SET007 Axioms: SET007+353.ax
%------------------------------------------------------------------------------
% File : SET007+353 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Jordan's Property for Certain Subsets of the Plane
% 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 : jordan1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 66 ( 3 unt; 0 def)
% Number of atoms : 549 ( 96 equ)
% Maximal formula atoms : 21 ( 8 avg)
% Number of connectives : 533 ( 50 ~; 20 |; 192 &)
% ( 16 <=>; 255 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 12 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of predicates : 21 ( 19 usr; 1 prp; 0-3 aty)
% Number of functors : 38 ( 38 usr; 7 con; 0-4 aty)
% Number of variables : 287 ( 256 !; 31 ?)
% 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_jordan1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> u1_struct_0(k3_pre_topc(A,B)) = B ) ) ).
fof(t2_jordan1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( ~ v3_struct_0(D)
& v2_pre_topc(D)
& l1_pre_topc(D)
& v1_connsp_1(D)
& ? [E] :
( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(D),u1_struct_0(A))
& m2_relset_1(E,u1_struct_0(D),u1_struct_0(A))
& v5_pre_topc(E,D,A)
& r2_hidden(B,k2_relat_1(E))
& r2_hidden(C,k2_relat_1(E)) ) ) ) )
=> v1_connsp_1(A) ) ) ).
fof(t3_jordan1,axiom,
$true ).
fof(t4_jordan1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k22_borsuk_1),u1_struct_0(A))
& m2_relset_1(D,u1_struct_0(k22_borsuk_1),u1_struct_0(A))
& v5_pre_topc(D,k22_borsuk_1,A)
& B = k1_funct_1(D,np__0)
& C = k1_funct_1(D,np__1) ) ) )
=> v1_connsp_1(A) ) ) ).
fof(t5_jordan1,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,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_hidden(C,B)
& r2_hidden(D,B)
& C != D
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_pre_topc(A,B)))
& m2_relset_1(E,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_pre_topc(A,B))) )
=> ~ ( v5_pre_topc(E,k22_borsuk_1,k3_pre_topc(A,B))
& C = k1_funct_1(E,np__0)
& D = k1_funct_1(E,np__1) ) ) ) ) )
=> v2_connsp_1(B,A) ) ) ) ).
fof(t6_jordan1,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)))
=> ( ( v2_connsp_1(B,A)
& v2_connsp_1(C,A) )
=> ( r1_xboole_0(B,C)
| v2_connsp_1(k4_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) ) ).
fof(t7_jordan1,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] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v2_connsp_1(B,A)
& v2_connsp_1(C,A)
& v2_connsp_1(D,A) )
=> ( r1_xboole_0(B,C)
| r1_xboole_0(C,D)
| v2_connsp_1(k4_subset_1(u1_struct_0(A),k4_subset_1(u1_struct_0(A),B,C),D),A) ) ) ) ) ) ) ).
fof(t8_jordan1,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] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v2_connsp_1(B,A)
& v2_connsp_1(C,A)
& v2_connsp_1(D,A)
& v2_connsp_1(E,A) )
=> ( r1_xboole_0(B,C)
| r1_xboole_0(C,D)
| r1_xboole_0(D,E)
| v2_connsp_1(k4_subset_1(u1_struct_0(A),k4_subset_1(u1_struct_0(A),k4_subset_1(u1_struct_0(A),B,C),D),E),A) ) ) ) ) ) ) ) ).
fof(d1_jordan1,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_jordan1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r1_tarski(k3_topreal1(A,C,D),B) ) ) ) ) ) ) ).
fof(t9_jordan1,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_jordan1(B,A)
=> v2_connsp_1(B,k15_euclid(A)) ) ) ) ).
fof(t10_jordan1,axiom,
$true ).
fof(t11_jordan1,axiom,
$true ).
fof(d2_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( v2_jordan1(A)
<=> ( k3_subset_1(u1_struct_0(k15_euclid(np__2)),A) != k1_xboole_0
& ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& ? [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& k3_subset_1(u1_struct_0(k15_euclid(np__2)),A) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),B,C)
& r1_xboole_0(B,C)
& k6_subset_1(u1_struct_0(k15_euclid(np__2)),k6_pre_topc(k15_euclid(np__2),B),B) = k6_subset_1(u1_struct_0(k15_euclid(np__2)),k6_pre_topc(k15_euclid(np__2),C),C)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),A)))))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),A)))))
=> ( ( D = B
& E = C )
=> ( r3_connsp_1(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),A)),D)
& r3_connsp_1(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),A)),E) ) ) ) ) ) ) ) ) ) ).
