SET007 Axioms: SET007+588.ax
%------------------------------------------------------------------------------
% File : SET007+588 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Compactness of the Bounded Closed Subsets of cal E^2_T
% 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 : topreal6 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 120 ( 10 unt; 0 def)
% Number of atoms : 619 ( 95 equ)
% Maximal formula atoms : 11 ( 5 avg)
% Number of connectives : 576 ( 77 ~; 9 |; 194 &)
% ( 7 <=>; 289 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 7 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of predicates : 55 ( 53 usr; 1 prp; 0-3 aty)
% Number of functors : 79 ( 79 usr; 10 con; 0-5 aty)
% Number of variables : 245 ( 237 !; 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(fc1_topreal6,axiom,
! [A,B] :
( ( l1_pre_topc(A)
& v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_pre_topc(k3_pre_topc(A,B))
& v6_group_1(k3_pre_topc(A,B)) ) ) ).
fof(rc1_topreal6,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& v1_pre_topc(A)
& v2_pre_topc(A)
& v2_compts_1(A)
& v6_group_1(A)
& v10_pscomp_1(A) ) ).
fof(cc1_topreal6,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_xboole_0(B)
=> v2_connsp_1(B,A) ) ) ) ).
fof(cc2_topreal6,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_finset_1(B)
=> v6_compts_1(B,A) ) ) ) ).
fof(cc3_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( ( ~ v1_xboole_0(A)
& v1_jordan1(A,np__2) )
=> v2_connsp_1(A,k15_euclid(np__2)) ) ) ).
fof(cc4_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( ~ v1_sppol_1(A)
=> ~ v1_xboole_0(A) ) ) ).
fof(cc5_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( ~ v2_sppol_1(A)
=> ~ v1_xboole_0(A) ) ) ).
fof(cc6_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( v2_topreal4(A,k15_euclid(np__2))
=> ( v3_pre_topc(A,k15_euclid(np__2))
& v2_connsp_1(A,k15_euclid(np__2)) ) ) ) ).
fof(cc7_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( ( v3_pre_topc(A,k15_euclid(np__2))
& v2_connsp_1(A,k15_euclid(np__2)) )
=> v2_topreal4(A,k15_euclid(np__2)) ) ) ).
fof(cc8_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( v1_xboole_0(A)
=> v1_sppol_1(A) ) ) ).
fof(cc9_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( v1_xboole_0(A)
=> v2_sppol_1(A) ) ) ).
fof(rc2_topreal6,axiom,
? [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& ~ v1_xboole_0(A)
& v2_connsp_1(A,k15_euclid(np__2))
& v1_jordan1(A,np__2) ) ).
fof(fc2_topreal6,axiom,
! [A,B] :
( ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
& m1_subset_1(B,u1_struct_0(k15_euclid(np__2))) )
=> ( ~ v1_xboole_0(k1_topreal1(np__2,A,B))
& v2_connsp_1(k1_topreal1(np__2,A,B),k15_euclid(np__2))
& v1_jordan1(k1_topreal1(np__2,A,B),np__2)
& v2_tops_1(k1_topreal1(np__2,A,B),k15_euclid(np__2)) ) ) ).
fof(fc3_topreal6,axiom,
( ~ v1_xboole_0(k2_topreal1)
& v2_connsp_1(k2_topreal1,k15_euclid(np__2))
& ~ v1_realset1(k2_topreal1)
& ~ v1_sppol_1(k2_topreal1)
& ~ v2_sppol_1(k2_topreal1)
& v1_topreal4(k2_topreal1) ) ).
fof(cc10_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( v1_topreal2(A)
=> v2_connsp_1(A,k15_euclid(np__2)) ) ) ).
fof(fc4_topreal6,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ( ~ v1_xboole_0(k7_pscomp_1(A))
& v1_membered(k7_pscomp_1(A))
& v2_membered(k7_pscomp_1(A))
& v2_rcomp_1(k7_pscomp_1(A)) ) ) ).
fof(fc5_topreal6,axiom,
! [A] :
( ( v2_seq_4(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ( v1_membered(k7_pscomp_1(A))
& v2_membered(k7_pscomp_1(A))
& v2_seq_4(k7_pscomp_1(A))
& v2_rcomp_1(k7_pscomp_1(A)) ) ) ).
