SET007 Axioms: SET007+926.ax
%------------------------------------------------------------------------------
% File : SET007+926 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Partial Product and Sum of Series and Related Basic Inequalities
% 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 : series_5 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 58 ( 0 unt; 0 def)
% Number of atoms : 415 ( 23 equ)
% Maximal formula atoms : 11 ( 7 avg)
% Number of connectives : 403 ( 46 ~; 0 |; 163 &)
% ( 0 <=>; 194 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 8 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of predicates : 8 ( 7 usr; 0 prp; 1-3 aty)
% Number of functors : 28 ( 28 usr; 12 con; 0-4 aty)
% Number of variables : 169 ( 169 !; 0 ?)
% 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_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> r1_xreal_0(np__4,k3_xcmplx_0(k2_xcmplx_0(A,B),k2_xcmplx_0(k7_xcmplx_0(np__1,A),k7_xcmplx_0(np__1,B)))) ) ) ).
fof(t2_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> r1_xreal_0(k2_xcmplx_0(k3_xcmplx_0(k2_newton(A,np__3),B),k3_xcmplx_0(A,k2_newton(B,np__3))),k2_xcmplx_0(k2_newton(A,np__4),k2_newton(B,np__4))) ) ) ).
fof(t3_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ~ ( ~ r1_xreal_0(B,A)
& r1_xreal_0(k7_xcmplx_0(k2_xcmplx_0(B,C),k2_xcmplx_0(A,C)),np__1) ) ) ) ) ).
fof(t4_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ~ ( ~ r1_xreal_0(B,A)
& r1_xreal_0(k8_square_1(k7_xcmplx_0(A,B)),k7_xcmplx_0(A,B)) ) ) ) ).
fof(t5_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ~ ( ~ r1_xreal_0(B,A)
& r1_xreal_0(k7_xcmplx_0(k2_xcmplx_0(B,k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),np__2))),k2_xcmplx_0(A,k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),np__2)))),k8_square_1(k7_xcmplx_0(A,B))) ) ) ) ).
fof(t6_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ~ ( ~ r1_xreal_0(B,A)
& r1_xreal_0(k7_xcmplx_0(k2_xcmplx_0(B,k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),np__2))),k2_xcmplx_0(A,k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),np__2)))),k7_xcmplx_0(A,B)) ) ) ) ).
fof(t7_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> r1_xreal_0(k7_xcmplx_0(np__2,k2_xcmplx_0(k7_xcmplx_0(np__1,A),k7_xcmplx_0(np__1,B))),k8_square_1(k3_xcmplx_0(A,B))) ) ) ).
fof(t8_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> r1_xreal_0(k7_xcmplx_0(k2_xcmplx_0(A,B),np__2),k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),np__2))) ) ) ).
fof(t9_series_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> r1_xreal_0(k2_xcmplx_0(A,B),k8_square_1(k3_xcmplx_0(np__2,k2_xcmplx_0(k5_square_1(A),k5_square_1(B))))) ) ) ).
fof(t10_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> r1_xreal_0(k7_xcmplx_0(np__2,k2_xcmplx_0(k7_xcmplx_0(np__1,A),k7_xcmplx_0(np__1,B))),k7_xcmplx_0(k2_xcmplx_0(A,B),np__2)) ) ) ).
fof(t11_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> r1_xreal_0(k8_square_1(k3_xcmplx_0(A,B)),k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),np__2))) ) ) ).
fof(t12_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> r1_xreal_0(k7_xcmplx_0(np__2,k2_xcmplx_0(k7_xcmplx_0(np__1,A),k7_xcmplx_0(np__1,B))),k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),np__2))) ) ) ).
fof(t13_series_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( ~ r1_xreal_0(np__1,k18_complex1(A))
& ~ r1_xreal_0(np__1,k18_complex1(B))
& ~ r1_xreal_0(k18_complex1(k7_xcmplx_0(k2_xcmplx_0(A,B),k2_xcmplx_0(np__1,k3_xcmplx_0(A,B)))),np__1) ) ) ) ).