fof(t47_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( v2_jordan1(A)
=> ( k3_subset_1(u1_struct_0(k15_euclid(np__2)),A) != k1_xboole_0
& ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& ? [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),A)))))
& ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),A)))))
& k3_subset_1(u1_struct_0(k15_euclid(np__2)),A) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),B,C)
& r1_xboole_0(B,C)
& k6_subset_1(u1_struct_0(k15_euclid(np__2)),k6_pre_topc(k15_euclid(np__2),B),B) = k6_subset_1(u1_struct_0(k15_euclid(np__2)),k6_pre_topc(k15_euclid(np__2),C),C)
& D = B
& E = C
& r3_connsp_1(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),A)),D)
& r3_connsp_1(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),A)),E)
& ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),A)))))
=> ~ ( r3_connsp_1(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),A)),F)
& F != D
& F != E ) ) ) ) ) ) ) ) ) ).
fof(t12_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> a_4_0_jordan1(A,B,C,D) = k3_xboole_0(k3_xboole_0(k3_xboole_0(a_1_0_jordan1(A),a_1_1_jordan1(B)),a_1_2_jordan1(C)),a_1_3_jordan1(D)) ) ) ) ) ).
fof(t13_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> a_4_1_jordan1(A,B,C,D) = k2_xboole_0(k2_xboole_0(k2_xboole_0(a_1_1_jordan1(A),a_1_3_jordan1(C)),a_1_0_jordan1(B)),a_1_2_jordan1(D)) ) ) ) ) ).
fof(t14_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( E = a_4_2_jordan1(A,B,C,D)
=> v1_jordan1(E,np__2) ) ) ) ) ) ) ).
fof(t15_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( E = a_4_2_jordan1(A,B,C,D)
=> v2_connsp_1(E,k15_euclid(np__2)) ) ) ) ) ) ) ).
fof(t16_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( B = a_1_0_jordan1(A)
=> v1_jordan1(B,np__2) ) ) ) ).
fof(t17_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( B = a_1_0_jordan1(A)
=> v2_connsp_1(B,k15_euclid(np__2)) ) ) ) ).
fof(t18_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( B = a_1_1_jordan1(A)
=> v1_jordan1(B,np__2) ) ) ) ).
fof(t19_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( B = a_1_1_jordan1(A)
=> v2_connsp_1(B,k15_euclid(np__2)) ) ) ) ).
fof(t20_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( B = a_1_2_jordan1(A)
=> v1_jordan1(B,np__2) ) ) ) ).
fof(t21_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( B = a_1_2_jordan1(A)
=> v2_connsp_1(B,k15_euclid(np__2)) ) ) ) ).
fof(t22_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( B = a_1_3_jordan1(A)
=> v1_jordan1(B,np__2) ) ) ) ).
fof(t23_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( B = a_1_3_jordan1(A)
=> v2_connsp_1(B,k15_euclid(np__2)) ) ) ) ).
fof(t24_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( E = a_4_3_jordan1(A,B,C,D)
=> v2_connsp_1(E,k15_euclid(np__2)) ) ) ) ) ) ) ).
fof(t25_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( B = a_1_0_jordan1(A)
=> v3_pre_topc(B,k15_euclid(np__2)) ) ) ) ).
fof(t26_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( B = a_1_1_jordan1(A)
=> v3_pre_topc(B,k15_euclid(np__2)) ) ) ) ).
fof(t27_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( B = a_1_2_jordan1(A)
=> v3_pre_topc(B,k15_euclid(np__2)) ) ) ) ).
fof(t28_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( B = a_1_3_jordan1(A)
=> v3_pre_topc(B,k15_euclid(np__2)) ) ) ) ).
fof(t29_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( E = a_4_2_jordan1(A,B,C,D)
=> v3_pre_topc(E,k15_euclid(np__2)) ) ) ) ) ) ) ).
fof(t30_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( E = a_4_3_jordan1(A,B,C,D)
=> v3_pre_topc(E,k15_euclid(np__2)) ) ) ) ) ) ) ).
fof(t31_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( ( E = a_4_2_jordan1(A,B,C,D)
& F = a_4_3_jordan1(A,B,C,D) )
=> r1_xboole_0(E,F) ) ) ) ) ) ) ) ).
fof(t32_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> a_4_4_jordan1(A,B,C,D) = a_4_0_jordan1(A,B,C,D) ) ) ) ) ).
fof(t33_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> a_4_5_jordan1(A,B,C,D) = a_4_1_jordan1(A,B,C,D) ) ) ) ) ).