fof(fc6_topreal6,axiom,
! [A] :
( ( v1_seq_4(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ( v1_membered(k7_pscomp_1(A))
& v2_membered(k7_pscomp_1(A))
& v1_seq_4(k7_pscomp_1(A))
& v2_rcomp_1(k7_pscomp_1(A)) ) ) ).
fof(fc7_topreal6,axiom,
( ~ v3_struct_0(k3_topmetr)
& v1_pre_topc(k3_topmetr)
& v2_pre_topc(k3_topmetr)
& v3_compts_1(k3_topmetr) ) ).
fof(cc11_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( v1_finset_1(A)
=> ( v1_membered(A)
& v2_membered(A)
& v1_rcomp_1(A) ) ) ) ).
fof(fc8_topreal6,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> ( v1_membered(k1_rcomp_1(A,B))
& v2_membered(k1_rcomp_1(A,B))
& v1_seq_4(k1_rcomp_1(A,B))
& v2_seq_4(k1_rcomp_1(A,B))
& v3_seq_4(k1_rcomp_1(A,B))
& v1_rcomp_1(k1_rcomp_1(A,B))
& v2_rcomp_1(k1_rcomp_1(A,B)) ) ) ).
fof(rc3_topreal6,axiom,
? [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
& ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_card_4(A)
& v1_membered(A)
& v2_membered(A)
& v1_seq_4(A)
& v2_seq_4(A)
& v3_seq_4(A)
& v1_rcomp_1(A) ) ).
fof(t1_topreal6,axiom,
$true ).
fof(t2_topreal6,axiom,
$true ).
fof(t3_topreal6,axiom,
$true ).
fof(t4_topreal6,axiom,
$true ).
fof(t5_topreal6,axiom,
$true ).
fof(t6_topreal6,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(np__0,B) )
=> r1_xreal_0(k8_square_1(k2_xcmplx_0(A,B)),k2_xcmplx_0(k8_square_1(A),k8_square_1(B))) ) ) ) ).
fof(t7_topreal6,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(A,B) )
=> r1_xreal_0(k18_complex1(A),k18_complex1(B)) ) ) ) ).
fof(t8_topreal6,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( r1_xreal_0(A,B)
& r1_xreal_0(B,np__0) )
=> r1_xreal_0(k18_complex1(B),k18_complex1(A)) ) ) ) ).
fof(t9_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k16_rvsum_1(k1_newton(np__0,A)) = np__1 ) ).
fof(t10_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k16_rvsum_1(k1_newton(np__1,A)) = A ) ).
fof(t11_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k16_rvsum_1(k1_newton(np__2,A)) = k4_real_1(A,A) ) ).
fof(t12_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k16_rvsum_1(k1_newton(k1_nat_1(B,np__1),A)) = k4_real_1(k16_rvsum_1(k1_newton(B,A)),A) ) ) ).
fof(t13_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( B != np__0
& A = np__0 )
<=> k16_rvsum_1(k1_newton(B,A)) = np__0 ) ) ) ).
fof(t14_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,C)
=> ( A = np__0
| k16_rvsum_1(k1_newton(k5_binarith(C,B),A)) = k6_real_1(k16_rvsum_1(k1_newton(C,A)),k16_rvsum_1(k1_newton(B,A))) ) ) ) ) ) ).
fof(t15_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,C)
=> ( A = np__0
| k3_newton(A,k5_binarith(C,B)) = k6_real_1(k3_newton(A,C),k3_newton(A,B)) ) ) ) ) ) ).
fof(t16_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k11_rvsum_1(k2_finseq_4(k1_numbers,A,B)) = k2_finseq_4(k1_numbers,k7_square_1(A),k7_square_1(B)) ) ) ).
fof(t17_topreal6,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m2_finseq_1(C,k1_numbers)
=> ( ( r2_hidden(A,k4_finseq_1(k3_euclid(C)))
& B = k2_seq_1(k5_numbers,k1_numbers,C,A) )
=> k2_seq_1(k5_numbers,k1_numbers,k3_euclid(C),A) = k18_complex1(B) ) ) ) ) ).