fof(t14_series_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> r1_xreal_0(k7_xcmplx_0(k18_complex1(k2_xcmplx_0(A,B)),k2_xcmplx_0(np__1,k18_complex1(k2_xcmplx_0(A,B)))),k2_xcmplx_0(k7_xcmplx_0(k18_complex1(A),k2_xcmplx_0(np__1,k18_complex1(A))),k7_xcmplx_0(k18_complex1(B),k2_xcmplx_0(np__1,k18_complex1(B))))) ) ) ).
fof(t15_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ! [D] :
( ( v1_xreal_0(D)
& v2_xreal_0(D) )
=> ~ r1_xreal_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k7_xcmplx_0(A,k2_xcmplx_0(k2_xcmplx_0(A,B),C)),k7_xcmplx_0(B,k2_xcmplx_0(k2_xcmplx_0(A,B),D))),k7_xcmplx_0(D,k2_xcmplx_0(k2_xcmplx_0(B,D),C))),k7_xcmplx_0(C,k2_xcmplx_0(k2_xcmplx_0(A,D),C))),np__1) ) ) ) ) ).
fof(t16_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ! [D] :
( ( v1_xreal_0(D)
& v2_xreal_0(D) )
=> ~ r1_xreal_0(np__2,k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k7_xcmplx_0(A,k2_xcmplx_0(k2_xcmplx_0(A,B),C)),k7_xcmplx_0(B,k2_xcmplx_0(k2_xcmplx_0(A,B),D))),k7_xcmplx_0(D,k2_xcmplx_0(k2_xcmplx_0(B,D),C))),k7_xcmplx_0(C,k2_xcmplx_0(k2_xcmplx_0(A,D),C)))) ) ) ) ) ).
fof(t17_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ~ ( ~ r1_xreal_0(k2_xcmplx_0(A,B),C)
& ~ r1_xreal_0(k2_xcmplx_0(B,C),A)
& ~ r1_xreal_0(k2_xcmplx_0(A,C),B)
& ~ r1_xreal_0(k7_xcmplx_0(np__9,k2_xcmplx_0(k2_xcmplx_0(A,B),C)),k2_xcmplx_0(k2_xcmplx_0(k7_xcmplx_0(np__1,k6_xcmplx_0(k2_xcmplx_0(A,B),C)),k7_xcmplx_0(np__1,k6_xcmplx_0(k2_xcmplx_0(B,C),A))),k7_xcmplx_0(np__1,k6_xcmplx_0(k2_xcmplx_0(C,A),B)))) ) ) ) ) ).
fof(t18_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ! [D] :
( ( v1_xreal_0(D)
& v2_xreal_0(D) )
=> r1_xreal_0(k2_xcmplx_0(k8_square_1(k3_xcmplx_0(A,C)),k8_square_1(k3_xcmplx_0(B,D))),k8_square_1(k3_xcmplx_0(k2_xcmplx_0(A,B),k2_xcmplx_0(C,D)))) ) ) ) ) ).
fof(t19_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ! [D] :
( ( v1_xreal_0(D)
& v2_xreal_0(D) )
=> r1_xreal_0(k3_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(np__4,A),B),C),D),k3_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(A,B),k3_xcmplx_0(C,D)),k2_xcmplx_0(k3_xcmplx_0(A,C),k3_xcmplx_0(B,D)))) ) ) ) ) ).
fof(t20_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> r1_xreal_0(np__3,k2_xcmplx_0(k2_xcmplx_0(k7_xcmplx_0(A,B),k7_xcmplx_0(B,C)),k7_xcmplx_0(C,A))) ) ) ) ).
