SET007 Axioms: SET007+858.ax
%------------------------------------------------------------------------------
% File : SET007+858 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Partial Sum of Some Series
% 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_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 42 ( 0 unt; 0 def)
% Number of atoms : 322 ( 92 equ)
% Maximal formula atoms : 10 ( 7 avg)
% Number of connectives : 280 ( 0 ~; 0 |; 90 &)
% ( 0 <=>; 190 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 7 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of predicates : 7 ( 6 usr; 0 prp; 1-3 aty)
% Number of functors : 31 ( 31 usr; 18 con; 0-4 aty)
% Number of variables : 126 ( 126 !; 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_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k18_complex1(k2_newton(k4_xcmplx_0(np__1),A)) = np__1 ) ).
fof(t2_series_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k2_newton(k2_xcmplx_0(A,np__1),np__3) = k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_newton(A,np__3),k3_xcmplx_0(np__3,k2_newton(A,np__2))),k3_xcmplx_0(np__3,A)),np__1)
& k2_newton(k2_xcmplx_0(A,np__1),np__4) = k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_newton(A,np__4),k3_xcmplx_0(np__4,k2_newton(A,np__3))),k3_xcmplx_0(np__6,k2_newton(A,np__2))),k3_xcmplx_0(np__4,A)),np__1)
& k2_newton(k2_xcmplx_0(A,np__1),np__5) = k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_newton(A,np__5),k3_xcmplx_0(np__5,k2_newton(A,np__4))),k3_xcmplx_0(np__10,k2_newton(A,np__3))),k3_xcmplx_0(np__10,k2_newton(A,np__2))),k3_xcmplx_0(np__5,A)),np__1) ) ) ).
fof(t3_series_2,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) = B )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> 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) ) ) ) ).
fof(t4_series_2,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) = k2_nat_1(np__2,B) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k2_nat_1(B,k1_nat_1(B,np__1)) ) ) ) ).
fof(t5_series_2,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) = k1_nat_1(k2_nat_1(np__2,B),np__1) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k3_newton(k1_nat_1(B,np__1),np__2) ) ) ) ).
fof(t6_series_2,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) = k2_nat_1(B,k1_nat_1(B,np__1)) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(B,np__2)),np__3) ) ) ) ).
fof(t7_series_2,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) = k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(B,np__2)) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k2_nat_1(k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(B,np__2)),k1_nat_1(B,np__3)),np__4) ) ) ) ).
fof(t8_series_2,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) = k2_nat_1(k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(B,np__2)),k1_nat_1(B,np__3)) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k2_nat_1(k2_nat_1(k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(B,np__2)),k1_nat_1(B,np__3)),k1_nat_1(B,np__4)),np__5) ) ) ) ).
fof(t9_series_2,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,k2_nat_1(B,k1_nat_1(B,np__1))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(np__1,k7_xcmplx_0(np__1,k1_nat_1(B,np__1))) ) ) ) ).
fof(t10_series_2,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,k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(B,np__2))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(k7_xcmplx_0(np__1,np__4),k7_xcmplx_0(np__1,k2_nat_1(k2_nat_1(np__2,k1_nat_1(B,np__1)),k1_nat_1(B,np__2)))) ) ) ) ).
fof(t11_series_2,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,k2_nat_1(k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(B,np__2)),k1_nat_1(B,np__3))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(k7_xcmplx_0(np__1,np__18),k7_xcmplx_0(np__1,k2_nat_1(k2_nat_1(k2_nat_1(np__3,k1_nat_1(B,np__1)),k1_nat_1(B,np__2)),k1_nat_1(B,np__3)))) ) ) ) ).
fof(t12_series_2,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) = k3_newton(B,np__2) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(k2_nat_1(np__2,B),np__1)),np__6) ) ) ) ).
fof(t13_series_2,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) = k3_xcmplx_0(k2_newton(k4_xcmplx_0(np__1),k1_nat_1(B,np__1)),k3_newton(B,np__2)) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(k2_newton(k4_xcmplx_0(np__1),k1_nat_1(B,np__1)),B),k1_nat_1(B,np__1)),np__2) ) ) ) ).