fof(t18_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k3_euclid(k2_finseq_4(k1_numbers,A,B)) = k2_finseq_4(k1_numbers,k18_complex1(A),k18_complex1(B)) ) ) ).
fof(t19_topreal6,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ( r1_xreal_0(A,B)
& r1_xreal_0(C,D) )
=> k3_real_1(k18_complex1(k6_xcmplx_0(B,A)),k18_complex1(k6_xcmplx_0(D,C))) = k2_xcmplx_0(k6_xcmplx_0(B,A),k6_xcmplx_0(D,C)) ) ) ) ) ) ).
fof(t20_topreal6,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(B,np__0)
=> r2_hidden(A,k2_rcomp_1(k6_xcmplx_0(A,B),k2_xcmplx_0(A,B))) ) ) ) ).
fof(t21_topreal6,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(np__0,B)
=> r2_hidden(A,k1_rcomp_1(k6_xcmplx_0(A,B),k2_xcmplx_0(A,B))) ) ) ) ).
fof(t22_topreal6,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(B,A)
=> ( k4_pscomp_1(k2_rcomp_1(A,B)) = A
& k3_pscomp_1(k2_rcomp_1(A,B)) = B ) ) ) ) ).
fof(t23_topreal6,axiom,
$true ).
fof(t24_topreal6,axiom,
! [A] :
( ( v3_seq_4(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> r1_tarski(A,k1_rcomp_1(k4_pscomp_1(A),k3_pscomp_1(A))) ) ).
fof(t25_topreal6,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ( ( r1_t_0topsp(A,B)
& v1_connsp_1(A) )
=> v1_connsp_1(B) ) ) ) ).
fof(t26_topreal6,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_finset_1(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)))
=> ( r2_hidden(C,B)
=> v6_compts_1(C,A) ) )
=> v6_compts_1(k5_setfam_1(u1_struct_0(A),B),A) ) ) ) ).
fof(t27_topreal6,axiom,
$true ).
fof(t28_topreal6,axiom,
$true ).
fof(t29_topreal6,axiom,
! [A,B,C,D,E,F] :
( ( r1_tarski(A,B)
& r1_tarski(C,D) )
=> r1_tarski(k4_card_3(k4_funct_4(E,F,A,C)),k4_card_3(k4_funct_4(E,F,B,D))) ) ).
fof(t30_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
=> m1_subset_1(k4_card_3(k5_funct_4(k1_zfmisc_1(k1_numbers),np__1,np__2,A,B)),k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) ) ) ).
fof(t31_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( k5_toprns_1(np__2,k23_euclid(np__0,A)) = k18_complex1(A)
& k5_toprns_1(np__2,k23_euclid(A,np__0)) = k18_complex1(A) ) ) ).
fof(t32_topreal6,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k14_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( ( A = k16_euclid(np__2)
& A = B )
=> ( B = k2_finseq_4(k5_numbers,np__0,np__0)
& k21_euclid(B) = np__0
& k22_euclid(B) = np__0 ) ) ) ) ).
fof(t33_topreal6,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k14_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k14_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( ( A = k16_euclid(np__2)
& B = C )
=> k4_metric_1(k14_euclid(np__2),A,B) = k5_toprns_1(np__2,C) ) ) ) ) ).
fof(t34_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> k18_euclid(A,np__2,B) = k23_euclid(k4_real_1(A,k21_euclid(B)),k4_real_1(A,k22_euclid(B))) ) ) ).
fof(t35_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ~ ( B = k17_euclid(np__2,k18_euclid(k5_real_1(np__1,A),np__2,C),k18_euclid(A,np__2,D))
& B != C
& r1_xreal_0(np__0,A)
& r1_xreal_0(A,np__0) ) ) ) ) ) ).
fof(t36_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ~ ( B = k17_euclid(np__2,k18_euclid(k5_real_1(np__1,A),np__2,C),k18_euclid(A,np__2,D))
& B != D
& r1_xreal_0(A,np__1)
& r1_xreal_0(np__1,A) ) ) ) ) ) ).
fof(t37_topreal6,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(A,k3_topreal1(np__2,B,C))
=> ( A = B
| A = C
| r1_xreal_0(k21_euclid(C),k21_euclid(B))
| ( ~ r1_xreal_0(k21_euclid(A),k21_euclid(B))
& ~ r1_xreal_0(k21_euclid(C),k21_euclid(A)) ) ) ) ) ) ) ).
