SET007 Axioms: SET007+133.ax
%------------------------------------------------------------------------------
% File : SET007+133 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Integer and Rational Exponents
% 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 : prepower [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 138 ( 17 unt; 0 def)
% Number of atoms : 664 ( 97 equ)
% Maximal formula atoms : 16 ( 4 avg)
% Number of connectives : 612 ( 86 ~; 18 |; 183 &)
% ( 7 <=>; 318 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 19 ( 17 usr; 1 prp; 0-3 aty)
% Number of functors : 38 ( 38 usr; 6 con; 0-4 aty)
% Number of variables : 276 ( 272 !; 4 ?)
% 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(fc1_prepower,axiom,
! [A] :
( v1_int_1(A)
=> ( v4_ordinal2(k16_complex1(A))
& v1_xcmplx_0(k16_complex1(A))
& v1_xreal_0(k16_complex1(A))
& ~ v3_xreal_0(k16_complex1(A))
& v1_int_1(k16_complex1(A))
& v1_rat_1(k16_complex1(A)) ) ) ).
fof(fc2_prepower,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_int_1(B) )
=> ( v1_xcmplx_0(k6_prepower(A,B))
& v1_xreal_0(k6_prepower(A,B)) ) ) ).
fof(fc3_prepower,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_rat_1(B) )
=> ( v1_xcmplx_0(k8_prepower(A,B))
& v1_xreal_0(k8_prepower(A,B)) ) ) ).
fof(rc1_prepower,axiom,
? [A] :
( m1_relset_1(A,k5_numbers,k1_numbers)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v1_seq_1(A)
& v1_prepower(A) ) ).
fof(t1_prepower,axiom,
$true ).
fof(t2_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(B)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_xreal_0(A,k2_seq_1(k5_numbers,k1_numbers,B,C)) ) )
=> r1_xreal_0(A,k2_seq_2(B)) ) ) ) ).
fof(t3_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(B)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,C),A) ) )
=> r1_xreal_0(k2_seq_2(B),A) ) ) ) ).
fof(d1_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( B = k2_prepower(A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k2_newton(A,C) ) ) ) ) ).
fof(t4_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( B = k2_prepower(A)
<=> ( k2_seq_1(k5_numbers,k1_numbers,B,np__0) = np__1
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,B,k1_nat_1(C,np__1)) = k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,B,C),A) ) ) ) ) ) ).
fof(t5_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ( A != np__0
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k2_prepower(A),B) != np__0 ) ) ) ).
fof(t6_prepower,axiom,
$true ).
fof(t7_prepower,axiom,
$true ).
fof(t8_prepower,axiom,
$true ).
fof(t9_prepower,axiom,
$true ).
fof(t10_prepower,axiom,
$true ).
fof(t11_prepower,axiom,
$true ).
fof(t12_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v4_ordinal2(B)
=> ~ ( np__0 != A
& np__0 = k2_newton(A,B) ) ) ) ).
fof(t13_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v4_ordinal2(B)
=> ~ ( ~ r1_xreal_0(A,np__0)
& r1_xreal_0(k2_newton(A,B),np__0) ) ) ) ).
fof(t14_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v4_ordinal2(B)
=> k2_newton(k7_xcmplx_0(np__1,A),B) = k7_xcmplx_0(np__1,k2_newton(A,B)) ) ) ).
fof(t15_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v4_ordinal2(C)
=> k2_newton(k7_xcmplx_0(A,B),C) = k7_xcmplx_0(k2_newton(A,C),k2_newton(B,C)) ) ) ) ).
fof(t16_prepower,axiom,
$true ).
fof(t17_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v4_ordinal2(C)
=> ( r1_xreal_0(A,B)
=> ( r1_xreal_0(A,np__0)
| r1_xreal_0(k2_newton(A,C),k2_newton(B,C)) ) ) ) ) ) ).
fof(t18_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v4_ordinal2(C)
=> ~ ( r1_xreal_0(np__0,A)
& ~ r1_xreal_0(B,A)
& r1_xreal_0(np__1,C)
& r1_xreal_0(k2_newton(B,C),k2_newton(A,C)) ) ) ) ) ).
