SET007 Axioms: SET007+684.ax
%------------------------------------------------------------------------------
% File : SET007+684 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Urysohn Lemma
% 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 : urysohn3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 49 ( 1 unt; 0 def)
% Number of atoms : 598 ( 63 equ)
% Maximal formula atoms : 25 ( 12 avg)
% Number of connectives : 621 ( 72 ~; 17 |; 287 &)
% ( 8 <=>; 237 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 14 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 28 ( 26 usr; 1 prp; 0-5 aty)
% Number of functors : 37 ( 37 usr; 13 con; 0-5 aty)
% Number of variables : 238 ( 230 !; 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(cc1_urysohn3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,A)
=> ( v1_xcmplx_0(B)
& v1_xreal_0(B) ) ) ) ).
fof(t1_urysohn3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v5_compts_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v4_pre_topc(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_xboole_0(B,C)
=> ( B = k6_measure5
| ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ? [E] :
( v1_funct_1(E)
& v1_funct_2(E,k3_urysohn1(D),k1_pcomps_1(u1_struct_0(A)))
& m2_relset_1(E,k3_urysohn1(D),k1_pcomps_1(u1_struct_0(A)))
& r1_tarski(B,k1_funct_1(E,np__0))
& C = k4_xboole_0(k2_pre_topc(A),k1_funct_1(E,np__1))
& ! [F] :
( m2_subset_1(F,k1_numbers,k3_urysohn1(D))
=> ! [G] :
( m2_subset_1(G,k1_numbers,k3_urysohn1(D))
=> ( ~ r1_xreal_0(G,F)
=> ( v3_pre_topc(k6_urysohn1(A,k3_urysohn1(D),E,F),A)
& v3_pre_topc(k6_urysohn1(A,k3_urysohn1(D),E,G),A)
& r1_tarski(k6_pre_topc(A,k6_urysohn1(A,k3_urysohn1(D),E,F)),k6_urysohn1(A,k3_urysohn1(D),E,G)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d1_urysohn3,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] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( v5_compts_1(A)
& v4_pre_topc(B,A)
& v4_pre_topc(C,A)
& r1_xboole_0(B,C) )
=> ( B = k6_measure5
| ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k3_urysohn1(D),k1_pcomps_1(u1_struct_0(A)))
& m2_relset_1(E,k3_urysohn1(D),k1_pcomps_1(u1_struct_0(A))) )
=> ( m1_urysohn3(E,A,B,C,D)
<=> ( r1_tarski(B,k1_funct_1(E,np__0))
& C = k4_xboole_0(k2_pre_topc(A),k1_funct_1(E,np__1))
& ! [F] :
( m2_subset_1(F,k1_numbers,k3_urysohn1(D))
=> ! [G] :
( m2_subset_1(G,k1_numbers,k3_urysohn1(D))
=> ( ~ r1_xreal_0(G,F)
=> ( v3_pre_topc(k6_urysohn1(A,k3_urysohn1(D),E,F),A)
& v3_pre_topc(k6_urysohn1(A,k3_urysohn1(D),E,G),A)
& r1_tarski(k6_pre_topc(A,k6_urysohn1(A,k3_urysohn1(D),E,F)),k6_urysohn1(A,k3_urysohn1(D),E,G)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t2_urysohn3,axiom,
$true ).
fof(t3_urysohn3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v5_compts_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v4_pre_topc(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_xboole_0(B,C)
=> ( B = k6_measure5
| ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m1_urysohn3(E,A,B,C,D)
=> ? [F] :
( m1_urysohn3(F,A,B,C,k1_nat_1(D,np__1))
& ! [G] :
( m2_subset_1(G,k1_numbers,k3_urysohn1(k1_nat_1(D,np__1)))
=> ( r2_hidden(G,k3_urysohn1(D))
=> k6_urysohn1(A,k3_urysohn1(k1_nat_1(D,np__1)),F,G) = k1_funct_1(E,G) ) ) ) ) ) ) ) ) ) ) ).
