SET007 Axioms: SET007+86.ax
%------------------------------------------------------------------------------
% File : SET007+86 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Convergent Real Sequences
% Version : [Urb08] axioms.
% English : Upper and Lower Bound of Sets of Real Numbers
% 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 : seq_4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 79 ( 13 unt; 0 def)
% Number of atoms : 529 ( 50 equ)
% Maximal formula atoms : 20 ( 6 avg)
% Number of connectives : 522 ( 72 ~; 18 |; 198 &)
% ( 7 <=>; 227 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 30 ( 28 usr; 1 prp; 0-3 aty)
% Number of functors : 32 ( 32 usr; 4 con; 0-4 aty)
% Number of variables : 178 ( 165 !; 13 ?)
% 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(cc1_seq_4,axiom,
! [A] :
( m1_relset_1(A,k5_numbers,k1_numbers)
=> ( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v5_seqm_3(A) )
=> ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& v1_seq_1(A)
& v4_seq_2(A) ) ) ) ).
fof(t1_seq_4,axiom,
$true ).
fof(t2_seq_4,axiom,
$true ).
fof(t3_seq_4,axiom,
$true ).
fof(t4_seq_4,axiom,
$true ).
fof(t5_seq_4,axiom,
$true ).
fof(t6_seq_4,axiom,
$true ).
fof(t7_seq_4,axiom,
$true ).
fof(t8_seq_4,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
=> ~ ( ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ~ ( r2_hidden(C,A)
& r2_hidden(D,B)
& r1_xreal_0(D,C) ) ) )
& ! [C] :
( v1_xreal_0(C)
=> ? [D] :
( v1_xreal_0(D)
& ? [E] :
( v1_xreal_0(E)
& r2_hidden(D,A)
& r2_hidden(E,B)
& ~ ( r1_xreal_0(D,C)
& r1_xreal_0(C,E) ) ) ) ) ) ) ) ).
fof(t9_seq_4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
=> ( ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(C,B)
=> r2_hidden(k2_xcmplx_0(C,A),B) ) )
=> ( r1_xreal_0(A,np__0)
| ! [C] :
( v1_xreal_0(C)
=> ~ r2_hidden(C,B) )
| ! [C] :
( v1_xreal_0(C)
=> ? [D] :
( v1_xreal_0(D)
& r2_hidden(D,B)
& ~ r1_xreal_0(D,C) ) ) ) ) ) ) ).
fof(t10_seq_4,axiom,
! [A] :
( v1_xreal_0(A)
=> ~ ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_xreal_0(B,A) ) ) ).
fof(d1_seq_4,axiom,
! [A] :
( v2_membered(A)
=> ( v1_seq_4(A)
<=> ? [B] :
( v1_xreal_0(B)
& ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(C,A)
=> r1_xreal_0(C,B) ) ) ) ) ) ).
fof(d2_seq_4,axiom,
! [A] :
( v2_membered(A)
=> ( v2_seq_4(A)
<=> ? [B] :
( v1_xreal_0(B)
& ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(C,A)
=> r1_xreal_0(B,C) ) ) ) ) ) ).
fof(d3_seq_4,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( v3_seq_4(A)
<=> ( v2_seq_4(A)
& v1_seq_4(A) ) ) ) ).
fof(t11_seq_4,axiom,
$true ).
fof(t12_seq_4,axiom,
$true ).
fof(t13_seq_4,axiom,
$true ).
fof(t14_seq_4,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( v3_seq_4(A)
<=> ? [B] :
( v1_xreal_0(B)
& ~ r1_xreal_0(B,np__0)
& ! [C] :
( v1_xreal_0(C)
=> ~ ( r2_hidden(C,A)
& r1_xreal_0(B,k18_complex1(C)) ) ) ) ) ) ).
fof(t15_seq_4,axiom,
! [A] :
( v1_xreal_0(A)
=> v3_seq_4(k1_seq_4(A)) ) ).
