_xcmplx_0(k2_rat_1(A),k1_rat_1(B))) )
& ~ ( ~ r1_xreal_0(k3_xcmplx_0(k2_rat_1(B),k1_rat_1(A)),k3_xcmplx_0(k2_rat_1(A),k1_rat_1(B)))
& r1_xreal_0(B,A) ) ) ) ) ).
fof(t87_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ( k1_rat_1(k4_xcmplx_0(A)) = k1_rat_1(A)
& k2_rat_1(k4_xcmplx_0(A)) = k4_xcmplx_0(k2_rat_1(A)) ) ) ).
fof(t88_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ! [B] :
( v1_rat_1(B)
=> ( ( B = k7_xcmplx_0(np__1,A)
=> ( r1_xreal_0(A,np__0)
| ( k2_rat_1(B) = k1_rat_1(A)
& k1_rat_1(B) = k2_rat_1(A) ) ) )
& ( ( k2_rat_1(B) = k1_rat_1(A)
& k1_rat_1(B) = k2_rat_1(A) )
=> ( ~ r1_xreal_0(A,np__0)
& B = k7_xcmplx_0(np__1,A) ) ) ) ) ) ).
fof(t89_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> ! [B] :
( v1_rat_1(B)
=> ( ( B = k7_xcmplx_0(np__1,A)
=> ( r1_xreal_0(np__0,A)
| ( k2_rat_1(B) = k1_real_1(k1_rat_1(A))
& k1_rat_1(B) = k4_xcmplx_0(k2_rat_1(A)) ) ) )
& ( ( k2_rat_1(B) = k1_real_1(k1_rat_1(A))
& k1_rat_1(B) = k4_xcmplx_0(k2_rat_1(A)) )
=> ( ~ r1_xreal_0(np__0,A)
& B = k7_xcmplx_0(np__1,A) ) ) ) ) ) ).
fof(dt_k1_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> m2_subset_1(k1_rat_1(A),k1_numbers,k5_numbers) ) ).
fof(dt_k2_rat_1,axiom,
! [A] :
( v1_rat_1(A)
=> v1_int_1(k2_rat_1(A)) ) ).
%------------------------------------------------------------------------------