fof(t4_urysohn3,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] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m1_urysohn3(E,A,B,C,D)
=> m1_subset_1(E,k4_partfun1(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)))) ) ) ) ) ) ).
fof(t5_urysohn3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v5_compts_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v4_pre_topc(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( B != k6_measure5
& r1_xboole_0(B,C)
& ! [D] :
( m1_seqfunc(D,k4_urysohn1,k1_pcomps_1(u1_struct_0(A)))
=> ~ ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( m1_urysohn3(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,E),A,B,C,E)
& ! [F] :
( m1_subset_1(F,k1_relat_1(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,E)))
=> k1_funct_1(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,E),F) = k1_funct_1(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,k1_nat_1(E,np__1)),F) ) ) ) ) ) ) ) ) ).
fof(d2_urysohn3,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] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v5_compts_1(A)
& v4_pre_topc(B,A)
& v4_pre_topc(C,A)
& r1_xboole_0(B,C) )
=> ( B = k6_measure5
| ! [D] :
( m1_seqfunc(D,k4_urysohn1,k1_pcomps_1(u1_struct_0(A)))
=> ( m2_urysohn3(D,A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( m1_urysohn3(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,E),A,B,C,E)
& ! [F] :
( m1_subset_1(F,k1_relat_1(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,E)))
=> k1_funct_1(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,E),F) = k1_funct_1(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,k1_nat_1(E,np__1)),F) ) ) ) ) ) ) ) ) ) ) ).
fof(d3_urysohn3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( r2_hidden(A,k4_urysohn1)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( B = k3_urysohn3(A)
<=> ( ( r2_hidden(A,k3_urysohn1(np__0))
=> B = np__0 )
& ( B = np__0
=> r2_hidden(A,k3_urysohn1(np__0)) )
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(A,k3_urysohn1(k1_nat_1(C,np__1)))
=> ( r2_hidden(A,k3_urysohn1(C))
| B = k1_nat_1(C,np__1) ) ) ) ) ) ) ) ) ).
fof(t6_urysohn3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( r2_hidden(A,k4_urysohn1)
=> r2_hidden(A,k3_urysohn1(k3_urysohn3(A))) ) ) ).
fof(t7_urysohn3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( r2_hidden(A,k4_urysohn1)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(k3_urysohn3(A),B)
=> r2_hidden(A,k3_urysohn1(B)) ) ) ) ) ).
fof(t8_urysohn3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(A,k3_urysohn1(B))
=> r1_xreal_0(k3_urysohn3(A),B) ) ) ) ).
fof(t9_urysohn3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v5_compts_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v4_pre_topc(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_xboole_0(B,C)
=> ( B = k6_measure5
| ! [D] :
( m2_urysohn3(D,A,B,C)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k4_urysohn1)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> k1_funct_1(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,k3_urysohn3(E)),E) = k1_funct_1(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,k1_nat_1(k3_urysohn3(E),F)),E) ) ) ) ) ) ) ) ) ) ).
fof(t10_urysohn3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v5_compts_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v4_pre_topc(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_xboole_0(B,C)
=> ( B = k6_measure5
| ! [D] :
( m2_urysohn3(D,A,B,C)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( r2_hidden(E,k4_urysohn1)
& ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
=> ? [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
& r2_hidden(E,k3_urysohn1(G))
& F != k1_funct_1(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,G),E) ) ) ) ) ) ) ) ) ) ) ).