fof(t16_seq_4,axiom,
! [A] :
( v2_membered(A)
=> ~ ( ~ v1_xboole_0(A)
& v1_seq_4(A)
& ! [B] :
( v1_xreal_0(B)
=> ~ ( ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(C,A)
=> r1_xreal_0(C,B) ) )
& ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( v1_xreal_0(D)
=> ~ ( r2_hidden(D,A)
& ~ r1_xreal_0(D,k6_xcmplx_0(B,C)) ) ) ) ) ) ) ) ) ).
fof(t17_seq_4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v2_membered(C)
=> ( ( ! [D] :
( v1_xreal_0(D)
=> ( r2_hidden(D,C)
=> r1_xreal_0(D,A) ) )
& ! [D] :
( v1_xreal_0(D)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( v1_xreal_0(E)
=> ~ ( r2_hidden(E,C)
& ~ r1_xreal_0(E,k6_xcmplx_0(A,D)) ) ) ) )
& ! [D] :
( v1_xreal_0(D)
=> ( r2_hidden(D,C)
=> r1_xreal_0(D,B) ) )
& ! [D] :
( v1_xreal_0(D)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( v1_xreal_0(E)
=> ~ ( r2_hidden(E,C)
& ~ r1_xreal_0(E,k6_xcmplx_0(B,D)) ) ) ) ) )
=> A = B ) ) ) ) ).
fof(t18_seq_4,axiom,
! [A] :
( v2_membered(A)
=> ~ ( ~ v1_xboole_0(A)
& v2_seq_4(A)
& ! [B] :
( v1_xreal_0(B)
=> ~ ( ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(C,A)
=> r1_xreal_0(B,C) ) )
& ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( v1_xreal_0(D)
=> ~ ( r2_hidden(D,A)
& ~ r1_xreal_0(k2_xcmplx_0(B,C),D) ) ) ) ) ) ) ) ) ).
fof(t19_seq_4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v2_membered(C)
=> ( ( ! [D] :
( v1_xreal_0(D)
=> ( r2_hidden(D,C)
=> r1_xreal_0(A,D) ) )
& ! [D] :
( v1_xreal_0(D)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( v1_xreal_0(E)
=> ~ ( r2_hidden(E,C)
& ~ r1_xreal_0(k2_xcmplx_0(A,D),E) ) ) ) )
& ! [D] :
( v1_xreal_0(D)
=> ( r2_hidden(D,C)
=> r1_xreal_0(B,D) ) )
& ! [D] :
( v1_xreal_0(D)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( v1_xreal_0(E)
=> ~ ( r2_hidden(E,C)
& ~ r1_xreal_0(k2_xcmplx_0(B,D),E) ) ) ) ) )
=> A = B ) ) ) ) ).
fof(d4_seq_4,axiom,
! [A] :
( v2_membered(A)
=> ( v1_seq_4(A)
=> ( v1_xboole_0(A)
| ! [B] :
( v1_xreal_0(B)
=> ( B = k2_seq_4(A)
<=> ( ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(C,A)
=> r1_xreal_0(C,B) ) )
& ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( v1_xreal_0(D)
=> ~ ( r2_hidden(D,A)
& ~ r1_xreal_0(D,k6_xcmplx_0(B,C)) ) ) ) ) ) ) ) ) ) ) ).
fof(d5_seq_4,axiom,
! [A] :
( v2_membered(A)
=> ( v2_seq_4(A)
=> ( v1_xboole_0(A)
| ! [B] :
( v1_xreal_0(B)
=> ( B = k3_seq_4(A)
<=> ( ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(C,A)
=> r1_xreal_0(B,C) ) )
& ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( v1_xreal_0(D)
=> ~ ( r2_hidden(D,A)
& ~ r1_xreal_0(k2_xcmplx_0(B,C),D) ) ) ) ) ) ) ) ) ) ) ).
fof(t20_seq_4,axiom,
$true ).
fof(t21_seq_4,axiom,
$true ).
fof(t22_seq_4,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k5_seq_4(k1_seq_4(A)) = A
& k4_seq_4(k1_seq_4(A)) = A ) ) ).
fof(t23_seq_4,axiom,
! [A] :
( v1_xreal_0(A)
=> k5_seq_4(k1_seq_4(A)) = k4_seq_4(k1_seq_4(A)) ) ).
