SET007 Axioms: SET007+845.ax
%------------------------------------------------------------------------------
% File : SET007+845 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Nagata-Smirnov Theorem. Part I
% 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_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 54 ( 0 unt; 0 def)
% Number of atoms : 558 ( 23 equ)
% Maximal formula atoms : 25 ( 10 avg)
% Number of connectives : 571 ( 67 ~; 6 |; 299 &)
% ( 12 <=>; 187 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 10 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of predicates : 57 ( 56 usr; 0 prp; 1-4 aty)
% Number of functors : 43 ( 43 usr; 4 con; 0-5 aty)
% Number of variables : 180 ( 170 !; 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(rc1_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
& v1_nagata_1(B,A) ) ) ).
fof(rc2_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
& v1_xboole_0(B)
& v1_finset_1(B)
& v1_membered(B)
& v2_membered(B)
& v3_membered(B)
& v4_membered(B)
& v5_membered(B)
& v1_card_4(B)
& v1_nagata_1(B,A) ) ) ).
fof(fc1_nagata_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_nagata_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) )
=> v1_nagata_1(k3_pcomps_1(A,B),A) ) ).
fof(rc3_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
& v1_relat_1(B)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
& v2_nagata_1(B,A) ) ) ).
fof(cc1_nagata_1,axiom,
! [A] :
( ~ v1_card_4(A)
=> ~ v1_xboole_0(A) ) ).
fof(rc4_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
& v1_relat_1(B)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
& v3_nagata_1(B,A) ) ) ).
fof(d1_nagata_1,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v1_nagata_1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( v3_pre_topc(D,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& r2_hidden(C,D)
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_hidden(E,B)
& r2_hidden(F,B) )
=> ( r1_xboole_0(D,E)
| r1_xboole_0(D,F)
| E = F ) ) ) ) ) ) ) ) ) ).
fof(t1_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( ? [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& B = k6_domain_1(k1_zfmisc_1(u1_struct_0(A)),C) )
=> v1_nagata_1(B,A) ) ) ) ).
fof(t2_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( ( r1_tarski(B,C)
& v1_nagata_1(C,A) )
=> v1_nagata_1(B,A) ) ) ) ) ).
fof(t3_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v1_nagata_1(B,A)
=> v1_nagata_1(k5_subset_1(k1_zfmisc_1(u1_struct_0(A)),B,C),A) ) ) ) ) ).
fof(t4_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v1_nagata_1(B,A)
=> v1_nagata_1(k6_subset_1(k1_zfmisc_1(u1_struct_0(A)),B,C),A) ) ) ) ) ).
fof(t5_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( ( v1_nagata_1(B,A)
& v1_nagata_1(C,A)
& k3_setfam_1(B,C) = D )
=> v1_nagata_1(D,A) ) ) ) ) ) ).
fof(t6_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v1_nagata_1(B,A)
& r2_hidden(C,B)
& r2_hidden(D,B)
& C != D
& ~ r1_xboole_0(C,D) ) ) ) ) ) ).
fof(t7_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v1_nagata_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( v3_pre_topc(D,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& r2_hidden(C,D)
& v1_realset1(k4_xboole_0(k3_setfam_1(k6_domain_1(k1_zfmisc_1(u1_struct_0(A)),D),B),k1_tarski(k1_xboole_0))) ) ) ) ) ) ).
fof(t8_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v1_nagata_1(B,A)
<=> ( ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( v3_pre_topc(D,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& r2_hidden(C,D)
& v1_realset1(k4_xboole_0(k3_setfam_1(k6_domain_1(k1_zfmisc_1(u1_struct_0(A)),D),B),k1_tarski(k1_xboole_0))) ) )
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(C,B)
& r2_hidden(D,B)
& C != D
& ~ r1_xboole_0(C,D) ) ) ) ) ) ) ) ).
fof(t9_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v1_nagata_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> k6_pre_topc(A,k5_subset_1(u1_struct_0(A),C,D)) = k5_subset_1(u1_struct_0(A),k6_pre_topc(A,C),k6_pre_topc(A,D)) ) ) ) ) ) ) ).
fof(t10_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v1_nagata_1(B,A)
=> k6_pre_topc(A,k5_setfam_1(u1_struct_0(A),B)) = k5_setfam_1(u1_struct_0(A),k3_pcomps_1(A,B)) ) ) ) ).
fof(t11_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v1_nagata_1(B,A)
=> v1_pcomps_1(B,A) ) ) ) ).
fof(d2_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(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)))) )
=> ( v2_nagata_1(B,A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> v1_nagata_1(k1_nagata_1(A,B,C),A) ) ) ) ) ).