fof(t11_urysohn3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v5_compts_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v4_pre_topc(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_xboole_0(B,C)
=> ( B = k6_measure5
| ! [D] :
( m2_urysohn3(D,A,B,C)
=> ? [E] :
( v1_funct_1(E)
& v1_funct_2(E,k5_urysohn1,k1_pcomps_1(u1_struct_0(A)))
& m2_relset_1(E,k5_urysohn1,k1_pcomps_1(u1_struct_0(A)))
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( r2_hidden(F,k5_urysohn1)
=> ( ( r2_hidden(F,k1_urysohn1)
=> k1_funct_1(E,F) = k6_measure5 )
& ( r2_hidden(F,k2_urysohn1)
=> k1_funct_1(E,F) = u1_struct_0(A) )
& ( r2_hidden(F,k4_urysohn1)
=> ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( r2_hidden(F,k3_urysohn1(G))
=> k1_funct_1(E,F) = k1_funct_1(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,G),F) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d4_urysohn3,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] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v5_compts_1(A)
& v4_pre_topc(B,A)
& v4_pre_topc(C,A)
& r1_xboole_0(B,C) )
=> ( B = k6_measure5
| ! [D] :
( m2_urysohn3(D,A,B,C)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_urysohn1,k1_pcomps_1(u1_struct_0(A)))
& m2_relset_1(E,k5_urysohn1,k1_pcomps_1(u1_struct_0(A))) )
=> ( E = k4_urysohn3(A,B,C,D)
<=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( r2_hidden(F,k5_urysohn1)
=> ( ( r2_hidden(F,k1_urysohn1)
=> k1_funct_1(E,F) = k6_measure5 )
& ( r2_hidden(F,k2_urysohn1)
=> k1_funct_1(E,F) = u1_struct_0(A) )
& ( r2_hidden(F,k4_urysohn1)
=> ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( r2_hidden(F,k3_urysohn1(G))
=> k1_funct_1(E,F) = k1_funct_1(k1_seqfunc(k4_urysohn1,k1_pcomps_1(u1_struct_0(A)),D,G),F) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t12_urysohn3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v5_compts_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v4_pre_topc(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_xboole_0(B,C)
=> ( B = k6_measure5
| ! [D] :
( m2_urysohn3(D,A,B,C)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( F = k1_funct_1(k4_urysohn3(A,B,C,D),E)
& r2_hidden(E,k5_urysohn1) )
=> v3_pre_topc(F,A) ) ) ) ) ) ) ) ) ) ).
fof(t13_urysohn3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v5_compts_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v4_pre_topc(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_xboole_0(B,C)
=> ( B = k6_measure5
| ! [D] :
( m2_urysohn3(D,A,B,C)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( ( r2_hidden(E,k5_urysohn1)
& r2_hidden(F,k5_urysohn1) )
=> ( r1_xreal_0(F,E)
| ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A)))
=> ( G = k1_funct_1(k4_urysohn3(A,B,C,D),E)
=> r1_tarski(k6_pre_topc(A,G),k1_funct_1(k4_urysohn3(A,B,C,D),F)) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t14_urysohn3,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] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m2_urysohn3(D,A,B,C)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ? [F] :
( m1_subset_1(F,k1_zfmisc_1(k3_supinf_1))
& ! [G] :
( r2_hidden(G,F)
<=> ( r2_hidden(G,k4_urysohn1)
& ! [H] :
( m1_subset_1(H,k1_numbers)
=> ~ ( H = G
& r2_hidden(E,k1_funct_1(k4_urysohn3(A,B,C,D),H)) ) ) ) ) ) ) ) ) ) ) ).
fof(d5_urysohn3,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] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m2_urysohn3(D,A,B,C)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(k3_supinf_1))
=> ( F = k6_urysohn3(A,B,C,D,E)
<=> ! [G] :
( r2_hidden(G,F)
<=> ( r2_hidden(G,k4_urysohn1)
& ! [H] :
( m1_subset_1(H,k1_numbers)
=> ~ ( H = G
& r2_hidden(E,k1_funct_1(k4_urysohn3(A,B,C,D),H)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t15_urysohn3,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] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m2_urysohn3(D,A,B,C)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> r1_tarski(k6_urysohn3(A,B,C,D,E),k4_urysohn1) ) ) ) ) ) ).
