SET007 Axioms: SET007+145.ax
%------------------------------------------------------------------------------
% File : SET007+145 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : 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_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 56 ( 6 unt; 0 def)
% Number of atoms : 417 ( 43 equ)
% Maximal formula atoms : 14 ( 7 avg)
% Number of connectives : 382 ( 21 ~; 11 |; 188 &)
% ( 6 <=>; 156 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 7 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 16 ( 14 usr; 1 prp; 0-3 aty)
% Number of functors : 28 ( 28 usr; 5 con; 0-4 aty)
% Number of variables : 129 ( 121 !; 8 ?)
% 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_1,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)
=> k8_funct_2(k5_numbers,k1_numbers,B,C) = k3_power(A,k1_nat_1(C,np__1)) )
=> ( r1_xreal_0(A,np__0)
| r1_xreal_0(np__1,A)
| ( v4_seq_2(B)
& k2_seq_2(B) = np__0 ) ) ) ) ) ).
fof(t2_series_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ( B != np__0
=> k4_power(k18_complex1(B),A) = k18_complex1(k3_power(B,A)) ) ) ) ).
fof(t3_series_1,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)
=> k8_funct_2(k5_numbers,k1_numbers,B,C) = k3_power(A,k1_nat_1(C,np__1)) )
=> ( r1_xreal_0(np__1,k18_complex1(A))
| ( v4_seq_2(B)
& k2_seq_2(B) = np__0 ) ) ) ) ) ).
fof(d1_series_1,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) )
=> ( B = k1_series_1(A)
<=> ( k8_funct_2(k5_numbers,k1_numbers,B,np__0) = k8_funct_2(k5_numbers,k1_numbers,A,np__0)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,k1_numbers,B,k1_nat_1(C,np__1)) = k3_real_1(k8_funct_2(k5_numbers,k1_numbers,B,C),k8_funct_2(k5_numbers,k1_numbers,A,k1_nat_1(C,np__1))) ) ) ) ) ) ).
fof(d2_series_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v1_series_1(A)
<=> v4_seq_2(k1_series_1(A)) ) ) ).
fof(d3_series_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> k2_series_1(A) = k2_seq_2(k1_series_1(A)) ) ).
fof(t4_series_1,axiom,
$true ).
fof(t5_series_1,axiom,
$true ).
fof(t6_series_1,axiom,
$true ).
fof(t7_series_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v1_series_1(A)
=> ( v4_seq_2(A)
& k2_seq_2(A) = np__0 ) ) ) ).
fof(t8_series_1,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) )
=> k9_seq_1(k1_series_1(A),k1_series_1(B)) = k1_series_1(k9_seq_1(A,B)) ) ) ).
fof(t9_series_1,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) )
=> k10_seq_1(k1_series_1(A),k1_series_1(B)) = k1_series_1(k10_seq_1(A,B)) ) ) ).
fof(t10_series_1,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) )
=> ( ( v1_series_1(A)
& v1_series_1(B) )
=> ( v1_series_1(k9_seq_1(A,B))
& k2_series_1(k9_seq_1(A,B)) = k3_real_1(k2_series_1(A),k2_series_1(B)) ) ) ) ) ).
fof(t11_series_1,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) )
=> ( ( v1_series_1(A)
& v1_series_1(B) )
=> ( v1_series_1(k10_seq_1(A,B))
& k2_series_1(k10_seq_1(A,B)) = k5_real_1(k2_series_1(A),k2_series_1(B)) ) ) ) ) ).
fof(t12_series_1,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) )
=> k1_series_1(k14_seq_1(B,A)) = k14_seq_1(k1_series_1(B),A) ) ) ).
fof(t13_series_1,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) )
=> ( v1_series_1(B)
=> ( v1_series_1(k14_seq_1(B,A))
& k2_series_1(k14_seq_1(B,A)) = k3_xcmplx_0(A,k2_series_1(B)) ) ) ) ) ).
