SET007 Axioms: SET007+138.ax
%------------------------------------------------------------------------------
% File : SET007+138 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Conjugate, Bounded Complex, and Convergent Complex Sequences
% 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 : comseq_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 59 ( 0 unt; 0 def)
% Number of atoms : 479 ( 56 equ)
% Maximal formula atoms : 15 ( 8 avg)
% Number of connectives : 448 ( 28 ~; 8 |; 261 &)
% ( 6 <=>; 145 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 16 ( 15 usr; 0 prp; 1-3 aty)
% Number of functors : 33 ( 33 usr; 6 con; 0-4 aty)
% Number of variables : 116 ( 106 !; 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(fc1_comseq_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,A,k2_numbers)
& m1_relset_1(B,A,k2_numbers) )
=> ( ~ v1_xboole_0(k1_comseq_2(A,B))
& v1_relat_1(k1_comseq_2(A,B))
& v1_funct_1(k1_comseq_2(A,B))
& v1_funct_2(k1_comseq_2(A,B),A,k2_numbers)
& v1_partfun1(k1_comseq_2(A,B),A,k2_numbers) ) ) ).
fof(rc1_comseq_2,axiom,
? [A] :
( m1_relset_1(A,k5_numbers,k2_numbers)
& ~ v1_xboole_0(A)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& v1_partfun1(A,k5_numbers,k2_numbers)
& v1_comseq_2(A) ) ).
fof(rc2_comseq_2,axiom,
? [A] :
( m1_relset_1(A,k5_numbers,k2_numbers)
& ~ v1_xboole_0(A)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& v1_partfun1(A,k5_numbers,k2_numbers)
& v2_comseq_2(A) ) ).
fof(fc2_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& v2_comseq_2(A)
& m1_relset_1(A,k5_numbers,k2_numbers) )
=> ( ~ v1_xboole_0(k9_comseq_1(k5_numbers,A))
& v1_relat_1(k9_comseq_1(k5_numbers,A))
& v1_funct_1(k9_comseq_1(k5_numbers,A))
& v1_funct_2(k9_comseq_1(k5_numbers,A),k5_numbers,k1_numbers)
& v4_seq_2(k9_comseq_1(k5_numbers,A))
& v1_partfun1(k9_comseq_1(k5_numbers,A),k5_numbers,k1_numbers) ) ) ).
fof(fc3_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& v2_comseq_2(A)
& m1_relset_1(A,k5_numbers,k2_numbers) )
=> ( ~ v1_xboole_0(k1_comseq_2(k5_numbers,A))
& v1_relat_1(k1_comseq_2(k5_numbers,A))
& v1_funct_1(k1_comseq_2(k5_numbers,A))
& v1_funct_2(k1_comseq_2(k5_numbers,A),k5_numbers,k2_numbers)
& v1_partfun1(k1_comseq_2(k5_numbers,A),k5_numbers,k2_numbers)
& v2_comseq_2(k1_comseq_2(k5_numbers,A)) ) ) ).
fof(cc1_comseq_2,axiom,
! [A] :
( m1_relset_1(A,k5_numbers,k2_numbers)
=> ( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& v2_comseq_2(A) )
=> ( ~ v1_xboole_0(A)
& v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& v1_partfun1(A,k5_numbers,k2_numbers)
& v1_comseq_2(A) ) ) ) ).
fof(cc2_comseq_2,axiom,
! [A] :
( m1_relset_1(A,k5_numbers,k2_numbers)
=> ( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& ~ v1_comseq_2(A) )
=> ( ~ v1_xboole_0(A)
& v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& v1_partfun1(A,k5_numbers,k2_numbers)
& ~ v2_comseq_2(A) ) ) ) ).
fof(t1_comseq_2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ~ ( A != k5_complex1
& B != k5_complex1
& k17_complex1(k11_complex1(k12_complex1(A),k12_complex1(B))) != k6_real_1(k17_complex1(k11_complex1(A,B)),k4_real_1(k17_complex1(A),k17_complex1(B))) ) ) ) ).
fof(t2_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ~ r1_xreal_0(C,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(D,B)
& r1_xreal_0(C,k17_complex1(k1_comseq_1(A,D))) ) ) ) ) ) ).