fof(t16_urysohn3,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] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m2_urysohn3(D,A,B,C)
=> ? [E] :
( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(k3_topmetr))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(k3_topmetr))
& ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( k6_urysohn3(A,B,C,D,F) = k6_measure5
=> k7_urysohn3(A,k3_topmetr,E,F) = np__0 )
& ! [G] :
( ( ~ v1_xboole_0(G)
& m1_subset_1(G,k1_zfmisc_1(k3_supinf_1)) )
=> ( G = k6_urysohn3(A,B,C,D,F)
=> k7_urysohn3(A,k3_topmetr,E,F) = k9_supinf_1(G) ) ) ) ) ) ) ) ) ) ).
fof(d6_urysohn3,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] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m2_urysohn3(D,A,B,C)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(k3_topmetr))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(k3_topmetr)) )
=> ( E = k8_urysohn3(A,B,C,D)
<=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( k6_urysohn3(A,B,C,D,F) = k6_measure5
=> k7_urysohn3(A,k3_topmetr,E,F) = np__0 )
& ! [G] :
( ( ~ v1_xboole_0(G)
& m1_subset_1(G,k1_zfmisc_1(k3_supinf_1)) )
=> ( G = k6_urysohn3(A,B,C,D,F)
=> k7_urysohn3(A,k3_topmetr,E,F) = k9_supinf_1(G) ) ) ) ) ) ) ) ) ) ) ).
fof(t17_urysohn3,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] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m2_urysohn3(D,A,B,C)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( ( ~ v1_xboole_0(F)
& m1_subset_1(F,k1_zfmisc_1(k3_supinf_1)) )
=> ( F = k6_urysohn3(A,B,C,D,E)
=> ! [G] :
( m1_subset_1(G,k3_supinf_1)
=> ( G = np__1
=> ( r1_supinf_1(k1_supinf_2,k9_supinf_1(F))
& r1_supinf_1(k9_supinf_1(F),G) ) ) ) ) ) ) ) ) ) ) ).
fof(t18_urysohn3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v5_compts_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v4_pre_topc(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_xboole_0(B,C)
=> ( B = k6_measure5
| ! [D] :
( m2_urysohn3(D,A,B,C)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_urysohn1)
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ~ r1_xreal_0(E,k9_urysohn3(A,k8_urysohn3(A,B,C,D),F))
=> r2_hidden(F,k5_urysohn3(k5_urysohn1,A,k4_urysohn3(A,B,C,D),E)) ) ) ) ) ) ) ) ) ) ).
fof(t19_urysohn3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v5_compts_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v4_pre_topc(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_xboole_0(B,C)
=> ( B = k6_measure5
| ! [D] :
( m2_urysohn3(D,A,B,C)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r2_hidden(E,k4_subset_1(k1_numbers,k4_urysohn1,k2_urysohn1))
=> ( r1_xreal_0(E,np__0)
| ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( r2_hidden(F,k1_funct_1(k4_urysohn3(A,B,C,D),E))
=> r1_xreal_0(k9_urysohn3(A,k8_urysohn3(A,B,C,D),F),E) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t20_urysohn3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v5_compts_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v4_pre_topc(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_xboole_0(B,C)
=> ( B = k6_measure5
| ! [D] :
( m2_urysohn3(D,A,B,C)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_urysohn1)
=> ( ~ r1_xreal_0(E,np__0)
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ~ ( ~ r1_xreal_0(k9_urysohn3(A,k8_urysohn3(A,B,C,D),F),E)
& r2_hidden(F,k5_urysohn3(k5_urysohn1,A,k4_urysohn3(A,B,C,D),E)) ) ) ) ) ) ) ) ) ) ) ).
fof(t21_urysohn3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v5_compts_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v4_pre_topc(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_xboole_0(B,C)
=> ( B = k6_measure5
| ! [D] :
( m2_urysohn3(D,A,B,C)
=> ( v5_pre_topc(k8_urysohn3(A,B,C,D),A,k3_topmetr)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( r1_xreal_0(np__0,k9_urysohn3(A,k8_urysohn3(A,B,C,D),E))
& r1_xreal_0(k9_urysohn3(A,k8_urysohn3(A,B,C,D),E),np__1)
& ( r2_hidden(E,B)
=> k9_urysohn3(A,k8_urysohn3(A,B,C,D),E) = np__0 )
& ( r2_hidden(E,C)
=> k9_urysohn3(A,k8_urysohn3(A,B,C,D),E) = np__1 ) ) ) ) ) ) ) ) ) ) ).
