SET007 Axioms: SET007+922.ax
%------------------------------------------------------------------------------
% File : SET007+922 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Partial Sum and Partial Product 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_4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 30 ( 0 unt; 0 def)
% Number of atoms : 212 ( 64 equ)
% Maximal formula atoms : 12 ( 7 avg)
% Number of connectives : 183 ( 1 ~; 5 |; 49 &)
% ( 0 <=>; 128 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 8 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of predicates : 7 ( 6 usr; 0 prp; 1-3 aty)
% Number of functors : 25 ( 25 usr; 11 con; 0-4 aty)
% Number of variables : 93 ( 93 !; 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_4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> k2_newton(k2_xcmplx_0(k2_xcmplx_0(A,B),C),np__2) = k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_newton(A,np__2),k2_newton(B,np__2)),k2_newton(C,np__2)),k3_xcmplx_0(k3_xcmplx_0(np__2,A),B)),k3_xcmplx_0(k3_xcmplx_0(np__2,A),C)),k3_xcmplx_0(k3_xcmplx_0(np__2,B),C)) ) ) ) ).
fof(t2_series_4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k2_newton(k2_xcmplx_0(A,B),np__3) = k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_newton(A,np__3),k3_xcmplx_0(k3_xcmplx_0(np__3,k2_newton(A,np__2)),B)),k3_xcmplx_0(k3_xcmplx_0(np__3,k2_newton(B,np__2)),A)),k2_newton(B,np__3)) ) ) ).
fof(t3_series_4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> k2_newton(k2_xcmplx_0(k6_xcmplx_0(A,B),C),np__2) = k6_xcmplx_0(k2_xcmplx_0(k6_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_newton(A,np__2),k2_newton(B,np__2)),k2_newton(C,np__2)),k3_xcmplx_0(k3_xcmplx_0(np__2,A),B)),k3_xcmplx_0(k3_xcmplx_0(np__2,A),C)),k3_xcmplx_0(k3_xcmplx_0(np__2,B),C)) ) ) ) ).
fof(t4_series_4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> k2_newton(k6_xcmplx_0(k6_xcmplx_0(A,B),C),np__2) = k2_xcmplx_0(k6_xcmplx_0(k6_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_newton(A,np__2),k2_newton(B,np__2)),k2_newton(C,np__2)),k3_xcmplx_0(k3_xcmplx_0(np__2,A),B)),k3_xcmplx_0(k3_xcmplx_0(np__2,A),C)),k3_xcmplx_0(k3_xcmplx_0(np__2,B),C)) ) ) ) ).
fof(t5_series_4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k2_newton(k6_xcmplx_0(A,B),np__3) = k6_xcmplx_0(k2_xcmplx_0(k6_xcmplx_0(k2_newton(A,np__3),k3_xcmplx_0(k3_xcmplx_0(np__3,k2_newton(A,np__2)),B)),k3_xcmplx_0(k3_xcmplx_0(np__3,k2_newton(B,np__2)),A)),k2_newton(B,np__3)) ) ) ).
fof(t6_series_4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k2_newton(k2_xcmplx_0(A,B),np__4) = k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_newton(A,np__4),k3_xcmplx_0(k3_xcmplx_0(np__4,k2_newton(A,np__3)),B)),k3_xcmplx_0(k3_xcmplx_0(np__6,k2_newton(A,np__2)),k2_newton(B,np__2))),k3_xcmplx_0(k3_xcmplx_0(np__4,k2_newton(B,np__3)),A)),k2_newton(B,np__4)) ) ) ).
fof(t7_series_4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> k2_newton(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(A,B),C),D),np__2) = k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_newton(A,np__2),k2_newton(B,np__2)),k2_newton(C,np__2)),k2_newton(D,np__2)),k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(np__2,A),B),k3_xcmplx_0(k3_xcmplx_0(np__2,A),C)),k3_xcmplx_0(k3_xcmplx_0(np__2,A),D))),k2_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(np__2,B),C),k3_xcmplx_0(k3_xcmplx_0(np__2,B),D))),k3_xcmplx_0(k3_xcmplx_0(np__2,C),D)) ) ) ) ) ).