fof(t34_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> m1_subset_1(a_4_4_jordan1(A,B,C,D),k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) ) ) ) ) ).
fof(t35_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> m1_subset_1(a_4_5_jordan1(A,B,C,D),k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) ) ) ) ) ).
fof(t36_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( E = a_4_6_jordan1(A,B,C,D)
=> v2_connsp_1(E,k15_euclid(np__2)) ) ) ) ) ) ) ).
fof(t37_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( E = a_4_7_jordan1(A,B,C,D)
=> v2_connsp_1(E,k15_euclid(np__2)) ) ) ) ) ) ) ).
fof(t38_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( E = a_4_6_jordan1(A,B,C,D)
=> v3_pre_topc(E,k15_euclid(np__2)) ) ) ) ) ) ) ).
fof(t39_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( E = a_4_7_jordan1(A,B,C,D)
=> v3_pre_topc(E,k15_euclid(np__2)) ) ) ) ) ) ) ).
fof(t40_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( ( E = a_4_6_jordan1(A,B,C,D)
& F = a_4_7_jordan1(A,B,C,D) )
=> r1_xboole_0(E,F) ) ) ) ) ) ) ) ).
fof(t41_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( ( E = a_4_8_jordan1(A,B,C,D)
& F = a_4_6_jordan1(A,B,C,D)
& G = a_4_7_jordan1(A,B,C,D) )
=> ( r1_xreal_0(C,A)
| r1_xreal_0(D,B)
| ( k3_subset_1(u1_struct_0(k15_euclid(np__2)),E) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),F,G)
& k3_subset_1(u1_struct_0(k15_euclid(np__2)),E) != k1_xboole_0
& r1_xboole_0(F,G)
& ! [H] :
( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),E)))))
=> ! [I] :
( m1_subset_1(I,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),E)))))
=> ( ( H = F
& I = G )
=> ( r3_connsp_1(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),E)),H)
& r3_connsp_1(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),E)),I) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t42_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( ( E = a_4_8_jordan1(A,B,C,D)
& F = a_4_6_jordan1(A,B,C,D)
& G = a_4_7_jordan1(A,B,C,D) )
=> ( r1_xreal_0(C,A)
| r1_xreal_0(D,B)
| ( E = k6_subset_1(u1_struct_0(k15_euclid(np__2)),k6_pre_topc(k15_euclid(np__2),F),F)
& E = k6_subset_1(u1_struct_0(k15_euclid(np__2)),k6_pre_topc(k15_euclid(np__2),G),G) ) ) ) ) ) ) ) ) ) ) ).
fof(t43_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( ( E = a_4_9_jordan1(A,B,C,D)
& F = a_4_4_jordan1(A,B,C,D) )
=> r1_tarski(F,k2_pre_topc(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),E)))) ) ) ) ) ) ) ) ).
fof(t44_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( ( E = a_4_9_jordan1(A,B,C,D)
& F = a_4_4_jordan1(A,B,C,D) )
=> m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),E))))) ) ) ) ) ) ) ) ).
fof(t45_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( ( E = a_4_9_jordan1(A,B,C,D)
& F = a_4_5_jordan1(A,B,C,D) )
=> ( r1_xreal_0(B,A)
| r1_xreal_0(D,C)
| r1_tarski(F,k2_pre_topc(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),E)))) ) ) ) ) ) ) ) ) ).
fof(t46_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( ( E = a_4_9_jordan1(A,B,C,D)
& F = a_4_5_jordan1(A,B,C,D) )
=> ( r1_xreal_0(B,A)
| r1_xreal_0(D,C)
| m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),E))))) ) ) ) ) ) ) ) ) ).
fof(t48_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( E = a_4_9_jordan1(A,B,C,D)
=> ( r1_xreal_0(B,A)
| r1_xreal_0(D,C)
| v2_jordan1(E) ) ) ) ) ) ) ) ).
fof(t49_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( ( E = a_4_8_jordan1(A,B,C,D)
& F = a_4_7_jordan1(A,B,C,D) )
=> ( r1_xreal_0(C,A)
| r1_xreal_0(D,B)
| k6_pre_topc(k15_euclid(np__2),F) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),E,F) ) ) ) ) ) ) ) ) ).
fof(t50_jordan1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( ( E = a_4_8_jordan1(A,B,C,D)
& F = a_4_6_jordan1(A,B,C,D) )
=> ( r1_xreal_0(C,A)
| r1_xreal_0(D,B)
| k6_pre_topc(k15_euclid(np__2),F) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),E,F) ) ) ) ) ) ) ) ) ).