fof(t24_seq_4,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( v3_seq_4(A)
=> ( v1_xboole_0(A)
| r1_xreal_0(k5_seq_4(A),k4_seq_4(A)) ) ) ) ).
fof(t25_seq_4,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( v3_seq_4(A)
=> ( v1_xboole_0(A)
| ( ~ ( ? [B] :
( v1_xreal_0(B)
& ? [C] :
( v1_xreal_0(C)
& r2_hidden(B,A)
& r2_hidden(C,A)
& C != B ) )
& r1_xreal_0(k4_seq_4(A),k5_seq_4(A)) )
& ~ ( ~ r1_xreal_0(k4_seq_4(A),k5_seq_4(A))
& ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( r2_hidden(B,A)
& r2_hidden(C,A)
& C != B ) ) ) ) ) ) ) ) ).
fof(t26_seq_4,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v4_seq_2(A)
=> v4_seq_2(k22_seq_1(A)) ) ) ).
fof(t27_seq_4,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v4_seq_2(A)
=> k2_seq_2(k22_seq_1(A)) = k18_complex1(k2_seq_2(A)) ) ) ).
fof(t28_seq_4,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(k22_seq_1(A))
& k2_seq_2(k22_seq_1(A)) = np__0 )
=> ( v4_seq_2(A)
& k2_seq_2(A) = np__0 ) ) ) ).
fof(t29_seq_4,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) )
=> ( ( m1_seqm_3(A,B)
& v4_seq_2(B) )
=> v4_seq_2(A) ) ) ) ).
fof(t30_seq_4,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) )
=> ( ( m1_seqm_3(A,B)
& v4_seq_2(B) )
=> k2_seq_2(A) = k2_seq_2(B) ) ) ) ).
fof(t31_seq_4,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_seq_2(A)
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& r1_xreal_0(C,D)
& k8_funct_2(k5_numbers,k1_numbers,B,D) != k8_funct_2(k5_numbers,k1_numbers,A,D) ) )
| v4_seq_2(B) ) ) ) ) ).
fof(t32_seq_4,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_seq_2(A)
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& r1_xreal_0(C,D)
& k8_funct_2(k5_numbers,k1_numbers,B,D) != k8_funct_2(k5_numbers,k1_numbers,A,D) ) )
| k2_seq_2(A) = k2_seq_2(B) ) ) ) ) ).
fof(t33_seq_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) )
=> ( v4_seq_2(B)
=> ( v4_seq_2(k1_seqm_3(B,A))
& k2_seq_2(k1_seqm_3(B,A)) = k2_seq_2(B) ) ) ) ) ).
fof(t34_seq_4,axiom,
$true ).
fof(t35_seq_4,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_seq_2(A)
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> A != k1_seqm_3(B,C) )
| v4_seq_2(B) ) ) ) ) ).
fof(t36_seq_4,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_seq_2(A)
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> A != k1_seqm_3(B,C) )
| k2_seq_2(B) = k2_seq_2(A) ) ) ) ) ).
fof(t37_seq_4,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ~ ( v4_seq_2(A)
& k2_seq_2(A) != np__0
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ v2_relat_1(k1_seqm_3(A,B)) ) ) ) ).
fof(t38_seq_4,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ~ ( v4_seq_2(A)
& k2_seq_2(A) != np__0
& ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ~ ( m1_seqm_3(B,A)
& v2_relat_1(B) ) ) ) ) ).
fof(t39_seq_4,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v5_seqm_3(A)
=> v4_seq_2(A) ) ) ).
fof(t40_seq_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) )
=> ( ( ( v5_seqm_3(B)
& r2_hidden(A,k2_relat_1(B)) )
| ( v5_seqm_3(B)
& ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& k8_funct_2(k5_numbers,k1_numbers,B,C) = A ) ) )
=> k2_seq_2(B) = A ) ) ) ).
fof(t41_seq_4,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v5_seqm_3(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_2(A) = k8_funct_2(k5_numbers,k1_numbers,A,B) ) ) ) ).