fof(t8_series_4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> k2_newton(k2_xcmplx_0(k2_xcmplx_0(A,B),C),np__3) = k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_newton(A,np__3),k2_newton(B,np__3)),k2_newton(C,np__3)),k2_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(np__3,k2_newton(A,np__2)),B),k3_xcmplx_0(k3_xcmplx_0(np__3,k2_newton(A,np__2)),C))),k2_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(np__3,k2_newton(B,np__2)),A),k3_xcmplx_0(k3_xcmplx_0(np__3,k2_newton(B,np__2)),C))),k2_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(np__3,k2_newton(C,np__2)),A),k3_xcmplx_0(k3_xcmplx_0(np__3,k2_newton(C,np__2)),B))),k3_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(np__6,A),B),C)) ) ) ) ).
fof(t9_series_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ( B != np__0
=> k2_newton(k2_xcmplx_0(k2_newton(k7_xcmplx_0(np__1,B),k1_nat_1(A,np__1)),k2_newton(B,k1_nat_1(A,np__1))),np__2) = k2_xcmplx_0(k2_xcmplx_0(k2_newton(k7_xcmplx_0(np__1,B),k1_nat_1(k2_nat_1(np__2,A),np__2)),k2_newton(B,k1_nat_1(k2_nat_1(np__2,A),np__2))),np__2) ) ) ) ).
fof(t10_series_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [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_newton(B,D) )
=> ( B = np__1
| k2_seq_1(k5_numbers,k1_numbers,k1_series_1(C),A) = k7_xcmplx_0(k6_xcmplx_0(np__1,k2_newton(B,k1_nat_1(A,np__1))),k6_xcmplx_0(np__1,B)) ) ) ) ) ) ).
fof(t11_series_4,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)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k2_newton(k7_xcmplx_0(np__1,A),C) )
=> ( A = np__1
| A = np__0
| ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(B),C) = k7_xcmplx_0(k6_xcmplx_0(k2_newton(k7_xcmplx_0(np__1,A),C),A),k6_xcmplx_0(np__1,A)) ) ) ) ) ) ).
fof(t12_series_4,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) = k2_xcmplx_0(k2_xcmplx_0(k3_prepower(np__10,C),k2_nat_1(np__2,C)),np__1) )
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(B),A) = k2_xcmplx_0(k6_xcmplx_0(k7_xcmplx_0(k3_prepower(np__10,k1_nat_1(A,np__1)),np__9),k7_xcmplx_0(np__1,np__9)),k3_prepower(k1_nat_1(A,np__1),np__2)) ) ) ) ).
fof(t13_series_4,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) = k2_xcmplx_0(k6_xcmplx_0(k2_nat_1(np__2,C),np__1),k2_newton(k7_xcmplx_0(np__1,np__2),C)) )
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(B),A) = k6_xcmplx_0(k2_xcmplx_0(k3_prepower(A,np__2),np__1),k2_newton(k7_xcmplx_0(np__1,np__2),A)) ) ) ) ).
fof(t14_series_4,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) = k3_xcmplx_0(C,k2_newton(k7_xcmplx_0(np__1,np__2),C)) )
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(B),A) = k6_xcmplx_0(np__2,k3_xcmplx_0(k1_nat_1(np__2,A),k2_newton(k7_xcmplx_0(np__1,np__2),A))) ) ) ) ).
fof(t15_series_4,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_newton(k2_xcmplx_0(k2_newton(k7_xcmplx_0(np__1,np__2),B),k3_prepower(np__2,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(k2_xcmplx_0(k2_xcmplx_0(k4_xcmplx_0(k7_xcmplx_0(k2_newton(k7_xcmplx_0(np__1,np__4),B),np__3)),k7_xcmplx_0(k3_prepower(np__4,k1_nat_1(B,np__1)),np__3)),k2_nat_1(np__2,B)),np__3) ) ) ) ).
fof(t16_series_4,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_newton(k2_xcmplx_0(k2_newton(k7_xcmplx_0(np__1,np__3),B),k3_prepower(np__3,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(k2_xcmplx_0(k2_xcmplx_0(k4_xcmplx_0(k7_xcmplx_0(k2_newton(k7_xcmplx_0(np__1,np__9),B),np__8)),k7_xcmplx_0(k3_prepower(np__9,k1_nat_1(B,np__1)),np__8)),k2_nat_1(np__2,B)),np__3) ) ) ) ).
fof(t17_series_4,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_prepower(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_xcmplx_0(k6_xcmplx_0(k3_xcmplx_0(B,k3_prepower(np__2,k1_nat_1(B,np__1))),k3_prepower(np__2,k1_nat_1(B,np__1))),np__2) ) ) ) ).