fof(d3_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(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)))) )
=> ( v3_nagata_1(B,A)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> v1_pcomps_1(k1_nagata_1(A,B,C),A) ) ) ) ) ).
fof(d4_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v4_nagata_1(B,A)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
& v2_nagata_1(C,A)
& m2_relset_1(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
& B = k2_nagata_1(A,C) ) ) ) ) ).
fof(t12_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(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)))) )
=> ( v2_nagata_1(B,A)
=> v3_nagata_1(B,A) ) ) ) ).
fof(t13_nagata_1,axiom,
! [A] :
( ~ v1_card_4(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k2_pcomps_1(k2_zfmisc_1(A,A))))))
& v1_pcomps_1(B,k2_pcomps_1(k2_zfmisc_1(A,A)))
& ~ v4_nagata_1(B,k2_pcomps_1(k2_zfmisc_1(A,A))) ) ) ).
fof(d5_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(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)
<=> ( v2_nagata_1(B,A)
& m1_cantor_1(k2_nagata_1(A,B),A) ) ) ) ) ).
fof(d6_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(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_nagata_1(B,A)
& m1_cantor_1(k2_nagata_1(A,B),A) ) ) ) ) ).
fof(t14_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( v1_xreal_0(B)
=> ( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r2_hidden(k6_subset_1(u1_struct_0(A),k2_pre_topc(A),k10_metric_1(A,C,B)),k4_pcomps_1(A)) ) ) ) ) ).
fof(t15_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v3_pcomps_1(A)
=> ( v4_compts_1(A)
& v1_urysohn1(A) ) ) ) ).
fof(t16_nagata_1,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)))) )
=> ~ v6_nagata_1(B,A) ) ) ) ).
fof(t17_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_pcomps_1(u1_struct_0(A)))
& m2_relset_1(B,k5_numbers,k1_pcomps_1(u1_struct_0(A))) )
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> v3_pre_topc(k5_urysohn3(k5_numbers,A,B,C),A) )
=> v3_pre_topc(k2_prob_1(u1_struct_0(A),B),A) ) ) ) ).
fof(t18_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( ( v3_pre_topc(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( v4_pre_topc(B,A)
& v3_pre_topc(C,A)
& r1_tarski(B,C)
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_pcomps_1(u1_struct_0(A)))
& m2_relset_1(D,k5_numbers,k1_pcomps_1(u1_struct_0(A))) )
=> ~ ( r1_tarski(B,k2_prob_1(u1_struct_0(A),D))
& r1_tarski(k2_prob_1(u1_struct_0(A),D),C)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r1_tarski(k6_pre_topc(A,k5_urysohn3(k5_numbers,A,D,E)),C)
& v3_pre_topc(k5_urysohn3(k5_numbers,A,D,E),A) ) ) ) ) ) ) )
=> v5_compts_1(A) ) ) ).
fof(t19_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v4_compts_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)))) )
=> ( m1_cantor_1(k2_nagata_1(A,B),A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( v3_pre_topc(C,A)
& r2_hidden(D,C)
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(D,E)
& r1_tarski(k6_pre_topc(A,E),C)
& r2_hidden(E,k2_nagata_1(A,B)) ) ) ) ) ) ) ) ) ) ).
fof(t20_nagata_1,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) )
| v5_compts_1(A) ) ) ) ).
fof(d7_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& m2_relset_1(B,u1_struct_0(A),k1_numbers) )
=> ! [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] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),k1_numbers)
& m2_relset_1(D,u1_struct_0(A),k1_numbers) )
=> ( D = k3_nagata_1(A,B,C)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),k1_numbers,D,E) = k3_real_1(k8_funct_2(u1_struct_0(A),k1_numbers,B,E),k8_funct_2(u1_struct_0(A),k1_numbers,C,E)) ) ) ) ) ) ) ).
fof(t21_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& m2_relset_1(B,u1_struct_0(A),k1_numbers) )
=> ( v9_pscomp_1(B,A)
=> ! [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) )
=> ( ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,C,k8_borsuk_1(A,A,D,E)) = k18_complex1(k5_real_1(k8_funct_2(u1_struct_0(A),k1_numbers,B,D),k8_funct_2(u1_struct_0(A),k1_numbers,B,E))) ) )
=> v9_pscomp_1(C,k6_borsuk_1(A,A)) ) ) ) ) ) ).
fof(t22_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& m2_relset_1(B,u1_struct_0(A),k1_numbers) )
=> ! [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) )
=> ( ( v9_pscomp_1(B,A)
& v9_pscomp_1(C,A) )
=> v9_pscomp_1(k3_nagata_1(A,B,C),A) ) ) ) ) ).