fof(t14_series_1,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)
=> k8_funct_2(k5_numbers,k1_numbers,B,C) = k8_funct_2(k5_numbers,k1_numbers,A,np__0) )
=> k1_series_1(k1_seqm_3(A,np__1)) = k10_seq_1(k1_seqm_3(k1_series_1(A),np__1),B) ) ) ) ).
fof(t15_series_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v1_series_1(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> v1_series_1(k1_seqm_3(A,B)) ) ) ) ).
fof(t16_series_1,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)
& v1_series_1(k1_seqm_3(A,B)) )
=> v1_series_1(A) ) ) ).
fof(t17_series_1,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(k8_funct_2(k5_numbers,k1_numbers,A,C),k8_funct_2(k5_numbers,k1_numbers,B,C)) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_xreal_0(k8_funct_2(k5_numbers,k1_numbers,k1_series_1(A),C),k8_funct_2(k5_numbers,k1_numbers,k1_series_1(B),C)) ) ) ) ) ).
fof(t18_series_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v1_series_1(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_series_1(A) = k3_real_1(k8_funct_2(k5_numbers,k1_numbers,k1_series_1(A),B),k2_series_1(k1_seqm_3(A,k1_nat_1(B,np__1)))) ) ) ) ).
fof(t19_series_1,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__0,k8_funct_2(k5_numbers,k1_numbers,A,B)) )
=> v3_seqm_3(k1_series_1(A)) ) ) ).
fof(t20_series_1,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__0,k8_funct_2(k5_numbers,k1_numbers,A,B)) )
=> ( v1_seq_2(k1_series_1(A))
<=> v1_series_1(A) ) ) ) ).
fof(t21_series_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ( v1_series_1(A)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_xreal_0(np__0,k8_funct_2(k5_numbers,k1_numbers,A,B)) ) )
=> r1_xreal_0(np__0,k2_series_1(A)) ) ) ).
fof(t22_series_1,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(np__0,k8_funct_2(k5_numbers,k1_numbers,A,C)) )
& v1_series_1(B) )
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& r1_xreal_0(C,D)
& ~ r1_xreal_0(k8_funct_2(k5_numbers,k1_numbers,A,D),k8_funct_2(k5_numbers,k1_numbers,B,D)) ) )
| v1_series_1(A) ) ) ) ) ).
fof(t23_series_1,axiom,
$true ).
fof(t24_series_1,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(np__0,k8_funct_2(k5_numbers,k1_numbers,A,C))
& r1_xreal_0(k8_funct_2(k5_numbers,k1_numbers,A,C),k8_funct_2(k5_numbers,k1_numbers,B,C)) ) )
& v1_series_1(B) )
=> ( v1_series_1(A)
& r1_xreal_0(k2_series_1(A),k2_series_1(B)) ) ) ) ) ).
fof(t25_series_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v1_series_1(A)
<=> ! [B] :
( v1_xreal_0(B)
=> ~ ( ~ r1_xreal_0(B,np__0)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& r1_xreal_0(C,D)
& r1_xreal_0(B,k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,k1_series_1(A),D),k8_funct_2(k5_numbers,k1_numbers,k1_series_1(A),C)))) ) ) ) ) ) ) ).
fof(t26_series_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ( B != np__1
=> k8_funct_2(k5_numbers,k1_numbers,k1_series_1(k2_prepower(B)),A) = k7_xcmplx_0(k6_xcmplx_0(np__1,k3_power(B,k1_nat_1(A,np__1))),k6_xcmplx_0(np__1,B)) ) ) ) ).
fof(t27_series_1,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)
=> k8_funct_2(k5_numbers,k1_numbers,B,k1_nat_1(C,np__1)) = k3_xcmplx_0(A,k8_funct_2(k5_numbers,k1_numbers,B,C)) )
=> ( A = np__1
| ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,k1_numbers,k1_series_1(B),C) = k7_xcmplx_0(k3_xcmplx_0(k8_funct_2(k5_numbers,k1_numbers,B,np__0),k6_xcmplx_0(np__1,k3_power(A,k1_nat_1(C,np__1)))),k6_xcmplx_0(np__1,A)) ) ) ) ) ) ).