fof(t14_series_2,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_newton(k6_xcmplx_0(k2_nat_1(np__2,B),np__1),np__2)
& 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)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k3_xcmplx_0(B,k6_xcmplx_0(k3_xcmplx_0(np__4,k3_newton(B,np__2)),np__1)),np__3) ) ) ) ) ).
fof(t15_series_2,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) = k3_newton(B,np__3) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k3_xcmplx_0(k3_newton(B,np__2),k3_newton(k1_nat_1(B,np__1),np__2)),np__4) ) ) ) ).
fof(t16_series_2,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_newton(k6_xcmplx_0(k2_nat_1(np__2,B),np__1),np__3)
& 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)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k3_xcmplx_0(k3_newton(B,np__2),k6_xcmplx_0(k3_xcmplx_0(np__2,k3_newton(B,np__2)),np__1)) ) ) ) ) ).
fof(t17_series_2,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) = k3_newton(B,np__4) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k3_xcmplx_0(k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(k2_nat_1(np__2,B),np__1)),k6_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(np__3,k3_newton(B,np__2)),k2_nat_1(np__3,B)),np__1)),np__30) ) ) ) ).
fof(t18_series_2,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) = k3_xcmplx_0(k2_newton(k4_xcmplx_0(np__1),k1_nat_1(B,np__1)),k3_newton(B,np__4)) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(k2_newton(k4_xcmplx_0(np__1),k1_nat_1(B,np__1)),B),k1_nat_1(B,np__1)),k6_xcmplx_0(k2_xcmplx_0(k3_newton(B,np__2),B),np__1)),np__2) ) ) ) ).
fof(t19_series_2,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) = k3_newton(B,np__5) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(k3_newton(B,np__2),k3_newton(k1_nat_1(B,np__1),np__2)),k6_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(np__2,k3_newton(B,np__2)),k2_nat_1(np__2,B)),np__1)),np__12) ) ) ) ).
fof(t20_series_2,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) = k3_newton(B,np__6) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k3_xcmplx_0(k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(k2_nat_1(np__2,B),np__1)),k2_xcmplx_0(k6_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(np__3,k3_newton(B,np__4)),k3_xcmplx_0(np__6,k3_newton(B,np__3))),k2_nat_1(np__3,B)),np__1)),np__42) ) ) ) ).
fof(t21_series_2,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) = k3_newton(B,np__7) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(k3_newton(B,np__2),k3_newton(k1_nat_1(B,np__1),np__2)),k2_xcmplx_0(k6_xcmplx_0(k6_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(np__3,k3_newton(B,np__4)),k3_xcmplx_0(np__6,k3_newton(B,np__3))),k3_newton(B,np__2)),k2_nat_1(np__4,B)),np__2)),np__24) ) ) ) ).
fof(t22_series_2,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) = k3_xcmplx_0(B,k3_newton(k1_nat_1(B,np__1),np__2)) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k2_nat_1(k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(B,np__2)),k1_nat_1(k2_nat_1(np__3,B),np__5)),np__12) ) ) ) ).
fof(t23_series_2,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) = k3_xcmplx_0(k3_xcmplx_0(B,k3_newton(k1_nat_1(B,np__1),np__2)),k1_nat_1(B,np__2)) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(k2_nat_1(k2_nat_1(k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(B,np__2)),k1_nat_1(B,np__3)),k1_nat_1(k2_nat_1(np__2,B),np__3)),np__10) ) ) ) ).
fof(t24_series_2,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) = k3_xcmplx_0(k2_nat_1(B,k1_nat_1(B,np__1)),k3_newton(np__2,B)) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(k3_xcmplx_0(k3_newton(np__2,k1_nat_1(B,np__1)),k2_xcmplx_0(k6_xcmplx_0(k3_newton(B,np__2),B),np__2)),np__4) ) ) ) ).
fof(t25_series_2,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) = k7_xcmplx_0(np__1,k3_xcmplx_0(k6_xcmplx_0(B,np__1),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__2,B)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(k6_xcmplx_0(k7_xcmplx_0(np__3,np__4),k7_xcmplx_0(np__1,k2_nat_1(np__2,B))),k7_xcmplx_0(np__1,k2_nat_1(np__2,k1_nat_1(B,np__1)))) ) ) ) ) ).