fof(t23_nagata_1,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(k1_fraenkel(u1_struct_0(A),k1_numbers),k1_fraenkel(u1_struct_0(A),k1_numbers)),k1_fraenkel(u1_struct_0(A),k1_numbers))
& m2_relset_1(B,k2_zfmisc_1(k1_fraenkel(u1_struct_0(A),k1_numbers),k1_fraenkel(u1_struct_0(A),k1_numbers)),k1_fraenkel(u1_struct_0(A),k1_numbers)) )
=> ( ! [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] :
( ( 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_binop_1(B,C,D) = k3_nagata_1(A,C,D) ) )
=> ( v1_setwiseo(B,k1_fraenkel(u1_struct_0(A),k1_numbers))
& v1_binop_1(B,k1_fraenkel(u1_struct_0(A),k1_numbers))
& v2_binop_1(B,k1_fraenkel(u1_struct_0(A),k1_numbers)) ) ) ) ) ).
fof(t24_nagata_1,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(k1_fraenkel(u1_struct_0(A),k1_numbers),k1_fraenkel(u1_struct_0(A),k1_numbers)),k1_fraenkel(u1_struct_0(A),k1_numbers))
& m2_relset_1(B,k2_zfmisc_1(k1_fraenkel(u1_struct_0(A),k1_numbers),k1_fraenkel(u1_struct_0(A),k1_numbers)),k1_fraenkel(u1_struct_0(A),k1_numbers)) )
=> ( ! [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] :
( ( 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_binop_1(B,C,D) = k3_nagata_1(A,C,D) ) )
=> ! [C] :
( m2_fraenkel(C,u1_struct_0(A),k1_numbers,k1_fraenkel(u1_struct_0(A),k1_numbers))
=> ( r3_binop_1(k1_fraenkel(u1_struct_0(A),k1_numbers),C,B)
=> v9_pscomp_1(C,A) ) ) ) ) ) ).
fof(d8_nagata_1,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] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A)))
& m2_relset_1(C,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A))) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)))
& m2_relset_1(D,u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B))) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(B)) )
=> ( E = k4_nagata_1(A,B,C,D)
<=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> k7_urysohn3(A,B,E,F) = k1_funct_1(k8_funct_2(u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)),D,F),F) ) ) ) ) ) ) ) ).
fof(t25_nagata_1,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(k1_fraenkel(u1_struct_0(A),k1_numbers),k1_fraenkel(u1_struct_0(A),k1_numbers)),k1_fraenkel(u1_struct_0(A),k1_numbers))
& m2_relset_1(B,k2_zfmisc_1(k1_fraenkel(u1_struct_0(A),k1_numbers),k1_fraenkel(u1_struct_0(A),k1_numbers)),k1_fraenkel(u1_struct_0(A),k1_numbers)) )
=> ( ! [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] :
( ( 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_binop_1(B,C,D) = k3_nagata_1(A,C,D) ) )
=> ! [C] :
( m2_finseq_1(C,k1_fraenkel(u1_struct_0(A),k1_numbers))
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(D,k3_finseq_1(C))
=> ( np__0 = D
| ( v1_funct_1(k1_funct_1(C,D))
& v1_funct_2(k1_funct_1(C,D),u1_struct_0(A),k1_numbers)
& v9_pscomp_1(k1_funct_1(C,D),A)
& m2_relset_1(k1_funct_1(C,D),u1_struct_0(A),k1_numbers) ) ) ) )
=> ( v1_funct_1(k2_finsop_1(k1_fraenkel(u1_struct_0(A),k1_numbers),C,B))
& v1_funct_2(k2_finsop_1(k1_fraenkel(u1_struct_0(A),k1_numbers),C,B),u1_struct_0(A),k1_numbers)
& v9_pscomp_1(k2_finsop_1(k1_fraenkel(u1_struct_0(A),k1_numbers),C,B),A)
& m2_relset_1(k2_finsop_1(k1_fraenkel(u1_struct_0(A),k1_numbers),C,B),u1_struct_0(A),k1_numbers) ) ) ) ) ) ) ).
