SET007 Axioms: SET007+853.ax
%------------------------------------------------------------------------------
% File : SET007+853 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Nagata-Smirnov Theorem. Part II
% 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 : nagata_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 29 ( 1 unt; 0 def)
% Number of atoms : 353 ( 30 equ)
% Maximal formula atoms : 37 ( 12 avg)
% Number of connectives : 365 ( 41 ~; 2 |; 178 &)
% ( 6 <=>; 138 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 11 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 36 ( 35 usr; 0 prp; 1-3 aty)
% Number of functors : 45 ( 45 usr; 6 con; 0-6 aty)
% Number of variables : 126 ( 117 !; 9 ?)
% 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_nagata_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> A != k4_real_1(k3_prepower(np__2,B),k1_nat_1(k2_nat_1(np__2,C),np__1)) ) ) ) ) ).
fof(d1_nagata_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers)
& m2_relset_1(A,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers) )
=> ( A = k1_nagata_2
<=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k8_funct_2(k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers,A,k1_domain_1(k5_numbers,k5_numbers,B,C)) = k5_real_1(k4_real_1(k3_prepower(np__2,B),k1_nat_1(k2_nat_1(np__2,C),np__1)),np__1) ) ) ) ) ).
fof(t2_nagata_2,axiom,
v3_funct_2(k1_nagata_2,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers) ).
fof(d2_nagata_2,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
& m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,k1_numbers)
& m2_relset_1(D,A,k1_numbers) )
=> ( D = k2_nagata_2(A,B,C)
<=> ! [E] :
( m1_subset_1(E,A)
=> k2_seq_1(A,k1_numbers,D,E) = k1_metric_1(A,A,B,C,E) ) ) ) ) ) ).
fof(t3_nagata_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B))))
=> ( v3_pre_topc(C,k6_borsuk_1(A,B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(B)))
=> ( ( F = k10_relset_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),u1_struct_0(A),k9_funct_3(u1_struct_0(A),u1_struct_0(B)),k3_xboole_0(C,k2_zfmisc_1(u1_struct_0(A),k1_struct_0(B,E))))
=> v3_pre_topc(F,A) )
& ( G = k10_relset_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),u1_struct_0(B),k10_funct_3(u1_struct_0(A),u1_struct_0(B)),k3_xboole_0(C,k2_zfmisc_1(k1_struct_0(A,D),u1_struct_0(B))))
=> v3_pre_topc(G,B) ) ) ) ) ) ) ) ) ) ) ).
fof(t4_nagata_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
& m2_relset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) )
=> ( ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)
& m2_relset_1(C,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) )
=> ( B = C
=> v9_pscomp_1(C,k6_borsuk_1(A,A)) ) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> v9_pscomp_1(k2_nagata_2(u1_struct_0(A),B,C),A) ) ) ) ) ).
fof(d3_nagata_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
& m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,k1_numbers)
& m2_relset_1(D,A,k1_numbers) )
=> ( D = k3_nagata_2(A,B,C)
<=> ! [E] :
( m1_subset_1(E,A)
=> k8_funct_2(A,k1_numbers,D,E) = k4_pscomp_1(k10_relset_1(A,k1_numbers,k2_nagata_2(A,B,E),C)) ) ) ) ) ) ) ).
fof(t5_nagata_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
& m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
=> ( r1_nagata_1(A,B)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> ! [D] :
( m1_subset_1(D,A)
=> r1_xreal_0(np__0,k8_funct_2(A,k1_numbers,k3_nagata_2(A,B,C),D)) ) ) ) ) ) ).
fof(t6_nagata_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
& m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
=> ( r1_nagata_1(A,B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,A)
=> ( r2_hidden(D,C)
=> k8_funct_2(A,k1_numbers,k3_nagata_2(A,B,C),D) = np__0 ) ) ) ) ) ) ).
fof(t7_nagata_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
& m2_relset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) )
=> ( r1_nagata_1(u1_struct_0(A),B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
=> r1_xreal_0(k18_complex1(k5_real_1(k8_funct_2(u1_struct_0(A),k1_numbers,k3_nagata_2(u1_struct_0(A),B,E),C),k8_funct_2(u1_struct_0(A),k1_numbers,k3_nagata_2(u1_struct_0(A),B,E),D))),k1_metric_1(u1_struct_0(A),u1_struct_0(A),B,C,D)) ) ) ) ) ) ) ).
fof(t8_nagata_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
& m2_relset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) )
=> ( ( r1_nagata_1(u1_struct_0(A),B)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> v9_pscomp_1(k2_nagata_2(u1_struct_0(A),B,C),A) ) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> v9_pscomp_1(k3_nagata_2(u1_struct_0(A),B,C),A) ) ) ) ) ).