fof(t19_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( r1_xreal_0(np__1,A)
=> r1_xreal_0(np__1,k2_newton(A,B)) ) ) ) ).
fof(t20_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( ( r1_xreal_0(np__1,A)
& r1_xreal_0(np__1,B) )
=> r1_xreal_0(A,k2_newton(A,B)) ) ) ) ).
fof(t21_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v4_ordinal2(B)
=> ~ ( ~ r1_xreal_0(A,np__1)
& r1_xreal_0(np__2,B)
& r1_xreal_0(k2_newton(A,B),A) ) ) ) ).
fof(t22_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( ( r1_xreal_0(A,np__1)
& r1_xreal_0(np__1,B) )
=> ( r1_xreal_0(A,np__0)
| r1_xreal_0(k2_newton(A,B),A) ) ) ) ) ).
fof(t23_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v4_ordinal2(B)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(np__1,A)
& r1_xreal_0(np__2,B)
& r1_xreal_0(A,k2_newton(A,B)) ) ) ) ).
fof(t24_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( ~ r1_xreal_0(A,k1_real_1(np__1))
=> r1_xreal_0(k2_xcmplx_0(np__1,k3_xcmplx_0(B,A)),k2_newton(k2_xcmplx_0(np__1,A),B)) ) ) ) ).
fof(t25_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v4_ordinal2(B)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(np__1,A)
& ~ r1_xreal_0(k2_newton(k2_xcmplx_0(np__1,A),B),k2_xcmplx_0(np__1,k3_xcmplx_0(k3_newton(np__3,B),A))) ) ) ) ).
fof(t26_prepower,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_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) = k3_prepower(k2_seq_1(k5_numbers,k1_numbers,B,D),A) ) )
=> ( v4_seq_2(C)
& k2_seq_2(C) = k3_prepower(k2_seq_2(B),A) ) ) ) ) ) ).
fof(d2_prepower,axiom,
$true ).
fof(d3_prepower,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(np__1,A)
=> ! [C] :
( v1_xreal_0(C)
=> ( ( ~ r1_xreal_0(B,np__0)
=> ( C = k4_prepower(A,B)
<=> ( k2_newton(C,A) = B
& ~ r1_xreal_0(C,np__0) ) ) )
& ( B = np__0
=> ( C = k4_prepower(A,B)
<=> C = np__0 ) ) ) ) ) ) ) ).
fof(t27_prepower,axiom,
$true ).
fof(t28_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(np__1,B) )
=> ( k2_newton(k4_prepower(B,A),B) = A
& k4_prepower(B,k2_newton(A,B)) = A ) ) ) ) ).
fof(t29_prepower,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> k5_prepower(A,np__1) = np__1 ) ) ).
fof(t30_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r1_xreal_0(np__0,A)
=> k4_prepower(np__1,A) = A ) ) ).
fof(t31_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(np__0,B)
& r1_xreal_0(np__1,C) )
=> k4_prepower(C,k3_xcmplx_0(A,B)) = k3_xcmplx_0(k4_prepower(C,A),k4_prepower(C,B)) ) ) ) ) ).
fof(t32_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,B)
=> ( r1_xreal_0(A,np__0)
| k4_prepower(B,k7_xcmplx_0(np__1,A)) = k7_xcmplx_0(np__1,k4_prepower(B,A)) ) ) ) ) ).
fof(t33_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(np__1,C) )
=> ( r1_xreal_0(B,np__0)
| k4_prepower(C,k7_xcmplx_0(A,B)) = k7_xcmplx_0(k4_prepower(C,A),k4_prepower(C,B)) ) ) ) ) ) ).
fof(t34_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(np__1,B)
& r1_xreal_0(np__1,C) )
=> k4_prepower(B,k4_prepower(C,A)) = k4_prepower(k2_nat_1(B,C),A) ) ) ) ) ).