fof(t26_series_2,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) = k7_xcmplx_0(np__1,k3_xcmplx_0(k6_xcmplx_0(k2_nat_1(np__2,B),np__1),k1_nat_1(k2_nat_1(np__2,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)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(B,k1_nat_1(k2_nat_1(np__2,B),np__1)) ) ) ) ) ).
fof(t27_series_2,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) = k7_xcmplx_0(np__1,k3_xcmplx_0(k6_xcmplx_0(k2_nat_1(np__3,B),np__2),k1_nat_1(k2_nat_1(np__3,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)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k7_xcmplx_0(B,k1_nat_1(k2_nat_1(np__3,B),np__1)) ) ) ) ) ).
fof(t28_series_2,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) = k7_xcmplx_0(np__1,k3_xcmplx_0(k3_xcmplx_0(k6_xcmplx_0(k2_nat_1(np__2,B),np__1),k1_nat_1(k2_nat_1(np__2,B),np__1)),k1_nat_1(k2_nat_1(np__2,B),np__3)))
& 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)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(k7_xcmplx_0(np__1,np__12),k7_xcmplx_0(np__1,k2_nat_1(k2_nat_1(np__4,k1_nat_1(k2_nat_1(np__2,B),np__1)),k1_nat_1(k2_nat_1(np__2,B),np__3)))) ) ) ) ) ).
fof(t29_series_2,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) = k7_xcmplx_0(np__1,k3_xcmplx_0(k3_xcmplx_0(k6_xcmplx_0(k2_nat_1(np__3,B),np__2),k1_nat_1(k2_nat_1(np__3,B),np__1)),k1_nat_1(k2_nat_1(np__3,B),np__4)))
& 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)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(k7_xcmplx_0(np__1,np__24),k7_xcmplx_0(np__1,k2_nat_1(k2_nat_1(np__6,k1_nat_1(k2_nat_1(np__3,B),np__1)),k1_nat_1(k2_nat_1(np__3,B),np__4)))) ) ) ) ) ).
fof(t30_series_2,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(k6_xcmplx_0(k2_nat_1(np__2,B),np__1),k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(B,np__2))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,B)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k2_xcmplx_0(k6_xcmplx_0(k7_xcmplx_0(np__3,np__4),k7_xcmplx_0(np__2,k1_nat_1(B,np__2))),k7_xcmplx_0(np__1,k2_nat_1(k2_nat_1(np__2,k1_nat_1(B,np__1)),k1_nat_1(B,np__2)))) ) ) ) ) ).
fof(t31_series_2,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(k1_nat_1(B,np__2),k2_nat_1(k2_nat_1(B,k1_nat_1(B,np__1)),k1_nat_1(B,np__3))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,B)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(k6_xcmplx_0(k6_xcmplx_0(k7_xcmplx_0(np__29,np__36),k7_xcmplx_0(np__1,k1_nat_1(B,np__3))),k7_xcmplx_0(np__3,k2_nat_1(k2_nat_1(np__2,k1_nat_1(B,np__2)),k1_nat_1(B,np__3)))),k7_xcmplx_0(np__4,k2_nat_1(k2_nat_1(k2_nat_1(np__3,k1_nat_1(B,np__1)),k1_nat_1(B,np__2)),k1_nat_1(B,np__3)))) ) ) ) ) ).
fof(t32_series_2,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(k3_xcmplx_0(k1_nat_1(B,np__1),k3_newton(np__2,B)),k2_nat_1(k1_nat_1(B,np__2),k1_nat_1(B,np__3))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(k7_xcmplx_0(k3_newton(np__2,k1_nat_1(B,np__1)),k1_nat_1(B,np__3)),k7_xcmplx_0(np__1,np__2)) ) ) ) ).