fof(t9_nagata_2,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
& m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
=> ( r1_pcomps_1(A,B)
=> r1_nagata_1(A,B) ) ) ).
fof(t11_nagata_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_seqfunc(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
=> ( ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
& m2_relset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
& k1_seqfunc(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,B,C) = D
& r1_nagata_1(u1_struct_0(A),D) ) )
& ! [C] :
( m1_subset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))
=> v1_series_1(k10_seqfunc(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),B,C)) ) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
& m2_relset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) )
=> ( ! [D] :
( m1_subset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))
=> k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,C,D) = k2_series_1(k10_seqfunc(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),B,D)) )
=> r1_nagata_1(u1_struct_0(A),C) ) ) ) ) ) ).
fof(t12_nagata_2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [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)
=> ( r1_xreal_0(D,B)
=> r1_xreal_0(k8_funct_2(k5_numbers,k1_numbers,C,D),A) ) )
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(D,B)
=> r1_xreal_0(k8_funct_2(k5_numbers,k1_numbers,k1_series_1(C),D),k4_real_1(A,k1_nat_1(D,np__1))) ) ) ) ) ) ) ).
fof(t13_nagata_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(k18_complex1(k8_funct_2(k5_numbers,k1_numbers,k1_series_1(A),B)),k8_funct_2(k5_numbers,k1_numbers,k1_series_1(k22_seq_1(A)),B)) ) ) ).
fof(t14_nagata_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_seqfunc(B,u1_struct_0(A),k1_numbers)
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),k1_numbers)
& m2_relset_1(D,u1_struct_0(A),k1_numbers)
& k1_seqfunc(u1_struct_0(A),k1_numbers,B,C) = D
& v9_pscomp_1(D,A)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> r1_xreal_0(np__0,k8_funct_2(u1_struct_0(A),k1_numbers,D,E)) ) ) )
=> ( ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ~ ( v1_series_1(C)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> r1_xreal_0(k8_funct_2(k5_numbers,k1_numbers,k10_seqfunc(u1_struct_0(A),B,E),D),k8_funct_2(k5_numbers,k1_numbers,C,D)) ) ) ) )
| ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),k1_numbers)
& m2_relset_1(C,u1_struct_0(A),k1_numbers) )
=> ( ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),k1_numbers,C,D) = k2_series_1(k10_seqfunc(u1_struct_0(A),B,D)) )
=> v9_pscomp_1(C,A) ) ) ) ) ) ) ).
fof(t15_nagata_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ? [E] :
( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
& m2_relset_1(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
& k1_seqfunc(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,C,D) = E
& r1_nagata_1(u1_struct_0(A),E)
& ! [F] :
( m1_subset_1(F,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))
=> r1_xreal_0(k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,E,F),B) )
& ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)
& m2_relset_1(F,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) )
=> ( E = F
=> v9_pscomp_1(F,k6_borsuk_1(A,A)) ) ) ) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
& m2_relset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) )
=> ( ! [E] :
( m1_subset_1(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))
=> k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,D,E) = k2_series_1(k11_seq_1(k2_prepower(k6_real_1(np__1,np__2)),k10_seqfunc(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),C,E))) )
=> ( r1_nagata_1(u1_struct_0(A),D)
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)
& m2_relset_1(E,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) )
=> ( D = E
=> v9_pscomp_1(E,k6_borsuk_1(A,A)) ) ) ) ) ) ) ) ) ) ).
fof(t16_nagata_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
& m2_relset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) )
=> ( ( r1_nagata_1(u1_struct_0(A),B)
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)
& m2_relset_1(C,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) )
=> ( B = C
=> v9_pscomp_1(C,k6_borsuk_1(A,A)) ) ) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,k6_pre_topc(A,C))
=> k8_funct_2(u1_struct_0(A),k1_numbers,k3_nagata_2(u1_struct_0(A),B,C),D) = np__0 ) ) ) ) ) ) ).