fof(t22_urysohn3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v5_compts_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v4_pre_topc(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( B != k6_measure5
& r1_xboole_0(B,C)
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(k3_topmetr))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(k3_topmetr)) )
=> ~ ( v5_pre_topc(D,A,k3_topmetr)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( r1_xreal_0(np__0,k9_urysohn3(A,D,E))
& r1_xreal_0(k9_urysohn3(A,D,E),np__1)
& ( r2_hidden(E,B)
=> k9_urysohn3(A,D,E) = np__0 )
& ( r2_hidden(E,C)
=> k9_urysohn3(A,D,E) = np__1 ) ) ) ) ) ) ) ) ) ).
fof(t23_urysohn3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v5_compts_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v4_pre_topc(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( r1_xboole_0(B,C)
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(k3_topmetr))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(k3_topmetr)) )
=> ~ ( v5_pre_topc(D,A,k3_topmetr)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( r1_xreal_0(np__0,k9_urysohn3(A,D,E))
& r1_xreal_0(k9_urysohn3(A,D,E),np__1)
& ( r2_hidden(E,B)
=> k9_urysohn3(A,D,E) = np__0 )
& ( r2_hidden(E,C)
=> k9_urysohn3(A,D,E) = np__1 ) ) ) ) ) ) ) ) ) ).
fof(t24_urysohn3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v2_compts_1(A)
& v3_compts_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v4_pre_topc(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( v4_pre_topc(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( r1_xboole_0(B,C)
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(k3_topmetr))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(k3_topmetr)) )
=> ~ ( v5_pre_topc(D,A,k3_topmetr)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( r1_xreal_0(np__0,k9_urysohn3(A,D,E))
& r1_xreal_0(k9_urysohn3(A,D,E),np__1)
& ( r2_hidden(E,B)
=> k9_urysohn3(A,D,E) = np__0 )
& ( r2_hidden(E,C)
=> k9_urysohn3(A,D,E) = np__1 ) ) ) ) ) ) ) ) ) ).
fof(dt_m1_urysohn3,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(D,k5_numbers) )
=> ! [E] :
( m1_urysohn3(E,A,B,C,D)
=> ( v1_funct_1(E)
& v1_funct_2(E,k3_urysohn1(D),k1_pcomps_1(u1_struct_0(A)))
& m2_relset_1(E,k3_urysohn1(D),k1_pcomps_1(u1_struct_0(A))) ) ) ) ).
fof(existence_m1_urysohn3,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(D,k5_numbers) )
=> ? [E] : m1_urysohn3(E,A,B,C,D) ) ).
fof(dt_m2_urysohn3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [D] :
( m2_urysohn3(D,A,B,C)
=> m1_seqfunc(D,k4_urysohn1,k1_pcomps_1(u1_struct_0(A))) ) ) ).
fof(existence_m2_urysohn3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ? [D] : m2_urysohn3(D,A,B,C) ) ).
fof(dt_k1_urysohn3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k4_partfun1(A,B))
& m1_relset_1(C,k5_numbers,k4_partfun1(A,B))
& m1_subset_1(D,k5_numbers) )
=> ( v1_funct_1(k1_urysohn3(A,B,C,D))
& m2_relset_1(k1_urysohn3(A,B,C,D),A,B) ) ) ).
fof(redefinition_k1_urysohn3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k4_partfun1(A,B))
& m1_relset_1(C,k5_numbers,k4_partfun1(A,B))
& m1_subset_1(D,k5_numbers) )
=> k1_urysohn3(A,B,C,D) = k1_funct_1(C,D) ) ).