fof(d1_comseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k2_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k2_numbers) )
=> ( C = k1_comseq_2(A,B)
<=> ( k4_relset_1(A,k2_numbers,C) = k4_relset_1(A,k2_numbers,B)
& ! [D] :
( m1_subset_1(D,A)
=> ( r2_hidden(D,k4_relset_1(A,k2_numbers,C))
=> k1_funct_1(C,D) = k15_complex1(k4_finseq_4(A,k2_numbers,B,D)) ) ) ) ) ) ) ) ).
fof(d2_comseq_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k2_numbers)
& m2_relset_1(B,A,k2_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k2_numbers) )
=> ( C = k1_comseq_2(A,B)
<=> ( k4_relset_1(A,k2_numbers,C) = A
& ! [D] :
( m1_subset_1(D,A)
=> k1_funct_1(C,D) = k15_complex1(k8_funct_2(A,k2_numbers,B,D)) ) ) ) ) ) ) ).
fof(t3_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ( v1_comseq_1(A)
=> v1_comseq_1(k1_comseq_2(k5_numbers,A)) ) ) ).
fof(t4_comseq_2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> k1_comseq_2(k5_numbers,k4_comseq_1(k5_numbers,B,A)) = k4_comseq_1(k5_numbers,k1_comseq_2(k5_numbers,B),k15_complex1(A)) ) ) ).
fof(t5_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> k1_comseq_2(k5_numbers,k3_comseq_1(k5_numbers,A,B)) = k3_comseq_1(k5_numbers,k1_comseq_2(k5_numbers,A),k1_comseq_2(k5_numbers,B)) ) ) ).
fof(t6_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> k7_comseq_1(k1_comseq_2(k5_numbers,A)) = k1_comseq_2(k5_numbers,k7_comseq_1(A)) ) ).
fof(t7_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> k1_comseq_2(k5_numbers,k8_comseq_1(A,B)) = k8_comseq_1(k1_comseq_2(k5_numbers,A),k1_comseq_2(k5_numbers,B)) ) ) ).
fof(d3_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ( v1_comseq_2(A)
<=> ? [B] :
( m1_subset_1(B,k1_numbers)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(B,k17_complex1(k1_comseq_1(A,C))) ) ) ) ) ).
fof(t8_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ( v1_comseq_2(A)
<=> ? [B] :
( m1_subset_1(B,k1_numbers)
& ~ r1_xreal_0(B,np__0)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(B,k17_complex1(k1_comseq_1(A,C))) ) ) ) ) ).
fof(d4_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ( v2_comseq_2(A)
<=> ? [B] :
( m1_subset_1(B,k2_numbers)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ? [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
& r1_xreal_0(D,E)
& r1_xreal_0(C,k17_complex1(k11_complex1(k1_comseq_1(A,E),B))) ) ) ) ) ) ) ) ).
fof(d5_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ( v2_comseq_2(A)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ( B = k2_comseq_2(A)
<=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ? [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
& r1_xreal_0(D,E)
& r1_xreal_0(C,k17_complex1(k11_complex1(k1_comseq_1(A,E),B))) ) ) ) ) ) ) ) ) ).
fof(t9_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ( ? [B] :
( m1_subset_1(B,k2_numbers)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k1_comseq_1(A,C) = B ) )
=> v2_comseq_2(A) ) ) ).
fof(t10_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k1_comseq_1(A,C) = B )
=> k2_comseq_2(A) = B ) ) ) ).
fof(t11_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ( v2_comseq_2(A)
=> k2_seq_2(k9_comseq_1(k5_numbers,A)) = k17_complex1(k2_comseq_2(A)) ) ) ).
fof(t12_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ( v2_comseq_2(A)
=> k2_comseq_2(k1_comseq_2(k5_numbers,A)) = k15_complex1(k2_comseq_2(A)) ) ) ).
fof(t13_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_2(A)
& v2_comseq_2(B) )
=> v2_comseq_2(k2_comseq_1(k5_numbers,A,B)) ) ) ) ).
fof(t14_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_2(A)
& v2_comseq_2(B) )
=> k2_comseq_2(k2_comseq_1(k5_numbers,A,B)) = k8_complex1(k2_comseq_2(A),k2_comseq_2(B)) ) ) ) ).
fof(t15_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_2(A)
& v2_comseq_2(B) )
=> k2_seq_2(k9_comseq_1(k5_numbers,k2_comseq_1(k5_numbers,A,B))) = k17_complex1(k8_complex1(k2_comseq_2(A),k2_comseq_2(B))) ) ) ) ).