fof(t21_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ( k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(A,B),k3_xcmplx_0(B,C)),k3_xcmplx_0(C,A)) = np__1
=> r1_xreal_0(k9_square_1(np__3),k2_xcmplx_0(k2_xcmplx_0(A,B),C)) ) ) ) ) ).
fof(t22_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> r1_xreal_0(np__3,k2_xcmplx_0(k2_xcmplx_0(k7_xcmplx_0(k6_xcmplx_0(k2_xcmplx_0(A,B),C),C),k7_xcmplx_0(k6_xcmplx_0(k2_xcmplx_0(B,C),A),A)),k7_xcmplx_0(k6_xcmplx_0(k2_xcmplx_0(C,A),B),B))) ) ) ) ).
fof(t23_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> r1_xreal_0(k5_square_1(k2_xcmplx_0(k8_square_1(k3_xcmplx_0(A,B)),k7_xcmplx_0(np__1,k8_square_1(k3_xcmplx_0(A,B))))),k3_xcmplx_0(k2_xcmplx_0(A,k7_xcmplx_0(np__1,A)),k2_xcmplx_0(B,k7_xcmplx_0(np__1,B)))) ) ) ).
fof(t24_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> r1_xreal_0(k2_xcmplx_0(k2_xcmplx_0(C,A),B),k2_xcmplx_0(k2_xcmplx_0(k7_xcmplx_0(k3_xcmplx_0(A,B),C),k7_xcmplx_0(k3_xcmplx_0(C,B),A)),k7_xcmplx_0(k3_xcmplx_0(C,A),B))) ) ) ) ).
fof(t25_series_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(A,B)
& ~ r1_xreal_0(B,C)
& r1_xreal_0(k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(k5_square_1(A),B),k3_xcmplx_0(k5_square_1(B),C)),k3_xcmplx_0(k5_square_1(C),A)),k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(A,k5_square_1(B)),k3_xcmplx_0(B,k5_square_1(C))),k3_xcmplx_0(C,k5_square_1(A)))) ) ) ) ) ).
fof(t26_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ~ ( ~ r1_xreal_0(A,B)
& ~ r1_xreal_0(B,C)
& r1_xreal_0(k7_xcmplx_0(B,k6_xcmplx_0(A,B)),k7_xcmplx_0(C,k6_xcmplx_0(A,C))) ) ) ) ) ).
fof(t27_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ! [D] :
( ( v1_xreal_0(D)
& v2_xreal_0(D) )
=> ~ ( ~ r1_xreal_0(A,B)
& ~ r1_xreal_0(C,D)
& r1_xreal_0(k7_xcmplx_0(C,k2_xcmplx_0(C,B)),k7_xcmplx_0(D,k2_xcmplx_0(D,A))) ) ) ) ) ) ).
fof(t28_series_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> r1_xreal_0(k2_xcmplx_0(k3_xcmplx_0(A,B),k3_xcmplx_0(C,D)),k3_xcmplx_0(k8_square_1(k2_xcmplx_0(k5_square_1(A),k5_square_1(C))),k8_square_1(k2_xcmplx_0(k5_square_1(B),k5_square_1(D))))) ) ) ) ) ).
fof(t29_series_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( v1_xreal_0(E)
=> ! [F] :
( v1_xreal_0(F)
=> r1_xreal_0(k5_square_1(k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(A,B),k3_xcmplx_0(C,D)),k3_xcmplx_0(E,F))),k3_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(C)),k5_square_1(E)),k2_xcmplx_0(k2_xcmplx_0(k5_square_1(B),k5_square_1(D)),k5_square_1(F)))) ) ) ) ) ) ) ).
fof(t30_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> r1_xreal_0(k7_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(np__9,A),B),C),k2_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),k5_square_1(C))),k2_xcmplx_0(k2_xcmplx_0(A,B),C)) ) ) ) ).
