SET007 Axioms: SET007+723.ax
%------------------------------------------------------------------------------
% File : SET007+723 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On the General Position of Special Polygons
% 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 : jordan12 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 41 ( 2 unt; 0 def)
% Number of atoms : 506 ( 45 equ)
% Maximal formula atoms : 47 ( 12 avg)
% Number of connectives : 557 ( 92 ~; 12 |; 297 &)
% ( 6 <=>; 150 =>; 0 <=; 0 <~>)
% Maximal formula depth : 33 ( 12 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of predicates : 33 ( 31 usr; 1 prp; 0-3 aty)
% Number of functors : 39 ( 39 usr; 5 con; 0-4 aty)
% Number of variables : 148 ( 137 !; 11 ?)
% 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_jordan12,axiom,
! [A] :
( m1_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( v3_topreal1(A)
=> v1_goboard5(A) ) ) ).
fof(t1_jordan12,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(A,np__1)
& r1_xreal_0(k5_binarith(A,np__1),np__0) ) ) ).
fof(t2_jordan12,axiom,
$true ).
fof(t3_jordan12,axiom,
~ v1_abian(np__1) ).
fof(t4_jordan12,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,C)
& r1_xreal_0(k1_nat_1(C,np__1),k3_finseq_1(B)) )
=> ( r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(A)),B,C),k2_relat_1(B))
& r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(A)),B,k1_nat_1(C,np__1)),k2_relat_1(B)) ) ) ) ) ) ).
fof(t5_jordan12,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( ( v2_topreal1(k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B))
& v1_goboard5(k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B))
& r1_xreal_0(np__2,k3_finseq_1(B)) )
=> ( v2_topreal1(A)
& v3_topreal1(A) ) ) ) ) ).
fof(t6_jordan12,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> r1_tarski(k5_topreal1(np__2,A),k5_topreal1(np__2,k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B))) ) ) ).
fof(d1_jordan12,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(A)))
=> ( r1_jordan12(A,B,C)
<=> ( r1_xboole_0(k5_topreal1(A,B),k2_relat_1(C))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,D)
=> ( r1_xreal_0(k3_finseq_1(C),D)
| v1_realset1(k5_subset_1(u1_struct_0(k15_euclid(A)),k5_topreal1(A,B),k4_topreal1(A,C,D))) ) ) ) ) ) ) ) ) ).
fof(d2_jordan12,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(A)))
=> ( r2_jordan12(A,B,C)
<=> ( r1_jordan12(A,B,C)
& r1_jordan12(A,C,B) ) ) ) ) ) ).
fof(t7_jordan12,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( r2_jordan12(np__2,B,C)
=> ! [D] :
( m2_finseq_1(D,u1_struct_0(k15_euclid(np__2)))
=> ( D = k7_relat_1(C,k2_finseq_1(A))
=> r2_jordan12(np__2,B,D) ) ) ) ) ) ) ).
fof(t8_jordan12,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m2_finseq_1(D,u1_struct_0(k15_euclid(np__2)))
=> ( r2_jordan12(np__2,k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B),k4_graph_2(u1_struct_0(k15_euclid(np__2)),C,D))
=> r2_jordan12(np__2,k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B),C) ) ) ) ) ) ).
fof(t9_jordan12,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(C))
& r2_jordan12(np__2,B,C) )
=> ( r2_hidden(k1_funct_1(C,A),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,B)))
& r2_hidden(k1_funct_1(C,k1_nat_1(A,np__1)),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,B))) ) ) ) ) ) ).
fof(t10_jordan12,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( r2_jordan12(np__2,A,B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,C)
& r1_xreal_0(k1_nat_1(C,np__1),k3_finseq_1(A))
& r1_xreal_0(np__1,D)
& r1_xreal_0(k1_nat_1(D,np__1),k3_finseq_1(B)) )
=> v1_realset1(k5_subset_1(u1_struct_0(k15_euclid(np__2)),k4_topreal1(np__2,A,C),k4_topreal1(np__2,B,D))) ) ) ) ) ) ) ).
fof(t12_jordan12,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( r2_jordan12(np__2,A,B)
=> v1_finset_1(k5_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A),k5_topreal1(np__2,B))) ) ) ) ).
fof(t13_jordan12,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( r2_jordan12(np__2,A,B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> v1_finset_1(k5_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A),k4_topreal1(np__2,B,C))) ) ) ) ) ).
fof(t14_jordan12,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& ~ v5_seqm_3(A)
& v1_topreal1(A)
& v2_topreal1(A)
& v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(A)
& v2_goboard5(A)
& m2_finseq_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)))
=> ~ ( r1_xboole_0(k3_topreal1(np__2,B,C),k5_topreal1(np__2,A))
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ~ ( r4_connsp_1(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)),D)
& r2_hidden(B,D)
& r2_hidden(C,D) ) ) ) ) ) ) ).