fof(t35_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(np__1,B)
& r1_xreal_0(np__1,C) )
=> k3_xcmplx_0(k4_prepower(B,A),k4_prepower(C,A)) = k4_prepower(k2_nat_1(B,C),k2_newton(A,k1_nat_1(B,C))) ) ) ) ) ).
fof(t36_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(A,B)
& r1_xreal_0(np__1,C) )
=> r1_xreal_0(k4_prepower(C,A),k4_prepower(C,B)) ) ) ) ) ).
fof(t37_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__0,A)
& ~ r1_xreal_0(B,A)
& r1_xreal_0(np__1,C)
& r1_xreal_0(k4_prepower(C,B),k4_prepower(C,A)) ) ) ) ) ).
fof(t38_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,A)
& r1_xreal_0(np__1,B) )
=> ( r1_xreal_0(np__1,k4_prepower(B,A))
& r1_xreal_0(k4_prepower(B,A),A) ) ) ) ) ).
fof(t39_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(np__1,B) )
=> ( r1_xreal_0(np__1,A)
| ( r1_xreal_0(A,k4_prepower(B,A))
& ~ r1_xreal_0(np__1,k4_prepower(B,A)) ) ) ) ) ) ).
fof(t40_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,B)
=> ( r1_xreal_0(A,np__0)
| r1_xreal_0(k6_xcmplx_0(k4_prepower(B,A),np__1),k7_xcmplx_0(k6_xcmplx_0(A,np__1),B)) ) ) ) ) ).
fof(t41_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r1_xreal_0(np__0,A)
=> k4_prepower(np__2,A) = k8_square_1(A) ) ) ).
fof(t42_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,C)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k4_prepower(C,A) ) )
=> ( r1_xreal_0(A,np__0)
| ( v4_seq_2(B)
& k2_seq_2(B) = np__1 ) ) ) ) ) ).
fof(d4_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_int_1(B)
=> ( ( r1_xreal_0(np__0,B)
=> k6_prepower(A,B) = k2_newton(A,k1_prepower(B)) )
& ( ~ r1_xreal_0(np__0,B)
=> k6_prepower(A,B) = k5_xcmplx_0(k2_newton(A,k1_prepower(B))) ) ) ) ) ).
fof(t43_prepower,axiom,
$true ).
fof(t44_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> k6_prepower(A,np__0) = np__1 ) ).
fof(t45_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> k6_prepower(A,np__1) = A ) ).
fof(t46_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v4_ordinal2(B)
=> k6_prepower(A,B) = k2_newton(A,B) ) ) ).
fof(t47_prepower,axiom,
! [A] :
( v1_int_1(A)
=> k7_prepower(np__1,A) = np__1 ) ).
fof(t48_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_int_1(B)
=> ~ ( A != np__0
& k6_prepower(A,B) = np__0 ) ) ) ).
fof(t49_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_int_1(B)
=> ~ ( ~ r1_xreal_0(A,np__0)
& r1_xreal_0(k6_prepower(A,B),np__0) ) ) ) ).
fof(t50_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_int_1(C)
=> k6_prepower(k3_xcmplx_0(A,B),C) = k3_xcmplx_0(k6_prepower(A,C),k6_prepower(B,C)) ) ) ) ).
fof(t51_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_int_1(B)
=> ( A != np__0
=> k6_prepower(A,k4_xcmplx_0(B)) = k7_xcmplx_0(np__1,k6_prepower(A,B)) ) ) ) ).
fof(t52_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_int_1(B)
=> k6_prepower(k7_xcmplx_0(np__1,A),B) = k7_xcmplx_0(np__1,k6_prepower(A,B)) ) ) ).
fof(t53_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ( A != np__0
=> k6_prepower(A,k6_xcmplx_0(B,C)) = k7_xcmplx_0(k2_newton(A,B),k2_newton(A,C)) ) ) ) ) ).
fof(t54_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( A != np__0
=> k6_prepower(A,k2_xcmplx_0(B,C)) = k3_xcmplx_0(k6_prepower(A,B),k6_prepower(A,C)) ) ) ) ) ).