fof(t31_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> r1_xreal_0(k2_xcmplx_0(k2_xcmplx_0(A,B),C),k2_xcmplx_0(k2_xcmplx_0(k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k3_xcmplx_0(A,B)),k5_square_1(B)),np__3)),k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k5_square_1(B),k3_xcmplx_0(B,C)),k5_square_1(C)),np__3))),k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k5_square_1(C),k3_xcmplx_0(C,A)),k5_square_1(A)),np__3)))) ) ) ) ).
fof(t32_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> r1_xreal_0(k2_xcmplx_0(k2_xcmplx_0(k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k3_xcmplx_0(A,B)),k5_square_1(B)),np__3)),k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k5_square_1(B),k3_xcmplx_0(B,C)),k5_square_1(C)),np__3))),k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k5_square_1(C),k3_xcmplx_0(C,A)),k5_square_1(A)),np__3))),k2_xcmplx_0(k2_xcmplx_0(k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),np__2)),k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(B),k5_square_1(C)),np__2))),k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(C),k5_square_1(A)),np__2)))) ) ) ) ).
fof(t33_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> r1_xreal_0(k2_xcmplx_0(k2_xcmplx_0(k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),np__2)),k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(B),k5_square_1(C)),np__2))),k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k5_square_1(C),k5_square_1(A)),np__2))),k8_square_1(k3_xcmplx_0(np__3,k2_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),k5_square_1(C))))) ) ) ) ).
fof(t34_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> r1_xreal_0(k8_square_1(k3_xcmplx_0(np__3,k2_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),k5_square_1(C)))),k2_xcmplx_0(k2_xcmplx_0(k7_xcmplx_0(k3_xcmplx_0(B,C),A),k7_xcmplx_0(k3_xcmplx_0(C,A),B)),k7_xcmplx_0(k3_xcmplx_0(A,B),C))) ) ) ) ).
fof(t35_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ( k2_xcmplx_0(A,B) = np__1
=> r1_xreal_0(np__9,k3_xcmplx_0(k6_xcmplx_0(k7_xcmplx_0(np__1,k5_square_1(A)),np__1),k6_xcmplx_0(k7_xcmplx_0(np__1,k5_square_1(B)),np__1))) ) ) ) ).
fof(t36_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ( k2_xcmplx_0(A,B) = np__1
=> r1_xreal_0(k7_xcmplx_0(np__17,np__4),k2_xcmplx_0(k3_xcmplx_0(A,B),k7_xcmplx_0(np__1,k3_xcmplx_0(A,B)))) ) ) ) ).
fof(t37_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ( k2_xcmplx_0(k2_xcmplx_0(A,B),C) = np__1
=> r1_xreal_0(np__9,k2_xcmplx_0(k2_xcmplx_0(k7_xcmplx_0(np__1,A),k7_xcmplx_0(np__1,B)),k7_xcmplx_0(np__1,C))) ) ) ) ) ).
fof(t38_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ( k2_xcmplx_0(k2_xcmplx_0(A,B),C) = np__1
=> r1_xreal_0(np__8,k3_xcmplx_0(k3_xcmplx_0(k6_xcmplx_0(k7_xcmplx_0(np__1,A),np__1),k6_xcmplx_0(k7_xcmplx_0(np__1,B),np__1)),k6_xcmplx_0(k7_xcmplx_0(np__1,C),np__1))) ) ) ) ) ).
fof(t39_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ( k2_xcmplx_0(k2_xcmplx_0(A,B),C) = np__1
=> r1_xreal_0(np__64,k3_xcmplx_0(k3_xcmplx_0(k2_xcmplx_0(np__1,k7_xcmplx_0(np__1,A)),k2_xcmplx_0(np__1,k7_xcmplx_0(np__1,B))),k2_xcmplx_0(np__1,k7_xcmplx_0(np__1,C)))) ) ) ) ) ).
