SET007 Axioms: SET007+144.ax
%------------------------------------------------------------------------------
% File : SET007+144 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Real Exponents and Logarithms
% 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 : power [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 83 ( 9 unt; 0 def)
% Number of atoms : 434 ( 104 equ)
% Maximal formula atoms : 20 ( 5 avg)
% Number of connectives : 482 ( 131 ~; 11 |; 126 &)
% ( 6 <=>; 208 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 8 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 12 ( 10 usr; 1 prp; 0-3 aty)
% Number of functors : 31 ( 31 usr; 7 con; 0-4 aty)
% Number of variables : 185 ( 175 !; 10 ?)
% 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_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& B = k2_nat_1(np__2,C) )
=> k2_newton(k4_xcmplx_0(A),B) = k2_newton(A,B) ) ) ) ).
fof(t2_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& B = k1_nat_1(k2_nat_1(np__2,C),np__1) )
=> k2_newton(k4_xcmplx_0(A),B) = k4_xcmplx_0(k2_newton(A,B)) ) ) ) ).
fof(t3_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ ( ~ r1_xreal_0(np__0,A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> B != k2_nat_1(np__2,C) ) )
=> r1_xreal_0(np__0,k2_newton(A,B)) ) ) ) ).
fof(d1_power,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( ( r1_xreal_0(np__0,B)
& r1_xreal_0(np__1,A) )
=> k1_power(A,B) = k4_prepower(A,B) )
& ~ ( ~ r1_xreal_0(np__0,B)
& ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = k1_nat_1(k2_nat_1(np__2,C),np__1) )
& k1_power(A,B) != k4_xcmplx_0(k4_prepower(A,k4_xcmplx_0(B))) ) ) ) ) ).
fof(t4_power,axiom,
$true ).
fof(t5_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ ( ~ ( r1_xreal_0(np__1,B)
& r1_xreal_0(np__0,A) )
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> B != k1_nat_1(k2_nat_1(np__2,C),np__1) ) )
=> ( k2_newton(k1_power(B,A),B) = A
& k1_power(B,k2_newton(A,B)) = A ) ) ) ) ).
fof(t6_power,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> k2_power(A,np__0) = np__0 ) ) ).
fof(t7_power,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> k2_power(A,np__1) = np__1 ) ) ).
fof(t8_power,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__0,k1_power(B,A)) ) ) ) ).
fof(t9_power,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& A = k1_nat_1(k2_nat_1(np__2,B),np__1) )
=> k2_power(A,k1_real_1(np__1)) = k1_real_1(np__1) ) ) ).
fof(t10_power,axiom,
! [A] :
( v1_xreal_0(A)
=> k1_power(np__1,A) = A ) ).
fof(t11_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& B = k1_nat_1(k2_nat_1(np__2,C),np__1) )
=> k1_power(B,A) = k4_xcmplx_0(k1_power(B,k4_xcmplx_0(A))) ) ) ) ).
fof(t12_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ ( ~ ( r1_xreal_0(np__1,C)
& r1_xreal_0(np__0,A)
& r1_xreal_0(np__0,B) )
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> C != k1_nat_1(k2_nat_1(np__2,D),np__1) ) )
=> k1_power(C,k3_xcmplx_0(A,B)) = k3_xcmplx_0(k1_power(C,A),k1_power(C,B)) ) ) ) ) ).
fof(t13_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( ( ~ r1_xreal_0(A,np__0)
& r1_xreal_0(np__1,B) )
| ( A != np__0
& ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& B = k1_nat_1(k2_nat_1(np__2,C),np__1) ) ) )
=> k1_power(B,k7_xcmplx_0(np__1,A)) = k7_xcmplx_0(np__1,k1_power(B,A)) ) ) ) ).
fof(t14_power,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,np__0)
& r1_xreal_0(np__1,C) )
| ( B != np__0
& ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& C = k1_nat_1(k2_nat_1(np__2,D),np__1) ) ) )
=> k1_power(C,k7_xcmplx_0(A,B)) = k7_xcmplx_0(k1_power(C,A),k1_power(C,B)) ) ) ) ) ).