fof(t55_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> k6_prepower(k6_prepower(A,B),C) = k6_prepower(A,k3_xcmplx_0(B,C)) ) ) ) ).
fof(t56_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( v1_int_1(C)
=> ( r1_xreal_0(np__1,B)
=> ( r1_xreal_0(A,np__0)
| k6_prepower(k4_prepower(B,A),C) = k4_prepower(B,k6_prepower(A,C)) ) ) ) ) ) ).
fof(d5_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_rat_1(B)
=> k8_prepower(A,B) = k4_prepower(k1_rat_1(B),k6_prepower(A,k2_rat_1(B))) ) ) ).
fof(t57_prepower,axiom,
$true ).
fof(t58_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_rat_1(B)
=> ( B = np__0
=> ( r1_xreal_0(A,np__0)
| k8_prepower(A,B) = np__1 ) ) ) ) ).
fof(t59_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_rat_1(B)
=> ( B = np__1
=> ( r1_xreal_0(A,np__0)
| k8_prepower(A,B) = A ) ) ) ) ).
fof(t60_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( v1_rat_1(C)
=> ( C = B
=> ( r1_xreal_0(A,np__0)
| k8_prepower(A,C) = k2_newton(A,B) ) ) ) ) ) ).
fof(t61_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( v1_rat_1(C)
=> ( ( r1_xreal_0(np__1,B)
& C = k2_real_1(B) )
=> ( r1_xreal_0(A,np__0)
| k8_prepower(A,C) = k4_prepower(B,A) ) ) ) ) ) ).
fof(t62_prepower,axiom,
! [A] :
( v1_rat_1(A)
=> k9_prepower(np__1,A) = np__1 ) ).
fof(t63_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_rat_1(B)
=> ~ ( ~ r1_xreal_0(A,np__0)
& r1_xreal_0(k8_prepower(A,B),np__0) ) ) ) ).
fof(t64_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_rat_1(B)
=> ! [C] :
( v1_rat_1(C)
=> ( ~ r1_xreal_0(A,np__0)
=> k3_xcmplx_0(k8_prepower(A,B),k8_prepower(A,C)) = k8_prepower(A,k2_xcmplx_0(B,C)) ) ) ) ) ).
fof(t65_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_rat_1(B)
=> ( ~ r1_xreal_0(A,np__0)
=> k7_xcmplx_0(np__1,k8_prepower(A,B)) = k8_prepower(A,k4_xcmplx_0(B)) ) ) ) ).
fof(t66_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_rat_1(B)
=> ! [C] :
( v1_rat_1(C)
=> ( ~ r1_xreal_0(A,np__0)
=> k7_xcmplx_0(k8_prepower(A,B),k8_prepower(A,C)) = k8_prepower(A,k6_xcmplx_0(B,C)) ) ) ) ) ).
fof(t67_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_rat_1(C)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(B,np__0)
& k8_prepower(k3_xcmplx_0(A,B),C) != k3_xcmplx_0(k8_prepower(A,C),k8_prepower(B,C)) ) ) ) ) ).
fof(t68_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_rat_1(B)
=> ( ~ r1_xreal_0(A,np__0)
=> k8_prepower(k7_xcmplx_0(np__1,A),B) = k7_xcmplx_0(np__1,k8_prepower(A,B)) ) ) ) ).
fof(t69_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_rat_1(C)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(B,np__0)
& k8_prepower(k7_xcmplx_0(A,B),C) != k7_xcmplx_0(k8_prepower(A,C),k8_prepower(B,C)) ) ) ) ) ).
fof(t70_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_rat_1(B)
=> ! [C] :
( v1_rat_1(C)
=> ( ~ r1_xreal_0(A,np__0)
=> k8_prepower(k8_prepower(A,B),C) = k8_prepower(A,k3_xcmplx_0(B,C)) ) ) ) ) ).