fof(t38_topreal6,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(A,k3_topreal1(np__2,B,C))
=> ( A = B
| A = C
| r1_xreal_0(k22_euclid(C),k22_euclid(B))
| ( ~ r1_xreal_0(k22_euclid(A),k22_euclid(B))
& ~ r1_xreal_0(k22_euclid(C),k22_euclid(A)) ) ) ) ) ) ) ).
fof(t39_topreal6,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ? [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
& ~ r1_xreal_0(k18_pscomp_1(A),k21_euclid(C))
& B != C ) ) ) ).
fof(t40_topreal6,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ? [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
& ~ r1_xreal_0(k21_euclid(C),k20_pscomp_1(A))
& B != C ) ) ) ).
fof(t41_topreal6,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ? [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
& ~ r1_xreal_0(k22_euclid(C),k19_pscomp_1(A))
& B != C ) ) ) ).
fof(t42_topreal6,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ? [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
& ~ r1_xreal_0(k21_pscomp_1(A),k22_euclid(C))
& B != C ) ) ) ).
fof(t43_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> r1_tarski(k3_topreal1(np__2,k24_pscomp_1(A),k25_pscomp_1(A)),k5_topreal1(np__2,k1_sprect_1(A))) ) ).
fof(t44_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> r1_tarski(k3_topreal1(np__2,k22_pscomp_1(A),k25_pscomp_1(A)),k5_topreal1(np__2,k1_sprect_1(A))) ) ).
fof(t45_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> r1_tarski(k3_topreal1(np__2,k22_pscomp_1(A),k23_pscomp_1(A)),k5_topreal1(np__2,k1_sprect_1(A))) ) ).
fof(t47_topreal6,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k14_euclid(np__2)))
=> ! [D] :
( v1_xreal_0(D)
=> ( ( C = A
& r2_hidden(B,k9_metric_1(k14_euclid(np__2),C,D)) )
=> ( ~ r1_xreal_0(k21_euclid(B),k6_xcmplx_0(k21_euclid(A),D))
& ~ r1_xreal_0(k2_xcmplx_0(k21_euclid(A),D),k21_euclid(B)) ) ) ) ) ) ) ).
fof(t48_topreal6,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k14_euclid(np__2)))
=> ! [D] :
( v1_xreal_0(D)
=> ( ( C = A
& r2_hidden(B,k9_metric_1(k14_euclid(np__2),C,D)) )
=> ( ~ r1_xreal_0(k22_euclid(B),k6_xcmplx_0(k22_euclid(A),D))
& ~ r1_xreal_0(k2_xcmplx_0(k22_euclid(A),D),k22_euclid(B)) ) ) ) ) ) ) ).
fof(t49_topreal6,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k14_euclid(np__2)))
=> ! [C] :
( v1_xreal_0(C)
=> ( A = B
=> r1_tarski(k4_card_3(k5_funct_4(k1_zfmisc_1(k1_numbers),np__1,np__2,k2_rcomp_1(k6_xcmplx_0(k21_euclid(A),k7_xcmplx_0(C,k9_square_1(np__2))),k2_xcmplx_0(k21_euclid(A),k7_xcmplx_0(C,k9_square_1(np__2)))),k2_rcomp_1(k6_xcmplx_0(k22_euclid(A),k7_xcmplx_0(C,k9_square_1(np__2))),k2_xcmplx_0(k22_euclid(A),k7_xcmplx_0(C,k9_square_1(np__2)))))),k9_metric_1(k14_euclid(np__2),B,C)) ) ) ) ) ).
fof(t50_topreal6,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k14_euclid(np__2)))
=> ! [C] :
( v1_xreal_0(C)
=> ( A = B
=> r1_tarski(k9_metric_1(k14_euclid(np__2),B,C),k4_card_3(k5_funct_4(k1_zfmisc_1(k1_numbers),np__1,np__2,k2_rcomp_1(k6_xcmplx_0(k21_euclid(A),C),k2_xcmplx_0(k21_euclid(A),C)),k2_rcomp_1(k6_xcmplx_0(k22_euclid(A),C),k2_xcmplx_0(k22_euclid(A),C))))) ) ) ) ) ).