fof(fraenkel_a_4_0_jordan1,axiom,
! [A,B,C,D,E] :
( ( m1_subset_1(B,k1_numbers)
& m1_subset_1(C,k1_numbers)
& m1_subset_1(D,k1_numbers)
& m1_subset_1(E,k1_numbers) )
=> ( r2_hidden(A,a_4_0_jordan1(B,C,D,E))
<=> ? [F,G] :
( m1_subset_1(F,k1_numbers)
& m1_subset_1(G,k1_numbers)
& A = k23_euclid(F,G)
& ~ r1_xreal_0(F,B)
& ~ r1_xreal_0(C,F)
& ~ r1_xreal_0(G,D)
& ~ r1_xreal_0(E,G) ) ) ) ).
fof(fraenkel_a_1_0_jordan1,axiom,
! [A,B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_hidden(A,a_1_0_jordan1(B))
<=> ? [C,D] :
( m1_subset_1(C,k1_numbers)
& m1_subset_1(D,k1_numbers)
& A = k23_euclid(C,D)
& ~ r1_xreal_0(C,B) ) ) ) ).
fof(fraenkel_a_1_1_jordan1,axiom,
! [A,B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_hidden(A,a_1_1_jordan1(B))
<=> ? [C,D] :
( m1_subset_1(C,k1_numbers)
& m1_subset_1(D,k1_numbers)
& A = k23_euclid(C,D)
& ~ r1_xreal_0(B,C) ) ) ) ).
fof(fraenkel_a_1_2_jordan1,axiom,
! [A,B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_hidden(A,a_1_2_jordan1(B))
<=> ? [C,D] :
( m1_subset_1(C,k1_numbers)
& m1_subset_1(D,k1_numbers)
& A = k23_euclid(C,D)
& ~ r1_xreal_0(D,B) ) ) ) ).
fof(fraenkel_a_1_3_jordan1,axiom,
! [A,B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_hidden(A,a_1_3_jordan1(B))
<=> ? [C,D] :
( m1_subset_1(C,k1_numbers)
& m1_subset_1(D,k1_numbers)
& A = k23_euclid(C,D)
& ~ r1_xreal_0(B,D) ) ) ) ).
fof(fraenkel_a_4_1_jordan1,axiom,
! [A,B,C,D,E] :
( ( m1_subset_1(B,k1_numbers)
& m1_subset_1(C,k1_numbers)
& m1_subset_1(D,k1_numbers)
& m1_subset_1(E,k1_numbers) )
=> ( r2_hidden(A,a_4_1_jordan1(B,C,D,E))
<=> ? [F,G] :
( m1_subset_1(F,k1_numbers)
& m1_subset_1(G,k1_numbers)
& A = k23_euclid(F,G)
& ~ ( r1_xreal_0(B,F)
& r1_xreal_0(F,C)
& r1_xreal_0(D,G)
& r1_xreal_0(G,E) ) ) ) ) ).
fof(fraenkel_a_4_2_jordan1,axiom,
! [A,B,C,D,E] :
( ( m1_subset_1(B,k1_numbers)
& m1_subset_1(C,k1_numbers)
& m1_subset_1(D,k1_numbers)
& m1_subset_1(E,k1_numbers) )
=> ( r2_hidden(A,a_4_2_jordan1(B,C,D,E))
<=> ? [F,G] :
( m1_subset_1(F,k1_numbers)
& m1_subset_1(G,k1_numbers)
& A = k23_euclid(F,G)
& ~ r1_xreal_0(F,B)
& ~ r1_xreal_0(D,F)
& ~ r1_xreal_0(G,C)
& ~ r1_xreal_0(E,G) ) ) ) ).
fof(fraenkel_a_4_3_jordan1,axiom,
! [A,B,C,D,E] :
( ( m1_subset_1(B,k1_numbers)
& m1_subset_1(C,k1_numbers)
& m1_subset_1(D,k1_numbers)
& m1_subset_1(E,k1_numbers) )
=> ( r2_hidden(A,a_4_3_jordan1(B,C,D,E))
<=> ? [F,G] :
( m1_subset_1(F,k1_numbers)
& m1_subset_1(G,k1_numbers)
& A = k23_euclid(F,G)
& ~ ( r1_xreal_0(B,F)
& r1_xreal_0(F,D)
& r1_xreal_0(C,G)
& r1_xreal_0(G,E) ) ) ) ) ).