fof(t71_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_rat_1(B)
=> ( ( r1_xreal_0(np__1,A)
& r1_xreal_0(np__0,B) )
=> r1_xreal_0(np__1,k8_prepower(A,B)) ) ) ) ).
fof(t72_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_rat_1(B)
=> ( ( r1_xreal_0(np__1,A)
& r1_xreal_0(B,np__0) )
=> r1_xreal_0(k8_prepower(A,B),np__1) ) ) ) ).
fof(t73_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_rat_1(B)
=> ~ ( ~ r1_xreal_0(A,np__1)
& ~ r1_xreal_0(B,np__0)
& r1_xreal_0(k8_prepower(A,B),np__1) ) ) ) ).
fof(t74_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_rat_1(B)
=> ! [C] :
( v1_rat_1(C)
=> ( ( r1_xreal_0(np__1,A)
& r1_xreal_0(C,B) )
=> r1_xreal_0(k8_prepower(A,C),k8_prepower(A,B)) ) ) ) ) ).
fof(t75_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_rat_1(B)
=> ! [C] :
( v1_rat_1(C)
=> ~ ( ~ r1_xreal_0(A,np__1)
& ~ r1_xreal_0(B,C)
& r1_xreal_0(k8_prepower(A,B),k8_prepower(A,C)) ) ) ) ) ).
fof(t76_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_rat_1(B)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(np__1,A)
& ~ r1_xreal_0(B,np__0)
& r1_xreal_0(np__1,k8_prepower(A,B)) ) ) ) ).
fof(t77_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_rat_1(B)
=> ( ( r1_xreal_0(A,np__1)
& r1_xreal_0(B,np__0) )
=> ( r1_xreal_0(A,np__0)
| r1_xreal_0(np__1,k8_prepower(A,B)) ) ) ) ) ).
fof(d6_prepower,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v1_prepower(A)
<=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> v1_rat_1(k2_seq_1(k5_numbers,k1_numbers,A,B)) ) ) ) ).
fof(t78_prepower,axiom,
$true ).
fof(t79_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v1_prepower(B)
& m2_relset_1(B,k5_numbers,k1_numbers)
& v4_seq_2(B)
& k2_seq_2(B) = A
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_xreal_0(k10_prepower(B,C),A) ) ) ) ).
fof(t80_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v1_prepower(B)
& m2_relset_1(B,k5_numbers,k1_numbers)
& v4_seq_2(B)
& k2_seq_2(B) = A
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_xreal_0(A,k10_prepower(B,C)) ) ) ) ).
fof(d7_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v1_prepower(B)
& 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) )
=> ( C = k11_prepower(A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,C,D) = k8_prepower(A,k10_prepower(B,D)) ) ) ) ) ) ).
fof(t81_prepower,axiom,
$true ).
fof(t82_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v1_prepower(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( v4_seq_2(B)
=> ( r1_xreal_0(A,np__0)
| v4_seq_2(k11_prepower(A,B)) ) ) ) ) ).
fof(t83_prepower,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v1_prepower(A)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v1_prepower(B)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( v1_xreal_0(C)
=> ( ( v4_seq_2(A)
& v4_seq_2(B)
& k2_seq_2(A) = k2_seq_2(B) )
=> ( r1_xreal_0(C,np__0)
| ( v4_seq_2(k11_prepower(C,A))
& v4_seq_2(k11_prepower(C,B))
& k2_seq_2(k11_prepower(C,A)) = k2_seq_2(k11_prepower(C,B)) ) ) ) ) ) ) ).
fof(d8_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(A,np__0)
=> ! [C] :
( v1_xreal_0(C)
=> ( C = k12_prepower(A,B)
<=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& v1_prepower(D)
& m2_relset_1(D,k5_numbers,k1_numbers)
& v4_seq_2(D)
& k2_seq_2(D) = B
& v4_seq_2(k11_prepower(A,D))
& k2_seq_2(k11_prepower(A,D)) = C ) ) ) ) ) ) ).
fof(t84_prepower,axiom,
$true ).