fof(t15_power,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) )
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( B = k1_nat_1(k2_nat_1(np__2,D),np__1)
& C = k1_nat_1(k2_nat_1(np__2,E),np__1) ) ) ) )
=> k1_power(B,k1_power(C,A)) = k1_power(k2_nat_1(B,C),A) ) ) ) ) ).
fof(t16_power,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) )
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( B = k1_nat_1(k2_nat_1(np__2,D),np__1)
& C = k1_nat_1(k2_nat_1(np__2,E),np__1) ) ) ) )
=> k3_xcmplx_0(k1_power(B,A),k1_power(C,A)) = k1_power(k2_nat_1(B,C),k2_newton(A,k1_nat_1(B,C))) ) ) ) ) ).
fof(t17_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,B)
=> ( ( ~ ( r1_xreal_0(np__0,A)
& r1_xreal_0(np__1,C) )
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> C != k1_nat_1(k2_nat_1(np__2,D),np__1) ) )
| r1_xreal_0(k1_power(C,A),k1_power(C,B)) ) ) ) ) ) ).
fof(t18_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(B,A)
& ~ ( ~ ( r1_xreal_0(np__0,A)
& r1_xreal_0(np__1,C) )
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> C != k1_nat_1(k2_nat_1(np__2,D),np__1) ) )
& r1_xreal_0(k1_power(C,B),k1_power(C,A)) ) ) ) ) ).
fof(t19_power,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,k1_power(B,A))
& r1_xreal_0(k1_power(B,A),A) ) ) ) ) ).
fof(t20_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,k1_real_1(np__1))
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> B != k1_nat_1(k2_nat_1(np__2,C),np__1) )
| ( r1_xreal_0(k1_power(B,A),k1_real_1(np__1))
& r1_xreal_0(A,k1_power(B,A)) ) ) ) ) ) ).
fof(t21_power,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,k1_power(B,A))
& ~ r1_xreal_0(np__1,k1_power(B,A)) ) ) ) ) ) ).
fof(t22_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,np__0)
=> ( r1_xreal_0(A,k1_real_1(np__1))
| ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> B != k1_nat_1(k2_nat_1(np__2,C),np__1) )
| ( r1_xreal_0(k1_power(B,A),A)
& ~ r1_xreal_0(k1_power(B,A),k1_real_1(np__1)) ) ) ) ) ) ).
fof(t23_power,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(k1_power(B,A),np__1),k7_xcmplx_0(k6_xcmplx_0(A,np__1),B)) ) ) ) ) ).
fof(t24_power,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( v1_xreal_0(B)
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,C)
=> k2_seq_1(k5_numbers,k1_numbers,A,C) = k1_power(C,B) ) )
=> ( r1_xreal_0(B,np__0)
| ( v4_seq_2(A)
& k2_seq_2(A) = np__1 ) ) ) ) ) ).
fof(d2_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ( ~ r1_xreal_0(A,np__0)
=> ( C = k3_power(A,B)
<=> C = k12_prepower(A,B) ) )
& ( A = np__0
=> ( r1_xreal_0(B,np__0)
| ( C = k3_power(A,B)
<=> C = np__0 ) ) )
& ( ( A = np__0
& B = np__0 )
=> ( C = k3_power(A,B)
<=> C = np__1 ) )
& ( v1_int_1(B)
=> ( r1_xreal_0(np__0,A)
| ( C = k3_power(A,B)
<=> ? [D] :
( v1_int_1(D)
& D = B
& C = k6_prepower(A,D) ) ) ) ) ) ) ) ) ).
fof(t25_power,axiom,
$true ).
fof(t26_power,axiom,
$true ).
fof(t27_power,axiom,
$true ).
fof(t28_power,axiom,
$true ).
fof(t29_power,axiom,
! [A] :
( v1_xreal_0(A)
=> k3_power(A,np__0) = np__1 ) ).
fof(t30_power,axiom,
! [A] :
( v1_xreal_0(A)
=> k3_power(A,np__1) = A ) ).
fof(t31_power,axiom,
! [A] :
( v1_xreal_0(A)
=> k3_power(np__1,A) = np__1 ) ).
fof(t32_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ~ r1_xreal_0(A,np__0)
=> k3_power(A,k2_xcmplx_0(B,C)) = k3_xcmplx_0(k3_power(A,B),k3_power(A,C)) ) ) ) ) ).