fof(t18_series_4,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(k1_nat_1(k2_nat_1(np__2,B),np__1),k3_prepower(np__3,B)) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) = k2_xcmplx_0(k3_xcmplx_0(B,k3_prepower(np__3,k1_nat_1(B,np__1))),np__1) ) ) ) ).
fof(t19_series_4,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)
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k3_xcmplx_0(C,k2_newton(A,C)) )
=> ( A = np__1
| ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(B),C) = k6_xcmplx_0(k7_xcmplx_0(k3_xcmplx_0(A,k6_xcmplx_0(np__1,k2_newton(A,C))),k2_newton(k6_xcmplx_0(np__1,A),np__2)),k7_xcmplx_0(k3_xcmplx_0(C,k2_newton(A,k1_nat_1(C,np__1))),k6_xcmplx_0(np__1,A))) ) ) ) ) ) ).
fof(t20_series_4,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,k2_xcmplx_0(k5_prepower(np__2,k1_nat_1(C,np__1)),k5_prepower(np__2,C))) )
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(B),A) = k5_prepower(np__2,k1_nat_1(A,np__1)) ) ) ) ).
fof(t21_series_4,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_xcmplx_0(k3_prepower(np__2,B),k2_newton(k7_xcmplx_0(np__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_xcmplx_0(k6_xcmplx_0(k3_prepower(np__2,k1_nat_1(B,np__1)),k2_newton(k7_xcmplx_0(np__1,np__2),B)),np__1) ) ) ) ).
fof(t22_series_4,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_xcmplx_0(k2_nat_1(k11_newton(B),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(k11_newton(k1_nat_1(B,np__1)),k7_xcmplx_0(np__1,k11_newton(k1_nat_1(B,np__1)))) ) ) ) ) ).
fof(t23_series_4,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) = k2_newton(k7_xcmplx_0(A,k6_xcmplx_0(A,np__1)),C)
& k2_seq_1(k5_numbers,k1_numbers,B,np__0) = np__0 ) ) )
=> ( A = np__1
| ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,C)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_1(B),C) = k3_xcmplx_0(A,k6_xcmplx_0(k2_newton(k7_xcmplx_0(A,k6_xcmplx_0(A,np__1)),C),np__1)) ) ) ) ) ) ) ).
fof(t24_series_4,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) = k3_xcmplx_0(k3_prepower(np__2,B),k7_xcmplx_0(k6_xcmplx_0(k2_nat_1(np__3,B),np__1),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) = k2_xcmplx_0(k3_xcmplx_0(k3_prepower(np__2,B),k7_xcmplx_0(k6_xcmplx_0(k2_nat_1(np__3,B),np__4),np__2)),np__2) ) ) ) ) ).
fof(t25_series_4,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(k1_nat_1(C,np__1),k1_nat_1(C,np__2)) )
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_3(B),A) = k7_xcmplx_0(np__1,k1_nat_1(A,np__2)) ) ) ) ).
fof(t26_series_4,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,k1_nat_1(C,np__1)) )
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_3(B),A) = k7_xcmplx_0(np__1,k11_newton(k1_nat_1(A,np__1))) ) ) ) ).
fof(t27_series_4,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) = B
& 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)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_3(A),B) = k11_newton(B) ) ) ) ) ).
fof(t28_series_4,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) = k7_xcmplx_0(A,C)
& k2_seq_1(k5_numbers,k1_numbers,B,np__0) = np__1 ) ) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,C)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_3(B),C) = k7_xcmplx_0(k2_newton(A,C),k11_newton(C)) ) ) ) ) ) ).
fof(t29_series_4,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) = A
& k2_seq_1(k5_numbers,k1_numbers,B,np__0) = np__1 ) ) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,C)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_3(B),C) = k2_newton(A,C) ) ) ) ) ) ).
fof(t30_series_4,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__2,B)
=> ( k2_seq_1(k5_numbers,k1_numbers,A,B) = k6_xcmplx_0(np__1,k7_xcmplx_0(np__1,k3_prepower(B,np__2)))
& k2_seq_1(k5_numbers,k1_numbers,A,np__0) = np__1
& k2_seq_1(k5_numbers,k1_numbers,A,np__1) = np__1 ) ) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__2,B)
=> k2_seq_1(k5_numbers,k1_numbers,k1_series_3(A),B) = k7_xcmplx_0(k1_nat_1(B,np__1),k2_nat_1(np__2,B)) ) ) ) ) ).
%------------------------------------------------------------------------------