fof(t85_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ( ~ r1_xreal_0(A,np__0)
=> k12_prepower(A,np__0) = np__1 ) ) ).
fof(t86_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ( ~ r1_xreal_0(A,np__0)
=> k12_prepower(A,np__1) = A ) ) ).
fof(t87_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> k12_prepower(np__1,A) = np__1 ) ).
fof(t88_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_rat_1(B)
=> ( ~ r1_xreal_0(A,np__0)
=> k12_prepower(A,B) = k8_prepower(A,B) ) ) ) ).
fof(t89_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ~ r1_xreal_0(A,np__0)
=> k12_prepower(A,k2_xcmplx_0(B,C)) = k3_xcmplx_0(k12_prepower(A,B),k12_prepower(A,C)) ) ) ) ) ).
fof(t90_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(A,np__0)
=> k12_prepower(A,k4_xcmplx_0(B)) = k7_xcmplx_0(np__1,k12_prepower(A,B)) ) ) ) ).
fof(t91_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ~ r1_xreal_0(A,np__0)
=> k12_prepower(A,k6_xcmplx_0(B,C)) = k7_xcmplx_0(k12_prepower(A,B),k12_prepower(A,C)) ) ) ) ) ).
fof(t92_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(B,np__0)
& k12_prepower(k3_xcmplx_0(A,B),C) != k3_xcmplx_0(k12_prepower(A,C),k12_prepower(B,C)) ) ) ) ) ).
fof(t93_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(A,np__0)
=> k12_prepower(k7_xcmplx_0(np__1,A),B) = k7_xcmplx_0(np__1,k12_prepower(A,B)) ) ) ) ).
fof(t94_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(B,np__0)
& k12_prepower(k7_xcmplx_0(A,B),C) != k7_xcmplx_0(k12_prepower(A,C),k12_prepower(B,C)) ) ) ) ) ).
fof(t95_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( ~ r1_xreal_0(A,np__0)
& r1_xreal_0(k12_prepower(A,B),np__0) ) ) ) ).
fof(t96_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ( r1_xreal_0(np__1,A)
& r1_xreal_0(C,B) )
=> r1_xreal_0(k12_prepower(A,C),k12_prepower(A,B)) ) ) ) ) ).
fof(t97_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(A,np__1)
& ~ r1_xreal_0(B,C)
& r1_xreal_0(k12_prepower(A,B),k12_prepower(A,C)) ) ) ) ) ).
fof(t98_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ( r1_xreal_0(A,np__1)
& r1_xreal_0(C,B) )
=> ( r1_xreal_0(A,np__0)
| r1_xreal_0(k12_prepower(A,B),k12_prepower(A,C)) ) ) ) ) ) ).
fof(t99_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( r1_xreal_0(np__1,A)
& r1_xreal_0(np__0,B) )
=> r1_xreal_0(np__1,k12_prepower(A,B)) ) ) ) ).
fof(t100_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( ~ r1_xreal_0(A,np__1)
& ~ r1_xreal_0(B,np__0)
& r1_xreal_0(k12_prepower(A,B),np__1) ) ) ) ).
fof(t101_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( r1_xreal_0(np__1,A)
& r1_xreal_0(B,np__0) )
=> r1_xreal_0(k12_prepower(A,B),np__1) ) ) ) ).
fof(t102_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( ~ r1_xreal_0(A,np__1)
& ~ r1_xreal_0(np__0,B)
& r1_xreal_0(np__1,k12_prepower(A,B)) ) ) ) ).
fof(t103_prepower,axiom,
! [A] :
( v1_rat_1(A)
=> ! [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)
& v4_seq_2(C)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,D),np__0)
& k2_seq_1(k5_numbers,k1_numbers,C,D) = k9_prepower(k2_seq_1(k5_numbers,k1_numbers,B,D),A) ) ) )
=> ( r1_xreal_0(k2_seq_2(B),np__0)
| k2_seq_2(C) = k9_prepower(k2_seq_2(B),A) ) ) ) ) ) ).