fof(t28_series_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( ~ r1_xreal_0(np__1,k18_complex1(A))
=> ( v1_series_1(k2_prepower(A))
& k2_series_1(k2_prepower(A)) = k7_xcmplx_0(np__1,k6_xcmplx_0(np__1,A)) ) ) ) ).
fof(t29_series_1,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)
=> k8_funct_2(k5_numbers,k1_numbers,B,k1_nat_1(C,np__1)) = k3_xcmplx_0(A,k8_funct_2(k5_numbers,k1_numbers,B,C)) )
=> ( r1_xreal_0(np__1,k18_complex1(A))
| ( v1_series_1(B)
& k2_series_1(B) = k7_xcmplx_0(k8_funct_2(k5_numbers,k1_numbers,B,np__0),k6_xcmplx_0(np__1,A)) ) ) ) ) ) ).
fof(t30_series_1,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(k8_funct_2(k5_numbers,k1_numbers,A,C),np__0)
& k8_funct_2(k5_numbers,k1_numbers,B,C) = k6_real_1(k8_funct_2(k5_numbers,k1_numbers,A,k1_nat_1(C,np__1)),k8_funct_2(k5_numbers,k1_numbers,A,C)) ) )
& v4_seq_2(B) )
=> ( r1_xreal_0(np__1,k2_seq_2(B))
| v1_series_1(A) ) ) ) ) ).
fof(t31_series_1,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(k8_funct_2(k5_numbers,k1_numbers,A,B),np__0) )
& ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,C)
=> r1_xreal_0(np__1,k6_real_1(k8_funct_2(k5_numbers,k1_numbers,A,k1_nat_1(C,np__1)),k8_funct_2(k5_numbers,k1_numbers,A,C))) ) ) )
& v1_series_1(A) ) ) ).
fof(t32_series_1,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(np__0,k8_funct_2(k5_numbers,k1_numbers,A,C))
& k8_funct_2(k5_numbers,k1_numbers,B,C) = k2_power(C,k8_funct_2(k5_numbers,k1_numbers,A,C)) ) )
& v4_seq_2(B) )
=> ( r1_xreal_0(np__1,k2_seq_2(B))
| v1_series_1(A) ) ) ) ) ).
fof(t33_series_1,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(np__0,k8_funct_2(k5_numbers,k1_numbers,A,C))
& k8_funct_2(k5_numbers,k1_numbers,B,C) = k2_power(C,k8_funct_2(k5_numbers,k1_numbers,A,C)) ) )
& ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,D)
=> r1_xreal_0(np__1,k8_funct_2(k5_numbers,k1_numbers,B,D)) ) ) )
& v1_series_1(A) ) ) ) ).
fof(t34_series_1,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(np__0,k8_funct_2(k5_numbers,k1_numbers,A,C))
& k8_funct_2(k5_numbers,k1_numbers,B,C) = k2_power(C,k8_funct_2(k5_numbers,k1_numbers,A,C)) ) )
& v4_seq_2(B)
& ~ r1_xreal_0(k2_seq_2(B),np__1)
& v1_series_1(A) ) ) ) ).
fof(t35_series_1,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) )
=> ( ( v4_seqm_3(A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__0,k8_funct_2(k5_numbers,k1_numbers,A,C))
& k8_funct_2(k5_numbers,k1_numbers,B,C) = k4_real_1(k3_series_1(np__2,C),k8_funct_2(k5_numbers,k1_numbers,A,k3_series_1(np__2,C))) ) ) )
=> ( v1_series_1(A)
<=> v1_series_1(B) ) ) ) ) ).
fof(t36_series_1,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)
=> k8_funct_2(k5_numbers,k1_numbers,B,C) = k7_xcmplx_0(np__1,k3_power(C,A)) ) )
=> ( r1_xreal_0(A,np__1)
| v1_series_1(B) ) ) ) ) ).
fof(t37_series_1,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) )
=> ~ ( r1_xreal_0(A,np__1)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,C)
=> k8_funct_2(k5_numbers,k1_numbers,B,C) = k7_xcmplx_0(np__1,k3_power(C,A)) ) )
& v1_series_1(B) ) ) ) ).