fof(t26_nagata_1,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] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)))
& m2_relset_1(C,u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B))) )
=> ( ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( v1_funct_1(k8_funct_2(u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)),C,D))
& v1_funct_2(k8_funct_2(u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)),C,D),u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(k8_funct_2(u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)),C,D),A,B)
& m2_relset_1(k8_funct_2(u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)),C,D),u1_struct_0(A),u1_struct_0(B)) ) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A)))
& m2_relset_1(D,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A))) )
=> ( ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( r2_hidden(E,k5_urysohn3(u1_struct_0(A),A,D,E))
& v3_pre_topc(k5_urysohn3(u1_struct_0(A),A,D,E),A)
& ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ( r2_hidden(F,k5_urysohn3(u1_struct_0(A),A,D,G))
=> k1_funct_1(k8_funct_2(u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)),C,F),F) = k1_funct_1(k8_funct_2(u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)),C,G),F) ) ) ) ) )
=> v5_pre_topc(k4_nagata_1(A,B,D,C),A,B) ) ) ) ) ) ) ).
fof(d9_nagata_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,k1_numbers)
& m2_relset_1(C,A,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,k1_numbers)
& m2_relset_1(D,A,k1_numbers) )
=> ( D = k5_nagata_1(A,B,C)
<=> ! [E] :
( r2_hidden(E,A)
=> k1_funct_1(D,E) = k1_square_1(B,k1_funct_1(C,E)) ) ) ) ) ) ).
fof(t27_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [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) )
=> ( v9_pscomp_1(C,A)
=> v9_pscomp_1(k5_nagata_1(u1_struct_0(A),B,C),A) ) ) ) ) ).
fof(d10_nagata_1,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_nagata_1(A,B)
<=> ( v2_metric_1(B,A)
& v4_metric_1(B,A)
& v5_metric_1(B,A) ) ) ) ).
fof(t28_nagata_1,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_nagata_1(A,B)
<=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ( k1_metric_1(A,A,B,C,C) = np__0
& r1_xreal_0(k1_metric_1(A,A,B,C,E),k3_real_1(k1_metric_1(A,A,B,C,D),k1_metric_1(A,A,B,E,D))) ) ) ) ) ) ) ).
fof(t29_nagata_1,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_nagata_1(A,B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> r1_xreal_0(np__0,k1_metric_1(A,A,B,C,D)) ) ) ) ) ).
fof(t30_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [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) )
=> ( r1_nagata_1(u1_struct_0(A),C)
=> ( r1_xreal_0(B,np__0)
| r1_nagata_1(u1_struct_0(A),k5_nagata_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),B,C)) ) ) ) ) ) ).
fof(t31_nagata_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [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) )
=> ( r1_pcomps_1(u1_struct_0(A),C)
=> ( r1_xreal_0(B,np__0)
| r1_pcomps_1(u1_struct_0(A),k5_nagata_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),B,C)) ) ) ) ) ) ).
fof(dt_k1_nagata_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
& m1_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
& m1_subset_1(C,k5_numbers) )
=> m1_subset_1(k1_nagata_1(A,B,C),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ).
fof(redefinition_k1_nagata_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
& m1_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
& m1_subset_1(C,k5_numbers) )
=> k1_nagata_1(A,B,C) = k1_funct_1(B,C) ) ).
fof(dt_k2_nagata_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
& m1_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) )
=> m1_subset_1(k2_nagata_1(A,B),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ).
fof(redefinition_k2_nagata_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A))))
& m1_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) )
=> k2_nagata_1(A,B) = k3_card_3(B) ) ).
fof(dt_k3_nagata_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& m1_relset_1(B,u1_struct_0(A),k1_numbers)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),k1_numbers)
& m1_relset_1(C,u1_struct_0(A),k1_numbers) )
=> ( v1_funct_1(k3_nagata_1(A,B,C))
& v1_funct_2(k3_nagata_1(A,B,C),u1_struct_0(A),k1_numbers)
& m2_relset_1(k3_nagata_1(A,B,C),u1_struct_0(A),k1_numbers) ) ) ).
fof(dt_k4_nagata_1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A)))
& m1_relset_1(C,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A)))
& v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)))
& m1_relset_1(D,u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B))) )
=> ( v1_funct_1(k4_nagata_1(A,B,C,D))
& v1_funct_2(k4_nagata_1(A,B,C,D),u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(k4_nagata_1(A,B,C,D),u1_struct_0(A),u1_struct_0(B)) ) ) ).
fof(dt_k5_nagata_1,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_numbers)
& v1_funct_1(C)
& v1_funct_2(C,A,k1_numbers)
& m1_relset_1(C,A,k1_numbers) )
=> ( v1_funct_1(k5_nagata_1(A,B,C))
& v1_funct_2(k5_nagata_1(A,B,C),A,k1_numbers)
& m2_relset_1(k5_nagata_1(A,B,C),A,k1_numbers) ) ) ).
%------------------------------------------------------------------------------