fof(t104_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [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)
& v4_seq_2(C)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,C,D) = k12_prepower(A,k2_seq_1(k5_numbers,k1_numbers,B,D)) ) )
=> ( r1_xreal_0(A,np__0)
| k2_seq_2(C) = k12_prepower(A,k2_seq_2(B)) ) ) ) ) ) ).
fof(t105_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ~ r1_xreal_0(A,np__0)
=> k12_prepower(k12_prepower(A,B),C) = k12_prepower(A,k3_xcmplx_0(B,C)) ) ) ) ) ).
fof(dt_k1_prepower,axiom,
! [A] :
( v1_int_1(A)
=> m2_subset_1(k1_prepower(A),k1_numbers,k5_numbers) ) ).
fof(projectivity_k1_prepower,axiom,
! [A] :
( v1_int_1(A)
=> k1_prepower(k1_prepower(A)) = k1_prepower(A) ) ).
fof(redefinition_k1_prepower,axiom,
! [A] :
( v1_int_1(A)
=> k1_prepower(A) = k16_complex1(A) ) ).
fof(dt_k2_prepower,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v1_funct_1(k2_prepower(A))
& v1_funct_2(k2_prepower(A),k5_numbers,k1_numbers)
& m2_relset_1(k2_prepower(A),k5_numbers,k1_numbers) ) ) ).
fof(dt_k3_prepower,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k5_numbers) )
=> m1_subset_1(k3_prepower(A,B),k1_numbers) ) ).
fof(redefinition_k3_prepower,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k5_numbers) )
=> k3_prepower(A,B) = k2_newton(A,B) ) ).
fof(dt_k4_prepower,axiom,
! [A,B] :
( ( v4_ordinal2(A)
& v1_xreal_0(B) )
=> v1_xreal_0(k4_prepower(A,B)) ) ).
fof(dt_k5_prepower,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_numbers) )
=> m1_subset_1(k5_prepower(A,B),k1_numbers) ) ).
fof(redefinition_k5_prepower,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_numbers) )
=> k5_prepower(A,B) = k4_prepower(A,B) ) ).
fof(dt_k6_prepower,axiom,
$true ).
fof(dt_k7_prepower,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& v1_int_1(B) )
=> m1_subset_1(k7_prepower(A,B),k1_numbers) ) ).
fof(redefinition_k7_prepower,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& v1_int_1(B) )
=> k7_prepower(A,B) = k6_prepower(A,B) ) ).
fof(dt_k8_prepower,axiom,
$true ).
fof(dt_k9_prepower,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& v1_rat_1(B) )
=> m1_subset_1(k9_prepower(A,B),k1_numbers) ) ).
fof(redefinition_k9_prepower,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& v1_rat_1(B) )
=> k9_prepower(A,B) = k8_prepower(A,B) ) ).
fof(dt_k10_prepower,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v1_prepower(A)
& m1_relset_1(A,k5_numbers,k1_numbers)
& m1_subset_1(B,k5_numbers) )
=> v1_rat_1(k10_prepower(A,B)) ) ).
fof(redefinition_k10_prepower,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v1_prepower(A)
& m1_relset_1(A,k5_numbers,k1_numbers)
& m1_subset_1(B,k5_numbers) )
=> k10_prepower(A,B) = k1_funct_1(A,B) ) ).
fof(dt_k11_prepower,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& v1_prepower(B)
& m1_relset_1(B,k5_numbers,k1_numbers) )
=> ( v1_funct_1(k11_prepower(A,B))
& v1_funct_2(k11_prepower(A,B),k5_numbers,k1_numbers)
& m2_relset_1(k11_prepower(A,B),k5_numbers,k1_numbers) ) ) ).
fof(dt_k12_prepower,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> v1_xreal_0(k12_prepower(A,B)) ) ).
fof(dt_k13_prepower,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> m1_subset_1(k13_prepower(A,B),k1_numbers) ) ).
fof(redefinition_k13_prepower,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> k13_prepower(A,B) = k12_prepower(A,B) ) ).
%------------------------------------------------------------------------------