fof(t51_topreal6,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k14_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [D] :
( v1_xreal_0(D)
=> ( ( C = k9_metric_1(k14_euclid(np__2),B,D)
& A = B )
=> k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k16_pscomp_1,C) = k2_rcomp_1(k6_xcmplx_0(k21_euclid(A),D),k2_xcmplx_0(k21_euclid(A),D)) ) ) ) ) ) ).
fof(t52_topreal6,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k14_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [D] :
( v1_xreal_0(D)
=> ( ( C = k9_metric_1(k14_euclid(np__2),B,D)
& A = B )
=> k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k17_pscomp_1,C) = k2_rcomp_1(k6_xcmplx_0(k22_euclid(A),D),k2_xcmplx_0(k22_euclid(A),D)) ) ) ) ) ) ).
fof(t53_topreal6,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k14_euclid(np__2)))
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [D] :
( v1_xreal_0(D)
=> ( ( C = k9_metric_1(k14_euclid(np__2),B,D)
& A = B )
=> k18_pscomp_1(C) = k6_xcmplx_0(k21_euclid(A),D) ) ) ) ) ) ).
fof(t54_topreal6,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k14_euclid(np__2)))
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [D] :
( v1_xreal_0(D)
=> ( ( C = k9_metric_1(k14_euclid(np__2),B,D)
& A = B )
=> k20_pscomp_1(C) = k2_xcmplx_0(k21_euclid(A),D) ) ) ) ) ) ).
fof(t55_topreal6,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k14_euclid(np__2)))
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [D] :
( v1_xreal_0(D)
=> ( ( C = k9_metric_1(k14_euclid(np__2),B,D)
& A = B )
=> k21_pscomp_1(C) = k6_xcmplx_0(k22_euclid(A),D) ) ) ) ) ) ).
fof(t56_topreal6,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k14_euclid(np__2)))
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [D] :
( v1_xreal_0(D)
=> ( ( C = k9_metric_1(k14_euclid(np__2),B,D)
& A = B )
=> k19_pscomp_1(C) = k2_xcmplx_0(k22_euclid(A),D) ) ) ) ) ) ).
fof(t57_topreal6,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k14_euclid(np__2)))
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( B = k9_metric_1(k14_euclid(np__2),A,C)
& v1_sppol_1(B) ) ) ) ) ).
fof(t58_topreal6,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k14_euclid(np__2)))
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( B = k9_metric_1(k14_euclid(np__2),A,C)
& v2_sppol_1(B) ) ) ) ) ).
fof(t59_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k14_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r2_hidden(C,k9_metric_1(k14_euclid(np__2),B,A))
& r2_hidden(k23_euclid(k5_real_1(k21_euclid(C),k4_real_1(np__2,A)),k22_euclid(C)),k9_metric_1(k14_euclid(np__2),B,A)) ) ) ) ) ).
fof(t60_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v6_compts_1(B,k15_euclid(np__2))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k14_euclid(np__2)))
=> ( C = k16_euclid(np__2)
=> ( r1_xreal_0(A,np__0)
| r1_tarski(B,k9_metric_1(k14_euclid(np__2),C,k3_real_1(k3_real_1(k3_real_1(k3_real_1(k18_complex1(k20_pscomp_1(B)),k18_complex1(k19_pscomp_1(B))),k18_complex1(k18_pscomp_1(B))),k18_complex1(k21_pscomp_1(B))),A))) ) ) ) ) ) ).
fof(t61_topreal6,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ( ~ r1_xreal_0(np__0,A)
=> k11_metric_1(B,C,A) = k1_xboole_0 ) ) ) ) ).
fof(t62_topreal6,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k11_metric_1(A,B,np__0) = k1_struct_0(A,B) ) ) ).
fof(t63_topreal6,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ( ~ r1_xreal_0(np__0,A)
=> k10_metric_1(B,C,A) = k1_xboole_0 ) ) ) ) ).
fof(t64_topreal6,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k10_metric_1(A,B,np__0) = k1_struct_0(A,B) ) ) ).
fof(t65_topreal6,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
=> ( D = k10_metric_1(A,B,C)
=> v4_pre_topc(D,k5_pcomps_1(A)) ) ) ) ) ) ).