fof(t42_seq_4,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v4_seq_2(A)
=> ( k2_seq_2(A) = np__0
| ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ( m1_seqm_3(B,A)
& v2_relat_1(B) )
=> k2_seq_2(k18_seq_1(B)) = k2_real_1(k2_seq_2(A)) ) ) ) ) ) ).
fof(t43_seq_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)
=> k8_funct_2(k5_numbers,k1_numbers,B,C) = k7_xcmplx_0(np__1,k2_xcmplx_0(C,A)) )
=> ( r1_xreal_0(A,np__0)
| v4_seq_2(B) ) ) ) ) ).
fof(t44_seq_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)
=> k8_funct_2(k5_numbers,k1_numbers,B,C) = k7_xcmplx_0(np__1,k2_xcmplx_0(C,A)) )
=> ( r1_xreal_0(A,np__0)
| k2_seq_2(B) = np__0 ) ) ) ) ).
fof(t45_seq_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)
=> k8_funct_2(k5_numbers,k1_numbers,A,B) = k6_real_1(np__1,k1_nat_1(B,np__1)) )
=> ( v4_seq_2(A)
& k2_seq_2(A) = np__0 ) ) ) ).
fof(t46_seq_4,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)
=> k8_funct_2(k5_numbers,k1_numbers,C,D) = k7_xcmplx_0(B,k2_xcmplx_0(D,A)) )
=> ( r1_xreal_0(A,np__0)
| ( v4_seq_2(C)
& k2_seq_2(C) = np__0 ) ) ) ) ) ) ).
fof(t47_seq_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)
=> k8_funct_2(k5_numbers,k1_numbers,B,C) = k7_xcmplx_0(np__1,k2_xcmplx_0(k2_nat_1(C,C),A)) )
=> ( r1_xreal_0(A,np__0)
| v4_seq_2(B) ) ) ) ) ).
fof(t48_seq_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)
=> k8_funct_2(k5_numbers,k1_numbers,B,C) = k7_xcmplx_0(np__1,k2_xcmplx_0(k2_nat_1(C,C),A)) )
=> ( r1_xreal_0(A,np__0)
| k2_seq_2(B) = np__0 ) ) ) ) ).
fof(t49_seq_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)
=> k8_funct_2(k5_numbers,k1_numbers,A,B) = k6_real_1(np__1,k1_nat_1(k2_nat_1(B,B),np__1)) )
=> ( v4_seq_2(A)
& k2_seq_2(A) = np__0 ) ) ) ).
fof(t50_seq_4,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)
=> k8_funct_2(k5_numbers,k1_numbers,C,D) = k7_xcmplx_0(B,k2_xcmplx_0(k2_nat_1(D,D),A)) )
=> ( r1_xreal_0(A,np__0)
| ( v4_seq_2(C)
& k2_seq_2(C) = np__0 ) ) ) ) ) ) ).
fof(t51_seq_4,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ( v3_seqm_3(A)
& v1_seq_2(A) )
=> v4_seq_2(A) ) ) ).
fof(t52_seq_4,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ( v4_seqm_3(A)
& v2_seq_2(A) )
=> v4_seq_2(A) ) ) ).
fof(t53_seq_4,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ( v6_seqm_3(A)
& v3_seq_2(A) )
=> v4_seq_2(A) ) ) ).
fof(t54_seq_4,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ( v1_seq_2(A)
& v3_seqm_3(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_xreal_0(k8_funct_2(k5_numbers,k1_numbers,A,B),k2_seq_2(A)) ) ) ) ).
fof(t55_seq_4,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ( v2_seq_2(A)
& v4_seqm_3(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_xreal_0(k2_seq_2(A),k8_funct_2(k5_numbers,k1_numbers,A,B)) ) ) ) ).
fof(t56_seq_4,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)
& v1_seqm_3(B)
& v7_seqm_3(B)
& m2_relset_1(B,k5_numbers,k1_numbers)
& v6_seqm_3(k3_seqm_3(B,A)) ) ) ).
fof(t57_seq_4,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ~ ( v3_seq_2(A)
& ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ~ ( m1_seqm_3(B,A)
& v4_seq_2(B) ) ) ) ) ).
fof(t58_seq_4,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v4_seq_2(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,A,D),k8_funct_2(k5_numbers,k1_numbers,A,C)))) ) ) ) ) ) ) ).