fof(t15_jordan12,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(C)
& ~ v5_seqm_3(C)
& v1_topreal1(C)
& v2_topreal1(C)
& v1_finseq_6(C,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(C)
& v2_goboard5(C)
& m2_finseq_1(C,u1_struct_0(k15_euclid(np__2))) )
=> ( ~ ( ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& r4_connsp_1(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,C)),D)
& r2_hidden(A,D)
& r2_hidden(B,D) )
& ~ ( r2_hidden(A,k3_goboard9(C))
& r2_hidden(B,k3_goboard9(C)) )
& ~ ( r2_hidden(A,k2_goboard9(C))
& r2_hidden(B,k2_goboard9(C)) ) )
& ~ ( ( ( r2_hidden(A,k3_goboard9(C))
& r2_hidden(B,k3_goboard9(C)) )
| ( r2_hidden(A,k2_goboard9(C))
& r2_hidden(B,k2_goboard9(C)) ) )
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ~ ( r4_connsp_1(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,C)),D)
& r2_hidden(A,D)
& r2_hidden(B,D) ) ) ) ) ) ).
fof(t16_jordan12,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(C)
& ~ v5_seqm_3(C)
& v1_topreal1(C)
& v2_topreal1(C)
& v1_finseq_6(C,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(C)
& v2_goboard5(C)
& m2_finseq_1(C,u1_struct_0(k15_euclid(np__2))) )
=> ( ( r2_hidden(A,k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,C)))
& r2_hidden(B,k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,C)))
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ~ ( r4_connsp_1(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,C)),D)
& r2_hidden(A,D)
& r2_hidden(B,D) ) ) )
<=> ( ( r2_hidden(A,k2_goboard9(C))
& r2_hidden(B,k3_goboard9(C)) )
| ( r2_hidden(A,k3_goboard9(C))
& r2_hidden(B,k2_goboard9(C)) ) ) ) ) ).
fof(t17_jordan12,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& ~ v5_seqm_3(A)
& v1_topreal1(A)
& v2_topreal1(A)
& v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(A)
& v2_goboard5(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ! [B,C,D] :
~ ( ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& r4_connsp_1(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)),E)
& r2_hidden(B,E)
& r2_hidden(C,E) )
& ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& r4_connsp_1(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)),E)
& r2_hidden(C,E)
& r2_hidden(D,E) )
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ~ ( r4_connsp_1(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)),E)
& r2_hidden(B,E)
& r2_hidden(D,E) ) ) ) ) ).
fof(t18_jordan12,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& ~ v5_seqm_3(A)
& v1_topreal1(A)
& v2_topreal1(A)
& v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(A)
& v2_goboard5(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ! [B,C,D] :
~ ( r2_hidden(B,k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)))
& r2_hidden(C,k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)))
& r2_hidden(D,k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)))
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ~ ( r4_connsp_1(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)),E)
& r2_hidden(B,E)
& r2_hidden(C,E) ) )
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ~ ( r4_connsp_1(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)),E)
& r2_hidden(C,E)
& r2_hidden(D,E) ) )
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ~ ( r4_connsp_1(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)),E)
& r2_hidden(B,E)
& r2_hidden(D,E) ) ) ) ) ).
fof(t19_jordan12,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_relat_1(B)
& v1_matrix_1(B)
& v3_goboard1(B)
& v4_goboard1(B)
& v5_goboard1(B)
& v6_goboard1(B)
& m2_finseq_1(B,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r1_xreal_0(A,k3_finseq_1(B))
=> v1_jordan1(k1_goboard5(B,A),np__2) ) ) ) ).
fof(t20_jordan12,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_relat_1(B)
& v1_matrix_1(B)
& v3_goboard1(B)
& v4_goboard1(B)
& v5_goboard1(B)
& v6_goboard1(B)
& m2_finseq_1(B,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r1_xreal_0(A,k1_matrix_1(B))
=> v1_jordan1(k2_goboard5(B,A),np__2) ) ) ) ).
fof(t21_jordan12,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v3_relat_1(C)
& v1_matrix_1(C)
& v3_goboard1(C)
& v4_goboard1(C)
& v5_goboard1(C)
& v6_goboard1(C)
& m2_finseq_1(C,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( ( r1_xreal_0(A,k3_finseq_1(C))
& r1_xreal_0(B,k1_matrix_1(C)) )
=> v1_jordan1(k3_goboard5(C,A,B),np__2) ) ) ) ) ).
fof(t22_jordan12,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& ~ v5_seqm_3(A)
& v1_topreal1(A)
& v2_topreal1(A)
& v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(A)
& v2_goboard5(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,B)
& r1_xreal_0(k1_nat_1(B,np__1),k3_finseq_1(A)) )
=> v1_jordan1(k5_goboard5(A,B),np__2) ) ) ) ).
fof(t23_jordan12,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& ~ v5_seqm_3(A)
& v1_topreal1(A)
& v2_topreal1(A)
& v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(A)
& v2_goboard5(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,B)
& r1_xreal_0(k1_nat_1(B,np__1),k3_finseq_1(A)) )
=> ( v1_jordan1(k4_gobrd13(A,k3_goboard2(A),B),np__2)
& v1_jordan1(k3_gobrd13(A,k3_goboard2(A),B),np__2) ) ) ) ) ).