fof(t40_series_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( k2_xcmplx_0(k2_xcmplx_0(A,B),C) = np__1
=> r1_xreal_0(k7_xcmplx_0(np__1,np__3),k2_xcmplx_0(k2_xcmplx_0(k5_square_1(A),k5_square_1(B)),k5_square_1(C))) ) ) ) ) ).
fof(t41_series_5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( k2_xcmplx_0(k2_xcmplx_0(A,B),C) = np__1
=> r1_xreal_0(k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(A,B),k3_xcmplx_0(B,C)),k3_xcmplx_0(C,A)),k7_xcmplx_0(np__1,np__3)) ) ) ) ) ).
fof(t42_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ( k3_xcmplx_0(k3_xcmplx_0(A,B),C) = np__1
=> r1_xreal_0(k2_xcmplx_0(k2_xcmplx_0(k8_square_1(A),k8_square_1(B)),k8_square_1(C)),k2_xcmplx_0(k2_xcmplx_0(k7_xcmplx_0(np__1,A),k7_xcmplx_0(np__1,B)),k7_xcmplx_0(np__1,C))) ) ) ) ) ).
fof(t43_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ~ ( ~ r1_xreal_0(A,B)
& ~ r1_xreal_0(B,C)
& r1_xreal_0(k3_xcmplx_0(k3_xcmplx_0(k3_power(A,k3_xcmplx_0(np__2,A)),k3_power(B,k3_xcmplx_0(np__2,B))),k3_power(C,k3_xcmplx_0(np__2,C))),k3_xcmplx_0(k3_xcmplx_0(k3_power(A,k2_xcmplx_0(B,C)),k3_power(B,k2_xcmplx_0(A,C))),k3_power(C,k2_xcmplx_0(A,B)))) ) ) ) ) ).
fof(t44_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,C)
=> r1_xreal_0(k2_xcmplx_0(k3_xcmplx_0(k2_newton(A,C),B),k3_xcmplx_0(A,k2_newton(B,C))),k2_xcmplx_0(k2_newton(A,k1_nat_1(C,np__1)),k2_newton(B,k1_nat_1(C,np__1)))) ) ) ) ) ).
fof(t45_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( ( v1_xreal_0(C)
& v2_xreal_0(C) )
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( k2_xcmplx_0(k5_square_1(A),k5_square_1(B)) = k5_square_1(C)
& r1_xreal_0(np__3,D)
& r1_xreal_0(k2_newton(C,k1_nat_1(D,np__2)),k2_xcmplx_0(k2_newton(A,k1_nat_1(D,np__2)),k2_newton(B,k1_nat_1(D,np__2)))) ) ) ) ) ) ).
fof(t46_series_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,A)
& r1_xreal_0(k2_newton(k2_xcmplx_0(np__1,k7_xcmplx_0(np__1,A)),k1_nat_1(A,np__1)),k2_newton(k2_xcmplx_0(np__1,k7_xcmplx_0(np__1,k1_nat_1(A,np__1))),A)) ) ) ).
fof(t47_series_5,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( ( v1_xreal_0(B)
& v2_xreal_0(B) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,C)
& r1_xreal_0(np__1,D) )
=> r1_xreal_0(k3_xcmplx_0(k2_xcmplx_0(k2_newton(A,D),k2_newton(B,D)),k2_xcmplx_0(k2_newton(A,C),k2_newton(B,C))),k3_xcmplx_0(np__2,k2_xcmplx_0(k2_newton(A,k1_nat_1(D,C)),k2_newton(B,k1_nat_1(D,C))))) ) ) ) ) ) ).
fof(t48_series_5,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) = k7_xcmplx_0(np__1,k9_square_1(k1_nat_1(B,np__1))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(k3_xcmplx_0(np__2,k9_square_1(k1_nat_1(B,np__1))),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B)) ) ) ) ).
fof(t49_series_5,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) = k7_xcmplx_0(np__1,k7_square_1(k1_nat_1(B,np__1))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B),k6_xcmplx_0(np__2,k7_xcmplx_0(np__1,k1_nat_1(B,np__1)))) ) ) ) ).