fof(t59_seq_4,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) )
=> ( ( v5_seqm_3(A)
& v4_seq_2(B) )
=> ( k2_seq_2(k9_seq_1(A,B)) = k3_real_1(k8_funct_2(k5_numbers,k1_numbers,A,np__0),k2_seq_2(B))
& k2_seq_2(k10_seq_1(A,B)) = k5_real_1(k8_funct_2(k5_numbers,k1_numbers,A,np__0),k2_seq_2(B))
& k2_seq_2(k10_seq_1(B,A)) = k5_real_1(k2_seq_2(B),k8_funct_2(k5_numbers,k1_numbers,A,np__0))
& k2_seq_2(k11_seq_1(A,B)) = k4_real_1(k8_funct_2(k5_numbers,k1_numbers,A,np__0),k2_seq_2(B)) ) ) ) ) ).
fof(t60_seq_4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v2_membered(A) )
=> ! [B] :
( v1_xreal_0(B)
=> ( ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(C,A)
=> r1_xreal_0(B,C) ) )
=> r1_xreal_0(B,k3_seq_4(A)) ) ) ) ).
fof(t61_seq_4,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v2_membered(B) )
=> ( ( ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(C,B)
=> r1_xreal_0(A,C) ) )
& ! [C] :
( v1_xreal_0(C)
=> ( ! [D] :
( v1_xreal_0(D)
=> ( r2_hidden(D,B)
=> r1_xreal_0(C,D) ) )
=> r1_xreal_0(C,A) ) ) )
=> A = k3_seq_4(B) ) ) ) ).
fof(t62_seq_4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v2_membered(A) )
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ! [D] :
( v1_xreal_0(D)
=> ( r2_hidden(D,A)
=> r1_xreal_0(D,C) ) )
=> r1_xreal_0(k2_seq_4(A),C) ) ) ) ) ).
fof(t63_seq_4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v2_membered(A) )
=> ! [B] :
( v1_xreal_0(B)
=> ( ( ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(C,A)
=> r1_xreal_0(C,B) ) )
& ! [C] :
( v1_xreal_0(C)
=> ( ! [D] :
( v1_xreal_0(D)
=> ( r2_hidden(D,A)
=> r1_xreal_0(D,C) ) )
=> r1_xreal_0(B,C) ) ) )
=> B = k2_seq_4(A) ) ) ) ).
fof(t64_seq_4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v2_membered(A) )
=> ! [B] :
( v2_membered(B)
=> ( ( r1_tarski(A,B)
& v2_seq_4(B) )
=> r1_xreal_0(k3_seq_4(B),k3_seq_4(A)) ) ) ) ).
fof(t65_seq_4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v2_membered(A) )
=> ! [B] :
( v2_membered(B)
=> ( ( r1_tarski(A,B)
& v1_seq_4(B) )
=> r1_xreal_0(k2_seq_4(A),k2_seq_4(B)) ) ) ) ).
fof(dt_k1_seq_4,axiom,
! [A] :
( v1_xreal_0(A)
=> m1_subset_1(k1_seq_4(A),k1_zfmisc_1(k1_numbers)) ) ).
fof(redefinition_k1_seq_4,axiom,
! [A] :
( v1_xreal_0(A)
=> k1_seq_4(A) = k1_tarski(A) ) ).
fof(dt_k2_seq_4,axiom,
! [A] :
( v2_membered(A)
=> v1_xreal_0(k2_seq_4(A)) ) ).
fof(dt_k3_seq_4,axiom,
! [A] :
( v2_membered(A)
=> v1_xreal_0(k3_seq_4(A)) ) ).
fof(dt_k4_seq_4,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> m1_subset_1(k4_seq_4(A),k1_numbers) ) ).
fof(redefinition_k4_seq_4,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> k4_seq_4(A) = k2_seq_4(A) ) ).
fof(dt_k5_seq_4,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> m1_subset_1(k5_seq_4(A),k1_numbers) ) ).
fof(redefinition_k5_seq_4,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> k5_seq_4(A) = k3_seq_4(A) ) ).
%------------------------------------------------------------------------------