fof(fraenkel_a_4_4_jordan1,axiom,
! [A,B,C,D,E] :
( ( m1_subset_1(B,k1_numbers)
& m1_subset_1(C,k1_numbers)
& m1_subset_1(D,k1_numbers)
& m1_subset_1(E,k1_numbers) )
=> ( r2_hidden(A,a_4_4_jordan1(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
& A = F
& ~ r1_xreal_0(k21_euclid(F),B)
& ~ r1_xreal_0(C,k21_euclid(F))
& ~ r1_xreal_0(k22_euclid(F),D)
& ~ r1_xreal_0(E,k22_euclid(F)) ) ) ) ).
fof(fraenkel_a_4_5_jordan1,axiom,
! [A,B,C,D,E] :
( ( m1_subset_1(B,k1_numbers)
& m1_subset_1(C,k1_numbers)
& m1_subset_1(D,k1_numbers)
& m1_subset_1(E,k1_numbers) )
=> ( r2_hidden(A,a_4_5_jordan1(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
& A = F
& ~ ( r1_xreal_0(B,k21_euclid(F))
& r1_xreal_0(k21_euclid(F),C)
& r1_xreal_0(D,k22_euclid(F))
& r1_xreal_0(k22_euclid(F),E) ) ) ) ) ).
fof(fraenkel_a_4_6_jordan1,axiom,
! [A,B,C,D,E] :
( ( m1_subset_1(B,k1_numbers)
& m1_subset_1(C,k1_numbers)
& m1_subset_1(D,k1_numbers)
& m1_subset_1(E,k1_numbers) )
=> ( r2_hidden(A,a_4_6_jordan1(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
& A = F
& ~ r1_xreal_0(k21_euclid(F),B)
& ~ r1_xreal_0(D,k21_euclid(F))
& ~ r1_xreal_0(k22_euclid(F),C)
& ~ r1_xreal_0(E,k22_euclid(F)) ) ) ) ).
fof(fraenkel_a_4_7_jordan1,axiom,
! [A,B,C,D,E] :
( ( m1_subset_1(B,k1_numbers)
& m1_subset_1(C,k1_numbers)
& m1_subset_1(D,k1_numbers)
& m1_subset_1(E,k1_numbers) )
=> ( r2_hidden(A,a_4_7_jordan1(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
& A = F
& ~ ( r1_xreal_0(B,k21_euclid(F))
& r1_xreal_0(k21_euclid(F),D)
& r1_xreal_0(C,k22_euclid(F))
& r1_xreal_0(k22_euclid(F),E) ) ) ) ) ).
fof(fraenkel_a_4_8_jordan1,axiom,
! [A,B,C,D,E] :
( ( m1_subset_1(B,k1_numbers)
& m1_subset_1(C,k1_numbers)
& m1_subset_1(D,k1_numbers)
& m1_subset_1(E,k1_numbers) )
=> ( r2_hidden(A,a_4_8_jordan1(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
& A = F
& ~ ( ~ ( k21_euclid(F) = B
& r1_xreal_0(k22_euclid(F),E)
& r1_xreal_0(C,k22_euclid(F)) )
& ~ ( r1_xreal_0(k21_euclid(F),D)
& r1_xreal_0(B,k21_euclid(F))
& k22_euclid(F) = E )
& ~ ( r1_xreal_0(k21_euclid(F),D)
& r1_xreal_0(B,k21_euclid(F))
& k22_euclid(F) = C )
& ~ ( k21_euclid(F) = D
& r1_xreal_0(k22_euclid(F),E)
& r1_xreal_0(C,k22_euclid(F)) ) ) ) ) ) ).
fof(fraenkel_a_4_9_jordan1,axiom,
! [A,B,C,D,E] :
( ( m1_subset_1(B,k1_numbers)
& m1_subset_1(C,k1_numbers)
& m1_subset_1(D,k1_numbers)
& m1_subset_1(E,k1_numbers) )
=> ( r2_hidden(A,a_4_9_jordan1(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
& A = F
& ~ ( ~ ( k21_euclid(F) = B
& r1_xreal_0(k22_euclid(F),E)
& r1_xreal_0(D,k22_euclid(F)) )
& ~ ( r1_xreal_0(k21_euclid(F),C)
& r1_xreal_0(B,k21_euclid(F))
& k22_euclid(F) = E )
& ~ ( r1_xreal_0(k21_euclid(F),C)
& r1_xreal_0(B,k21_euclid(F))
& k22_euclid(F) = D )
& ~ ( k21_euclid(F) = C
& r1_xreal_0(k22_euclid(F),E)
& r1_xreal_0(D,k22_euclid(F)) ) ) ) ) ) ).
%------------------------------------------------------------------------------