fof(d4_series_1,axiom,
$true ).
fof(d5_series_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v2_series_1(A)
<=> v1_series_1(k22_seq_1(A)) ) ) ).
fof(t38_series_1,axiom,
$true ).
fof(t39_series_1,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)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,C)
=> r1_xreal_0(k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,k1_series_1(A),C),k8_funct_2(k5_numbers,k1_numbers,k1_series_1(A),B))),k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,k1_series_1(k22_seq_1(A)),C),k8_funct_2(k5_numbers,k1_numbers,k1_series_1(k22_seq_1(A)),B)))) ) ) ) ) ).
fof(t40_series_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v2_series_1(A)
=> v1_series_1(A) ) ) ).
fof(t41_series_1,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__0,k8_funct_2(k5_numbers,k1_numbers,A,B)) )
& v1_series_1(A) )
=> v2_series_1(A) ) ) ).
fof(t42_series_1,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)
=> ( k8_funct_2(k5_numbers,k1_numbers,A,C) != np__0
& k8_funct_2(k5_numbers,k1_numbers,B,C) = k6_real_1(k8_funct_2(k5_numbers,k1_numbers,k22_seq_1(A),k1_nat_1(C,np__1)),k8_funct_2(k5_numbers,k1_numbers,k22_seq_1(A),C)) ) )
& v4_seq_2(B) )
=> ( r1_xreal_0(np__1,k2_seq_2(B))
| v2_series_1(A) ) ) ) ) ).
fof(t43_series_1,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) )
=> ~ ( ~ r1_xreal_0(A,np__0)
& ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,D)
=> r1_xreal_0(A,k18_complex1(k8_funct_2(k5_numbers,k1_numbers,B,D))) ) ) )
& v4_seq_2(B)
& k2_seq_2(B) = np__0 ) ) ) ).
fof(t44_series_1,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)
=> k8_funct_2(k5_numbers,k1_numbers,A,B) != np__0 )
& ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,C)
=> r1_xreal_0(np__1,k6_real_1(k8_funct_2(k5_numbers,k1_numbers,k22_seq_1(A),k1_nat_1(C,np__1)),k8_funct_2(k5_numbers,k1_numbers,k22_seq_1(A),C))) ) ) )
& v1_series_1(A) ) ) ).
fof(t45_series_1,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)
=> k8_funct_2(k5_numbers,k1_numbers,A,C) = k2_power(C,k8_funct_2(k5_numbers,k1_numbers,k22_seq_1(B),C)) )
& v4_seq_2(A) )
=> ( r1_xreal_0(np__1,k2_seq_2(A))
| v2_series_1(B) ) ) ) ) ).
fof(t46_series_1,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)
=> k8_funct_2(k5_numbers,k1_numbers,A,C) = k2_power(C,k8_funct_2(k5_numbers,k1_numbers,k22_seq_1(B),C)) )
& ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,D)
=> r1_xreal_0(np__1,k8_funct_2(k5_numbers,k1_numbers,A,D)) ) ) )
& v1_series_1(B) ) ) ) ).
fof(t47_series_1,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)
=> k8_funct_2(k5_numbers,k1_numbers,A,C) = k2_power(C,k8_funct_2(k5_numbers,k1_numbers,k22_seq_1(B),C)) )
& v4_seq_2(A)
& ~ r1_xreal_0(k2_seq_2(A),np__1)
& v1_series_1(B) ) ) ) ).
fof(dt_k1_series_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers) )
=> ( v1_funct_1(k1_series_1(A))
& v1_funct_2(k1_series_1(A),k5_numbers,k1_numbers)
& m2_relset_1(k1_series_1(A),k5_numbers,k1_numbers) ) ) ).
fof(dt_k2_series_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers) )
=> m1_subset_1(k2_series_1(A),k1_numbers) ) ).
fof(dt_k3_series_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> m2_subset_1(k3_series_1(A,B),k1_numbers,k5_numbers) ) ).
fof(redefinition_k3_series_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> k3_series_1(A,B) = k3_power(A,B) ) ).
%------------------------------------------------------------------------------