fof(t66_topreal6,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k14_euclid(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( v1_xreal_0(D)
=> ( C = k10_metric_1(k14_euclid(A),B,D)
=> v4_pre_topc(C,k15_euclid(A)) ) ) ) ) ) ).
fof(t67_topreal6,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( v1_xreal_0(C)
=> v6_tbsp_1(k10_metric_1(A,B,C),A) ) ) ) ).
fof(t68_topreal6,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
=> ( D = k11_metric_1(A,B,C)
=> v4_pre_topc(D,k5_pcomps_1(A)) ) ) ) ) ) ).
fof(t69_topreal6,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k14_euclid(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( v1_xreal_0(D)
=> ( C = k11_metric_1(k14_euclid(A),B,D)
=> v4_pre_topc(C,k15_euclid(A)) ) ) ) ) ) ).
fof(t70_topreal6,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( v1_xreal_0(C)
=> v6_tbsp_1(k11_metric_1(A,B,C),A) ) ) ) ).
fof(t71_topreal6,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( v1_jordan2c(B,A)
=> v1_jordan2c(k6_pre_topc(k15_euclid(A),B),A) ) ) ) ).
fof(t72_topreal6,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ( v5_tbsp_1(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> v6_tbsp_1(B,A) ) ) ) ).
fof(t73_topreal6,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(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)))
=> ~ ( u1_struct_0(A) = k4_subset_1(u1_struct_0(A),B,C)
& ~ v5_tbsp_1(A)
& v6_tbsp_1(B,A)
& v6_tbsp_1(C,A) ) ) ) ) ).
fof(t74_topreal6,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ~ ( r1_xreal_0(np__1,A)
& u1_struct_0(k15_euclid(A)) = k4_subset_1(u1_struct_0(k15_euclid(A)),B,C)
& v1_jordan2c(B,A)
& v1_jordan2c(C,A) ) ) ) ) ).
fof(t75_topreal6,axiom,
$true ).
fof(t76_topreal6,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( ( v1_jordan2c(B,A)
& v1_jordan2c(C,A) )
=> v1_jordan2c(k4_subset_1(u1_struct_0(k15_euclid(A)),B,C),A) ) ) ) ) ).
fof(t77_topreal6,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v2_seq_4(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> k4_pscomp_1(A) = k4_pscomp_1(k7_pscomp_1(A)) ) ).
fof(t78_topreal6,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_seq_4(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> k3_pscomp_1(A) = k3_pscomp_1(k7_pscomp_1(A)) ) ).
fof(t79_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
=> ( A = B
=> ( v2_rcomp_1(A)
<=> v4_pre_topc(B,k3_topmetr) ) ) ) ) ).
fof(t80_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
=> ( A = B
=> k7_pscomp_1(A) = k6_pre_topc(k3_topmetr,B) ) ) ) ).
fof(t81_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
=> ( A = B
=> ( v1_rcomp_1(A)
<=> v6_compts_1(B,k3_topmetr) ) ) ) ) ).
fof(t82_topreal6,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( A != B
<=> k7_pscomp_1(k2_rcomp_1(A,B)) = k1_rcomp_1(A,B) ) ) ) ).
fof(t83_topreal6,axiom,
! [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),u1_struct_0(k3_topmetr))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(k3_topmetr)) )
=> ( B = C
=> ( v9_pscomp_1(B,A)
<=> v5_pre_topc(C,A,k3_topmetr) ) ) ) ) ) ).
fof(t84_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k6_borsuk_1(k3_topmetr,k3_topmetr)),u1_struct_0(k15_euclid(np__2)))
& m2_relset_1(C,u1_struct_0(k6_borsuk_1(k3_topmetr,k3_topmetr)),u1_struct_0(k15_euclid(np__2))) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> k1_funct_1(C,k4_tarski(D,E)) = k2_finseq_4(k1_numbers,D,E) ) )
=> k4_pre_topc(k6_borsuk_1(k3_topmetr,k3_topmetr),k15_euclid(np__2),C,k2_zfmisc_1(A,B)) = k4_card_3(k5_funct_4(k1_zfmisc_1(k1_numbers),np__1,np__2,A,B)) ) ) ) ) ).