fof(t33_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(A,np__0)
=> k3_power(A,k4_xcmplx_0(B)) = k7_xcmplx_0(np__1,k3_power(A,B)) ) ) ) ).
fof(t34_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ~ r1_xreal_0(A,np__0)
=> k3_power(A,k6_xcmplx_0(B,C)) = k7_xcmplx_0(k3_power(A,B),k3_power(A,C)) ) ) ) ) ).
fof(t35_power,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)
& k3_power(k3_xcmplx_0(A,B),C) != k3_xcmplx_0(k3_power(A,C),k3_power(B,C)) ) ) ) ) ).
fof(t36_power,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)
& k3_power(k7_xcmplx_0(A,B),C) != k7_xcmplx_0(k3_power(A,C),k3_power(B,C)) ) ) ) ) ).
fof(t37_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(A,np__0)
=> k3_power(k7_xcmplx_0(np__1,A),B) = k3_power(A,k4_xcmplx_0(B)) ) ) ) ).
fof(t38_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ~ r1_xreal_0(A,np__0)
=> k3_power(k3_power(A,B),C) = k3_power(A,k3_xcmplx_0(B,C)) ) ) ) ) ).
fof(t39_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( ~ r1_xreal_0(A,np__0)
& r1_xreal_0(k3_power(A,B),np__0) ) ) ) ).
fof(t40_power,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(k3_power(A,B),np__1) ) ) ) ).
fof(t41_power,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,k3_power(A,B)) ) ) ) ).
fof(t42_power,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,A)
& ~ r1_xreal_0(C,np__0)
& r1_xreal_0(k3_power(B,C),k3_power(A,C)) ) ) ) ) ).
fof(t43_power,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,A)
& ~ r1_xreal_0(np__0,C)
& r1_xreal_0(k3_power(A,C),k3_power(B,C)) ) ) ) ) ).
fof(t44_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(B,A)
& ~ r1_xreal_0(C,np__1)
& r1_xreal_0(k3_power(C,B),k3_power(C,A)) ) ) ) ) ).
fof(t45_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(B,A)
& ~ r1_xreal_0(C,np__0)
& ~ r1_xreal_0(np__1,C)
& r1_xreal_0(k3_power(C,A),k3_power(C,B)) ) ) ) ) ).
fof(t46_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( A != np__0
=> k3_power(A,B) = k2_newton(A,B) ) ) ) ).
fof(t47_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,B)
=> k3_power(A,B) = k2_newton(A,B) ) ) ) ).
fof(t48_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( A != np__0
=> k3_power(A,B) = k2_newton(A,B) ) ) ) ).
fof(t49_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,B)
=> k3_power(A,B) = k2_newton(A,B) ) ) ) ).
fof(t50_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_int_1(B)
=> ( A != np__0
=> k3_power(A,B) = k6_prepower(A,B) ) ) ) ).
fof(t51_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_rat_1(B)
=> ( ~ r1_xreal_0(A,np__0)
=> k3_power(A,B) = k8_prepower(A,B) ) ) ) ).
fof(t52_power,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) )
=> k3_power(A,k6_real_1(np__1,B)) = k1_power(B,A) ) ) ) ).
fof(t53_power,axiom,
! [A] :
( v1_xreal_0(A)
=> k3_power(A,np__2) = k5_square_1(A) ) ).
fof(t54_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_int_1(B)
=> ~ ( A != np__0
& ? [C] :
( v1_int_1(C)
& B = k3_xcmplx_0(np__2,C) )
& k3_power(k4_xcmplx_0(A),B) != k3_power(A,B) ) ) ) ).
fof(t55_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_int_1(B)
=> ~ ( A != np__0
& ? [C] :
( v1_int_1(C)
& B = k2_xcmplx_0(k3_xcmplx_0(np__2,C),np__1) )
& k3_power(k4_xcmplx_0(A),B) != k4_xcmplx_0(k3_power(A,B)) ) ) ) ).
fof(t56_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,k1_real_1(np__1))
=> r1_xreal_0(k2_xcmplx_0(np__1,k3_xcmplx_0(B,A)),k3_power(k2_xcmplx_0(np__1,A),B)) ) ) ) ).