fof(dt_k2_urysohn3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k4_partfun1(A,B))
& m1_relset_1(C,k5_numbers,k4_partfun1(A,B))
& m1_subset_1(D,k5_numbers) )
=> m1_subset_1(k2_urysohn3(A,B,C,D),k4_partfun1(A,B)) ) ).
fof(redefinition_k2_urysohn3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k4_partfun1(A,B))
& m1_relset_1(C,k5_numbers,k4_partfun1(A,B))
& m1_subset_1(D,k5_numbers) )
=> k2_urysohn3(A,B,C,D) = k1_funct_1(C,D) ) ).
fof(dt_k3_urysohn3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> m2_subset_1(k3_urysohn3(A),k1_numbers,k5_numbers) ) ).
fof(dt_k4_urysohn3,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& m2_urysohn3(D,A,B,C) )
=> ( v1_funct_1(k4_urysohn3(A,B,C,D))
& v1_funct_2(k4_urysohn3(A,B,C,D),k5_urysohn1,k1_pcomps_1(u1_struct_0(A)))
& m2_relset_1(k4_urysohn3(A,B,C,D),k5_urysohn1,k1_pcomps_1(u1_struct_0(A))) ) ) ).
fof(dt_k5_urysohn3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v2_pre_topc(B)
& l1_pre_topc(B)
& v1_funct_1(C)
& v1_funct_2(C,A,k1_pcomps_1(u1_struct_0(B)))
& m1_relset_1(C,A,k1_pcomps_1(u1_struct_0(B)))
& m1_subset_1(D,A) )
=> m1_subset_1(k5_urysohn3(A,B,C,D),k1_zfmisc_1(u1_struct_0(B))) ) ).
fof(redefinition_k5_urysohn3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v2_pre_topc(B)
& l1_pre_topc(B)
& v1_funct_1(C)
& v1_funct_2(C,A,k1_pcomps_1(u1_struct_0(B)))
& m1_relset_1(C,A,k1_pcomps_1(u1_struct_0(B)))
& m1_subset_1(D,A) )
=> k5_urysohn3(A,B,C,D) = k1_funct_1(C,D) ) ).
fof(dt_k6_urysohn3,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& m2_urysohn3(D,A,B,C)
& m1_subset_1(E,u1_struct_0(A)) )
=> m1_subset_1(k6_urysohn3(A,B,C,D,E),k1_zfmisc_1(k3_supinf_1)) ) ).
fof(dt_k7_urysohn3,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),u1_struct_0(B))
& m1_relset_1(C,u1_struct_0(A),u1_struct_0(B))
& m1_subset_1(D,u1_struct_0(A)) )
=> m1_subset_1(k7_urysohn3(A,B,C,D),u1_struct_0(B)) ) ).
fof(redefinition_k7_urysohn3,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),u1_struct_0(B))
& m1_relset_1(C,u1_struct_0(A),u1_struct_0(B))
& m1_subset_1(D,u1_struct_0(A)) )
=> k7_urysohn3(A,B,C,D) = k1_funct_1(C,D) ) ).
fof(dt_k8_urysohn3,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& m2_urysohn3(D,A,B,C) )
=> ( v1_funct_1(k8_urysohn3(A,B,C,D))
& v1_funct_2(k8_urysohn3(A,B,C,D),u1_struct_0(A),u1_struct_0(k3_topmetr))
& m2_relset_1(k8_urysohn3(A,B,C,D),u1_struct_0(A),u1_struct_0(k3_topmetr)) ) ) ).
fof(dt_k9_urysohn3,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),u1_struct_0(k3_topmetr))
& m1_relset_1(B,u1_struct_0(A),u1_struct_0(k3_topmetr))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k9_urysohn3(A,B,C),k1_numbers) ) ).
fof(redefinition_k9_urysohn3,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),u1_struct_0(k3_topmetr))
& m1_relset_1(B,u1_struct_0(A),u1_struct_0(k3_topmetr))
& m1_subset_1(C,u1_struct_0(A)) )
=> k9_urysohn3(A,B,C) = k1_funct_1(B,C) ) ).
%------------------------------------------------------------------------------