fof(t85_topreal6,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,u1_struct_0(k6_borsuk_1(k3_topmetr,k3_topmetr)),u1_struct_0(k15_euclid(np__2)))
& m2_relset_1(A,u1_struct_0(k6_borsuk_1(k3_topmetr,k3_topmetr)),u1_struct_0(k15_euclid(np__2))) )
=> ( ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> k1_funct_1(A,k4_tarski(B,C)) = k2_finseq_4(k1_numbers,B,C) ) )
=> v3_tops_2(A,k6_borsuk_1(k3_topmetr,k3_topmetr),k15_euclid(np__2)) ) ) ).
fof(t86_topreal6,axiom,
r1_t_0topsp(k6_borsuk_1(k3_topmetr,k3_topmetr),k15_euclid(np__2)) ).
fof(t87_topreal6,axiom,
! [A] :
( ( v1_rcomp_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_rcomp_1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ( v6_compts_1(k4_card_3(k5_funct_4(k1_zfmisc_1(k1_numbers),np__1,np__2,A,B)),k15_euclid(np__2))
& m1_subset_1(k4_card_3(k5_funct_4(k1_zfmisc_1(k1_numbers),np__1,np__2,A,B)),k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) ) ) ) ).
fof(t88_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( ( v1_jordan2c(A,np__2)
& v4_pre_topc(A,k15_euclid(np__2)) )
=> v6_compts_1(A,k15_euclid(np__2)) ) ) ).
fof(t89_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( v1_jordan2c(A,np__2)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k15_euclid(np__2)),k1_numbers)
& v9_pscomp_1(B,k15_euclid(np__2))
& m2_relset_1(B,u1_struct_0(k15_euclid(np__2)),k1_numbers) )
=> r1_tarski(k7_pscomp_1(k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,B,A)),k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,B,k6_pre_topc(k15_euclid(np__2),A))) ) ) ) ).
fof(t90_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> r1_tarski(k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k16_pscomp_1,k6_pre_topc(k15_euclid(np__2),A)),k7_pscomp_1(k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k16_pscomp_1,A))) ) ).
fof(t91_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> r1_tarski(k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k17_pscomp_1,k6_pre_topc(k15_euclid(np__2),A)),k7_pscomp_1(k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k17_pscomp_1,A))) ) ).
fof(t92_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( v1_jordan2c(A,np__2)
=> k7_pscomp_1(k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k16_pscomp_1,A)) = k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k16_pscomp_1,k6_pre_topc(k15_euclid(np__2),A)) ) ) ).
fof(t93_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( v1_jordan2c(A,np__2)
=> k7_pscomp_1(k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k17_pscomp_1,A)) = k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k17_pscomp_1,k6_pre_topc(k15_euclid(np__2),A)) ) ) ).
fof(t94_topreal6,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( v1_jordan2c(A,np__2)
=> k18_pscomp_1(A) = k18_pscomp_1(k6_pre_topc(k15_euclid(np__2),A)) ) ) ).
fof(t95_topreal6,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( v1_jordan2c(A,np__2)
=> k20_pscomp_1(A) = k20_pscomp_1(k6_pre_topc(k15_euclid(np__2),A)) ) ) ).
fof(t96_topreal6,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( v1_jordan2c(A,np__2)
=> k19_pscomp_1(A) = k19_pscomp_1(k6_pre_topc(k15_euclid(np__2),A)) ) ) ).
fof(t97_topreal6,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( v1_jordan2c(A,np__2)
=> k21_pscomp_1(A) = k21_pscomp_1(k6_pre_topc(k15_euclid(np__2),A)) ) ) ).
fof(t46_topreal6,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( ~ v1_xboole_0(a_1_0_topreal6(A))
& v2_connsp_1(a_1_0_topreal6(A),k15_euclid(np__2))
& v1_jordan1(a_1_0_topreal6(A),np__2)
& m1_subset_1(a_1_0_topreal6(A),k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) ) ) ).
fof(fraenkel_a_1_0_topreal6,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( r2_hidden(A,a_1_0_topreal6(B))
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
& A = C
& ~ r1_xreal_0(k18_pscomp_1(B),k21_euclid(C)) ) ) ) ).
%------------------------------------------------------------------------------