fof(t50_series_5,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] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k7_xcmplx_0(np__1,k7_square_1(k1_nat_1(C,np__1))) )
& r1_xreal_0(np__2,k2_seq_1(k5_numbers,k1_numbers,k1_series_1(B),A)) ) ) ) ).
fof(t51_series_5,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)
=> ~ r1_xreal_0(np__1,k2_seq_1(k5_numbers,k1_numbers,A,B)) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(k1_nat_1(B,np__1),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B)) ) ) ) ).
fof(t52_series_5,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)
=> ( ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,B),np__0)
& ~ r1_xreal_0(np__1,k2_seq_1(k5_numbers,k1_numbers,A,B)) ) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_xreal_0(k6_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B),B),k2_seq_1(k5_numbers,k1_numbers,k1_series_3(A),B)) ) ) ) ).
fof(t53_series_5,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [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(k2_seq_1(k5_numbers,k1_numbers,A,C),np__0)
& k2_seq_1(k5_numbers,k1_numbers,B,C) = k7_xcmplx_0(np__1,k2_seq_1(k5_numbers,k1_numbers,A,C)) ) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(B),C),np__0) ) ) ) ) ).
fof(t54_series_5,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [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(k2_seq_1(k5_numbers,k1_numbers,A,C),np__0)
& k2_seq_1(k5_numbers,k1_numbers,B,C) = k7_xcmplx_0(np__1,k2_seq_1(k5_numbers,k1_numbers,A,C)) ) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_xreal_0(k7_square_1(k1_nat_1(C,np__1)),k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),C),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(B),C))) ) ) ) ) ).
fof(t55_series_5,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)
=> ( r1_xreal_0(np__1,B)
=> ( k2_seq_1(k5_numbers,k1_numbers,A,B) = k9_square_1(B)
& k2_seq_1(k5_numbers,k1_numbers,A,np__0) = np__0 ) ) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,B)
& r1_xreal_0(k3_xcmplx_0(k3_xcmplx_0(k7_xcmplx_0(np__1,np__6),k1_nat_1(k2_nat_1(np__4,B),np__3)),k9_square_1(B)),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B)) ) ) ) ) ).
fof(t56_series_5,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)
=> ( r1_xreal_0(np__1,B)
=> ( k2_seq_1(k5_numbers,k1_numbers,A,B) = k9_square_1(B)
& k2_seq_1(k5_numbers,k1_numbers,A,np__0) = np__0 ) ) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,B)
& r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B),k3_xcmplx_0(k3_xcmplx_0(k7_xcmplx_0(np__2,np__3),B),k9_square_1(B))) ) ) ) ) ).
fof(t57_series_5,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)
=> ( r1_xreal_0(np__1,B)
=> ( k2_seq_1(k5_numbers,k1_numbers,A,B) = k2_xcmplx_0(np__1,k7_xcmplx_0(np__1,k1_nat_1(k2_nat_1(np__2,B),np__1)))
& k2_seq_1(k5_numbers,k1_numbers,A,np__0) = np__1 ) ) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,B)
& r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_3(A),B),k3_xcmplx_0(k7_xcmplx_0(np__1,np__2),k9_square_1(k1_nat_1(k2_nat_1(np__2,B),np__3)))) ) ) ) ) ).
fof(t58_series_5,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)
=> ( r1_xreal_0(np__1,B)
=> ( k2_seq_1(k5_numbers,k1_numbers,A,B) = k9_square_1(k2_nat_1(B,k1_nat_1(B,np__1)))
& k2_seq_1(k5_numbers,k1_numbers,A,np__0) = np__0 ) ) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,B)
& r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B),k7_xcmplx_0(k2_nat_1(B,k1_nat_1(B,np__1)),np__2)) ) ) ) ) ).
%------------------------------------------------------------------------------