fof(t16_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_2(A)
& v2_comseq_2(B) )
=> k2_comseq_2(k1_comseq_2(k5_numbers,k2_comseq_1(k5_numbers,A,B))) = k8_complex1(k15_complex1(k2_comseq_2(A)),k15_complex1(k2_comseq_2(B))) ) ) ) ).
fof(t17_comseq_2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( v2_comseq_2(B)
=> v2_comseq_2(k4_comseq_1(k5_numbers,B,A)) ) ) ) ).
fof(t18_comseq_2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( v2_comseq_2(B)
=> k2_comseq_2(k4_comseq_1(k5_numbers,B,A)) = k9_complex1(A,k2_comseq_2(B)) ) ) ) ).
fof(t19_comseq_2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( v2_comseq_2(B)
=> k2_seq_2(k9_comseq_1(k5_numbers,k4_comseq_1(k5_numbers,B,A))) = k4_real_1(k17_complex1(A),k17_complex1(k2_comseq_2(B))) ) ) ) ).
fof(t20_comseq_2,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( v2_comseq_2(B)
=> k2_comseq_2(k1_comseq_2(k5_numbers,k4_comseq_1(k5_numbers,B,A))) = k9_complex1(k15_complex1(A),k15_complex1(k2_comseq_2(B))) ) ) ) ).
fof(t21_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ( v2_comseq_2(A)
=> v2_comseq_2(k5_comseq_1(k5_numbers,A)) ) ) ).
fof(t22_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ( v2_comseq_2(A)
=> k2_comseq_2(k5_comseq_1(k5_numbers,A)) = k10_complex1(k2_comseq_2(A)) ) ) ).
fof(t23_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ( v2_comseq_2(A)
=> k2_seq_2(k9_comseq_1(k5_numbers,k5_comseq_1(k5_numbers,A))) = k17_complex1(k2_comseq_2(A)) ) ) ).
fof(t24_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ( v2_comseq_2(A)
=> k2_comseq_2(k1_comseq_2(k5_numbers,k5_comseq_1(k5_numbers,A))) = k10_complex1(k15_complex1(k2_comseq_2(A))) ) ) ).
fof(t25_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_2(A)
& v2_comseq_2(B) )
=> v2_comseq_2(k6_comseq_1(k5_numbers,A,B)) ) ) ) ).
fof(t26_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_2(A)
& v2_comseq_2(B) )
=> k2_comseq_2(k6_comseq_1(k5_numbers,A,B)) = k11_complex1(k2_comseq_2(A),k2_comseq_2(B)) ) ) ) ).
fof(t27_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_2(A)
& v2_comseq_2(B) )
=> k2_seq_2(k9_comseq_1(k5_numbers,k6_comseq_1(k5_numbers,A,B))) = k17_complex1(k11_complex1(k2_comseq_2(A),k2_comseq_2(B))) ) ) ) ).
fof(t28_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_2(A)
& v2_comseq_2(B) )
=> k2_comseq_2(k1_comseq_2(k5_numbers,k6_comseq_1(k5_numbers,A,B))) = k11_complex1(k15_complex1(k2_comseq_2(A)),k15_complex1(k2_comseq_2(B))) ) ) ) ).
fof(t29_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& v2_comseq_2(A)
& m2_relset_1(A,k5_numbers,k2_numbers)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& v2_comseq_2(B)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> v2_comseq_2(k3_comseq_1(k5_numbers,A,B)) ) ) ) ).
fof(t30_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& v2_comseq_2(A)
& m2_relset_1(A,k5_numbers,k2_numbers)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& v2_comseq_2(B)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> k2_comseq_2(k3_comseq_1(k5_numbers,A,B)) = k9_complex1(k2_comseq_2(A),k2_comseq_2(B)) ) ) ) ).
fof(t31_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_2(A)
& v2_comseq_2(B) )
=> k2_seq_2(k9_comseq_1(k5_numbers,k3_comseq_1(k5_numbers,A,B))) = k4_real_1(k17_complex1(k2_comseq_2(A)),k17_complex1(k2_comseq_2(B))) ) ) ) ).
fof(t32_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_2(A)
& v2_comseq_2(B) )
=> k2_comseq_2(k1_comseq_2(k5_numbers,k3_comseq_1(k5_numbers,A,B))) = k9_complex1(k15_complex1(k2_comseq_2(A)),k15_complex1(k2_comseq_2(B))) ) ) ) ).