fof(t17_nagata_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v1_urysohn1(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
=> ( ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ? [E] :
( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
& m2_relset_1(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
& k1_seqfunc(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,C,D) = E
& r1_nagata_1(u1_struct_0(A),E)
& ! [F] :
( m1_subset_1(F,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))
=> r1_xreal_0(k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,E,F),B) )
& ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)
& m2_relset_1(F,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) )
=> ( E = F
=> v9_pscomp_1(F,k6_borsuk_1(A,A)) ) ) ) )
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( ~ r2_hidden(D,E)
& v4_pre_topc(E,A)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ? [G] :
( v1_funct_1(G)
& v1_funct_2(G,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
& m2_relset_1(G,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
& k1_seqfunc(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,C,F) = G
& r1_xreal_0(k8_funct_2(u1_struct_0(A),k1_numbers,k3_nagata_2(u1_struct_0(A),G,E),D),np__0) ) ) ) ) ) )
=> ( ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
& m2_relset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
& r1_pcomps_1(u1_struct_0(A),D)
& ! [E] :
( m1_subset_1(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))
=> k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,D,E) = k2_series_1(k11_seq_1(k2_prepower(k6_real_1(np__1,np__2)),k10_seqfunc(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),C,E))) ) )
& v3_pcomps_1(A) ) ) ) ) ) ) ).
fof(t18_nagata_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m2_relset_1(D,k2_zfmisc_1(A,A),A) )
=> ~ ( v2_funct_1(B)
& v2_funct_1(C)
& r1_tarski(k5_relset_1(k5_numbers,A,C),k5_relset_1(k5_numbers,A,B))
& v1_binop_1(D,A)
& v2_binop_1(D,A)
& ( v1_setwiseo(D,A)
| ( r1_xreal_0(np__1,k3_finseq_1(C))
& ~ r1_xreal_0(k3_finseq_1(B),k3_finseq_1(C)) ) )
& ! [E] :
( m2_finseq_1(E,A)
=> ~ ( v2_funct_1(E)
& k5_relset_1(k5_numbers,A,E) = k6_subset_1(A,k5_relset_1(k5_numbers,A,B),k5_relset_1(k5_numbers,A,C))
& k2_finsop_1(A,B,D) = k2_binop_1(A,A,A,D,k2_finsop_1(A,C,D),k2_finsop_1(A,E,D)) ) ) ) ) ) ) ) ).
fof(t19_nagata_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ( v4_compts_1(A)
& v1_urysohn1(A)
& ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
& m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
& v6_nagata_1(B,A) ) )
<=> v3_pcomps_1(A) ) ) ).
fof(t20_nagata_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v3_pcomps_1(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( r1_pre_topc(A,B)
& v1_tops_2(B,A)
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
& m2_relset_1(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) )
=> ~ ( v1_tops_2(k2_nagata_1(A,C),A)
& r1_pre_topc(A,k2_nagata_1(A,C))
& r1_setfam_1(k2_nagata_1(A,C),B)
& v2_nagata_1(C,A) ) ) ) ) ) ) ).
fof(t21_nagata_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ~ ( v3_pcomps_1(A)
& ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
& m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) )
=> ~ v5_nagata_1(B,A) ) ) ) ).
fof(t22_nagata_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ( v4_compts_1(A)
& v1_urysohn1(A)
& ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
& m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
& v5_nagata_1(B,A) ) )
<=> v3_pcomps_1(A) ) ) ).
fof(dt_k1_nagata_2,axiom,
( v1_funct_1(k1_nagata_2)
& v1_funct_2(k1_nagata_2,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers)
& m2_relset_1(k1_nagata_2,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers) ) ).
fof(dt_k2_nagata_2,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
& m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers)
& m1_subset_1(C,A) )
=> ( v1_funct_1(k2_nagata_2(A,B,C))
& v1_funct_2(k2_nagata_2(A,B,C),A,k1_numbers)
& m2_relset_1(k2_nagata_2(A,B,C),A,k1_numbers) ) ) ).
fof(dt_k3_nagata_2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
& m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers)
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> ( v1_funct_1(k3_nagata_2(A,B,C))
& v1_funct_2(k3_nagata_2(A,B,C),A,k1_numbers)
& m2_relset_1(k3_nagata_2(A,B,C),A,k1_numbers) ) ) ).
fof(t10_nagata_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
& m2_relset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) )
=> ( ( r1_pcomps_1(u1_struct_0(A),B)
& ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> k6_pre_topc(A,C) = a_3_0_nagata_2(A,B,C) ) )
=> v3_pcomps_1(A) ) ) ) ).
fof(fraenkel_a_3_0_nagata_2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(u1_struct_0(B),u1_struct_0(B)),k1_numbers)
& m2_relset_1(C,k2_zfmisc_1(u1_struct_0(B),u1_struct_0(B)),k1_numbers)
& ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_3_0_nagata_2(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = E
& k8_funct_2(u1_struct_0(B),k1_numbers,k3_nagata_2(u1_struct_0(B),C,D),E) = np__0 ) ) ) ).
%------------------------------------------------------------------------------