fof(t33_series_2,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(k3_xcmplx_0(k3_newton(B,np__2),k3_newton(np__4,B)),k2_nat_1(k1_nat_1(B,np__1),k1_nat_1(B,np__2))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,B)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k2_xcmplx_0(k7_xcmplx_0(np__2,np__3),k7_xcmplx_0(k3_xcmplx_0(k6_xcmplx_0(B,np__1),k3_newton(np__4,k1_nat_1(B,np__1))),k2_nat_1(np__3,k1_nat_1(B,np__2)))) ) ) ) ) ).
fof(t34_series_2,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(k1_nat_1(B,np__2),k3_xcmplx_0(k2_nat_1(B,k1_nat_1(B,np__1)),k3_newton(np__2,B))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,B)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(np__1,k7_xcmplx_0(np__1,k3_xcmplx_0(k1_nat_1(B,np__1),k3_newton(np__2,B)))) ) ) ) ) ).
fof(t35_series_2,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(k1_nat_1(k2_nat_1(np__2,B),np__3),k3_xcmplx_0(k2_nat_1(B,k1_nat_1(B,np__1)),k3_newton(np__3,B))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,B)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(np__1,k7_xcmplx_0(np__1,k3_xcmplx_0(k1_nat_1(B,np__1),k3_newton(np__3,B)))) ) ) ) ) ).
fof(t36_series_2,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(k3_xcmplx_0(k2_newton(k4_xcmplx_0(np__1),B),k3_newton(np__2,k1_nat_1(B,np__1))),k3_xcmplx_0(k2_xcmplx_0(k3_newton(np__2,k1_nat_1(B,np__1)),k2_newton(k4_xcmplx_0(np__1),k1_nat_1(B,np__1))),k2_xcmplx_0(k3_newton(np__2,k1_nat_1(B,np__2)),k2_newton(k4_xcmplx_0(np__1),k1_nat_1(B,np__2))))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k2_xcmplx_0(k7_xcmplx_0(np__1,np__3),k7_xcmplx_0(k2_newton(k4_xcmplx_0(np__1),k1_nat_1(B,np__2)),k3_xcmplx_0(np__3,k2_xcmplx_0(k3_newton(np__2,k1_nat_1(B,np__2)),k2_newton(k4_xcmplx_0(np__1),k1_nat_1(B,np__2)))))) ) ) ) ).
fof(t37_series_2,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) = k2_nat_1(k11_newton(B),B) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,B)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(k11_newton(k1_nat_1(B,np__1)),np__1) ) ) ) ) ).
fof(t38_series_2,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(B,k11_newton(k1_nat_1(B,np__1))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,B)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(np__1,k7_xcmplx_0(np__1,k11_newton(k1_nat_1(B,np__1)))) ) ) ) ) ).
fof(t39_series_2,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) = k7_xcmplx_0(k6_xcmplx_0(k2_xcmplx_0(k3_newton(B,np__2),B),np__1),k11_newton(k1_nat_1(B,np__2)))
& 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)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(k7_xcmplx_0(np__1,np__2),k7_xcmplx_0(k1_nat_1(B,np__1),k11_newton(k1_nat_1(B,np__2)))) ) ) ) ) ).
fof(t40_series_2,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(k3_xcmplx_0(B,k3_newton(np__2,B)),k11_newton(k1_nat_1(B,np__2))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,B)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k6_xcmplx_0(np__1,k7_xcmplx_0(k3_newton(np__2,k1_nat_1(B,np__1)),k11_newton(k1_nat_1(B,np__2)))) ) ) ) ) ).
fof(t41_series_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,C,D) = k2_xcmplx_0(k3_xcmplx_0(A,D),B) )
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(C),D) = k2_xcmplx_0(k2_xcmplx_0(k7_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(A,k1_nat_1(D,np__1)),D),np__2),k3_xcmplx_0(D,B)),B) ) ) ) ) ) ).
fof(t42_series_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,C,D) = k2_xcmplx_0(k3_xcmplx_0(A,D),B) )
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(C),D) = k7_xcmplx_0(k3_xcmplx_0(k1_nat_1(D,np__1),k2_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,C,np__0),k2_seq_1(k5_numbers,k1_numbers,C,D))),np__2) ) ) ) ) ) ).
%------------------------------------------------------------------------------