fof(t33_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ~ ( v2_comseq_2(A)
& k2_comseq_2(A) != k5_complex1
& ! [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(k17_complex1(k1_comseq_1(A,C)),k6_real_1(k17_complex1(k2_comseq_2(A)),np__2)) ) ) ) ) ).
fof(t34_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_2(A)
& v1_comseq_1(A) )
=> ( k2_comseq_2(A) = k5_complex1
| v2_comseq_2(k7_comseq_1(A)) ) ) ) ).
fof(t35_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_2(A)
& v1_comseq_1(A) )
=> ( k2_comseq_2(A) = k5_complex1
| k2_comseq_2(k7_comseq_1(A)) = k12_complex1(k2_comseq_2(A)) ) ) ) ).
fof(t36_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_2(A)
& v1_comseq_1(A) )
=> ( k2_comseq_2(A) = k5_complex1
| k2_seq_2(k9_comseq_1(k5_numbers,k7_comseq_1(A))) = k2_real_1(k17_complex1(k2_comseq_2(A))) ) ) ) ).
fof(t37_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_2(A)
& v1_comseq_1(A) )
=> ( k2_comseq_2(A) = k5_complex1
| k2_comseq_2(k1_comseq_2(k5_numbers,k7_comseq_1(A))) = k12_complex1(k15_complex1(k2_comseq_2(A))) ) ) ) ).
fof(t38_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_2(A)
& v2_comseq_2(B)
& v1_comseq_1(B) )
=> ( k2_comseq_2(B) = k5_complex1
| v2_comseq_2(k8_comseq_1(A,B)) ) ) ) ) ).
fof(t39_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_2(A)
& v2_comseq_2(B)
& v1_comseq_1(B) )
=> ( k2_comseq_2(B) = k5_complex1
| k2_comseq_2(k8_comseq_1(A,B)) = k13_complex1(k2_comseq_2(A),k2_comseq_2(B)) ) ) ) ) ).
fof(t40_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_2(A)
& v2_comseq_2(B)
& v1_comseq_1(B) )
=> ( k2_comseq_2(B) = k5_complex1
| k2_seq_2(k9_comseq_1(k5_numbers,k8_comseq_1(A,B))) = k6_real_1(k17_complex1(k2_comseq_2(A)),k17_complex1(k2_comseq_2(B))) ) ) ) ) ).
fof(t41_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_2(A)
& v2_comseq_2(B)
& v1_comseq_1(B) )
=> ( k2_comseq_2(B) = k5_complex1
| k2_comseq_2(k1_comseq_2(k5_numbers,k8_comseq_1(A,B))) = k13_complex1(k15_complex1(k2_comseq_2(A)),k15_complex1(k2_comseq_2(B))) ) ) ) ) ).
fof(t42_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_2(A)
& v1_comseq_2(B)
& k2_comseq_2(A) = k5_complex1 )
=> v2_comseq_2(k3_comseq_1(k5_numbers,A,B)) ) ) ) ).
fof(t43_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_2(A)
& v1_comseq_2(B)
& k2_comseq_2(A) = k5_complex1 )
=> k2_comseq_2(k3_comseq_1(k5_numbers,A,B)) = k5_complex1 ) ) ) ).
fof(t44_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_2(A)
& v1_comseq_2(B)
& k2_comseq_2(A) = k5_complex1 )
=> k2_seq_2(k9_comseq_1(k5_numbers,k3_comseq_1(k5_numbers,A,B))) = np__0 ) ) ) ).
fof(t45_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m2_relset_1(A,k5_numbers,k2_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers) )
=> ( ( v2_comseq_2(A)
& v1_comseq_2(B)
& k2_comseq_2(A) = k5_complex1 )
=> k2_comseq_2(k1_comseq_2(k5_numbers,k3_comseq_1(k5_numbers,A,B))) = k5_complex1 ) ) ) ).
fof(dt_k1_comseq_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& m1_relset_1(B,A,k2_numbers) )
=> ( v1_funct_1(k1_comseq_2(A,B))
& m2_relset_1(k1_comseq_2(A,B),A,k2_numbers) ) ) ).
fof(dt_k2_comseq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k2_numbers)
& m1_relset_1(A,k5_numbers,k2_numbers) )
=> m1_subset_1(k2_comseq_2(A),k2_numbers) ) ).
%------------------------------------------------------------------------------