fof(t57_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(A,np__0)
& A != np__1
& B != C
& k3_power(A,B) = k3_power(A,C) ) ) ) ) ).
fof(d3_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( ~ r1_xreal_0(A,np__0)
& A != np__1
& ~ r1_xreal_0(B,np__0)
& ~ ! [C] :
( v1_xreal_0(C)
=> ( C = k5_power(A,B)
<=> k3_power(A,C) = B ) ) ) ) ) ).
fof(t58_power,axiom,
$true ).
fof(t59_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ~ ( ~ r1_xreal_0(A,np__0)
& A != np__1
& k5_power(A,np__1) != np__0 ) ) ).
fof(t60_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ~ ( ~ r1_xreal_0(A,np__0)
& A != np__1
& k5_power(A,A) != np__1 ) ) ).
fof(t61_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(A,np__0)
& A != np__1
& ~ r1_xreal_0(B,np__0)
& ~ r1_xreal_0(C,np__0)
& k2_xcmplx_0(k5_power(A,B),k5_power(A,C)) != k5_power(A,k3_xcmplx_0(B,C)) ) ) ) ) ).
fof(t62_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(A,np__0)
& A != np__1
& ~ r1_xreal_0(B,np__0)
& ~ r1_xreal_0(C,np__0)
& k6_xcmplx_0(k5_power(A,B),k5_power(A,C)) != k5_power(A,k7_xcmplx_0(B,C)) ) ) ) ) ).
fof(t63_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(A,np__0)
& A != np__1
& ~ r1_xreal_0(B,np__0)
& k5_power(A,k3_power(B,C)) != k3_xcmplx_0(C,k5_power(A,B)) ) ) ) ) ).
fof(t64_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(A,np__0)
& A != np__1
& ~ r1_xreal_0(B,np__0)
& B != np__1
& ~ r1_xreal_0(C,np__0)
& k5_power(A,C) != k3_xcmplx_0(k5_power(A,B),k5_power(B,C)) ) ) ) ) ).
fof(t65_power,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,np__0)
& ~ r1_xreal_0(C,B)
& r1_xreal_0(k5_power(A,C),k5_power(A,B)) ) ) ) ) ).
fof(t66_power,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(np__1,A)
& ~ r1_xreal_0(B,np__0)
& ~ r1_xreal_0(C,B)
& r1_xreal_0(k5_power(A,B),k5_power(A,C)) ) ) ) ) ).
fof(t67_power,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,A,B) = k4_power(k3_real_1(np__1,k6_real_1(np__1,k1_nat_1(B,np__1))),k1_nat_1(B,np__1)) )
=> v4_seq_2(A) ) ) ).
fof(d4_power,axiom,
! [A] :
( v1_xreal_0(A)
=> ( A = k7_power
<=> ! [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)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k4_power(k3_real_1(np__1,k6_real_1(np__1,k1_nat_1(C,np__1))),k1_nat_1(C,np__1)) )
=> A = k2_seq_2(B) ) ) ) ) ).
fof(dt_k1_power,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& v1_xreal_0(B) )
=> v1_xreal_0(k1_power(A,B)) ) ).
fof(dt_k2_power,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_numbers) )
=> m1_subset_1(k2_power(A,B),k1_numbers) ) ).
fof(redefinition_k2_power,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_numbers) )
=> k2_power(A,B) = k1_power(A,B) ) ).
fof(dt_k3_power,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> v1_xreal_0(k3_power(A,B)) ) ).
fof(dt_k4_power,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> m1_subset_1(k4_power(A,B),k1_numbers) ) ).
fof(redefinition_k4_power,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> k4_power(A,B) = k3_power(A,B) ) ).
fof(dt_k5_power,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> v1_xreal_0(k5_power(A,B)) ) ).
fof(dt_k6_power,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> m1_subset_1(k6_power(A,B),k1_numbers) ) ).
fof(redefinition_k6_power,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> k6_power(A,B) = k5_power(A,B) ) ).
fof(dt_k7_power,axiom,
v1_xreal_0(k7_power) ).
fof(dt_k8_power,axiom,
m1_subset_1(k8_power,k1_numbers) ).
fof(redefinition_k8_power,axiom,
k8_power = k7_power ).
%------------------------------------------------------------------------------