fof(t24_jordan12,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] :
( ( ~ v1_xboole_0(C)
& ~ v5_seqm_3(C)
& v1_topreal1(C)
& v2_topreal1(C)
& v1_finseq_6(C,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(C)
& v2_goboard5(C)
& m2_finseq_1(C,u1_struct_0(k15_euclid(np__2))) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r2_hidden(D,k3_topreal1(np__2,A,B))
& ? [E] : k5_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,C),k3_topreal1(np__2,A,B)) = k1_tarski(E)
& ~ r2_hidden(D,k5_topreal1(np__2,C))
& ~ r1_xboole_0(k5_topreal1(np__2,C),k3_topreal1(np__2,A,D))
& ~ r1_xboole_0(k5_topreal1(np__2,C),k3_topreal1(np__2,D,B)) ) ) ) ) ) ).
fof(t25_jordan12,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)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ( ( v2_sppol_1(k3_topreal1(np__2,A,B))
& v2_sppol_1(k3_topreal1(np__2,C,D)) )
=> ( r2_subset_1(k3_topreal1(np__2,A,B),k3_topreal1(np__2,C,D))
| k21_euclid(A) = k21_euclid(C) ) ) ) ) ) ) ).
fof(t26_jordan12,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))
& k22_euclid(B) = k22_euclid(C)
& k22_euclid(C) = k22_euclid(A)
& ~ r2_hidden(B,k3_topreal1(np__2,A,C))
& ~ r2_hidden(C,k3_topreal1(np__2,A,B)) ) ) ) ) ).
fof(t27_jordan12,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))
& k21_euclid(B) = k21_euclid(C)
& k21_euclid(C) = k21_euclid(A)
& ~ r2_hidden(B,k3_topreal1(np__2,A,C))
& ~ r2_hidden(C,k3_topreal1(np__2,A,B)) ) ) ) ) ).
fof(t28_jordan12,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)))
=> ~ ( A != B
& A != C
& r2_hidden(A,k3_topreal1(np__2,B,C))
& r2_hidden(B,k3_topreal1(np__2,A,C)) ) ) ) ) ).
fof(t29_jordan12,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)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ~ ( ~ r2_hidden(D,k3_topreal1(np__2,B,C))
& r2_hidden(A,k3_topreal1(np__2,B,C))
& A != B
& A != C
& ( ( k21_euclid(B) = k21_euclid(C)
& k21_euclid(C) = k21_euclid(D) )
| ( k22_euclid(B) = k22_euclid(C)
& k22_euclid(C) = k22_euclid(D) ) )
& ~ r2_hidden(B,k3_topreal1(np__2,D,A))
& ~ r2_hidden(C,k3_topreal1(np__2,D,A)) ) ) ) ) ) ).
fof(t30_jordan12,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)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ~ ( ( ( k21_euclid(A) = k21_euclid(B)
& k21_euclid(C) = k21_euclid(D) )
| ( k22_euclid(A) = k22_euclid(B)
& k22_euclid(C) = k22_euclid(D) ) )
& k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_topreal1(np__2,A,B),k3_topreal1(np__2,C,D)) = k1_tarski(E)
& E != A
& E != B
& E != C ) ) ) ) ) ) ).
fof(t31_jordan12,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)))
=> ! [D] :
( ( ~ v1_xboole_0(D)
& ~ v5_seqm_3(D)
& v1_topreal1(D)
& v2_topreal1(D)
& v1_finseq_6(D,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(D)
& v2_goboard5(D)
& m2_finseq_1(D,u1_struct_0(k15_euclid(np__2))) )
=> ( k5_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,D),k3_topreal1(np__2,B,C)) = k1_tarski(A)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ~ ( ~ r2_hidden(E,k3_topreal1(np__2,B,C))
& ~ r2_hidden(B,k5_topreal1(np__2,D))
& ~ r2_hidden(C,k5_topreal1(np__2,D))
& ( ( k21_euclid(B) = k21_euclid(C)
& k21_euclid(B) = k21_euclid(E) )
| ( k22_euclid(B) = k22_euclid(C)
& k22_euclid(B) = k22_euclid(E) ) )
& ? [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
& r1_xreal_0(np__1,F)
& r1_xreal_0(k1_nat_1(F,np__1),k3_finseq_1(D))
& ( r2_hidden(E,k3_gobrd13(D,k3_goboard2(D),F))
| r2_hidden(E,k4_gobrd13(D,k3_goboard2(D),F)) )
& r2_hidden(A,k4_topreal1(np__2,D,F)) )
& ~ r2_hidden(E,k5_topreal1(np__2,D))
& ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ~ ( r4_connsp_1(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,D)),F)
& r2_hidden(E,F)
& r2_hidden(B,F) ) )
& ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ~ ( r4_connsp_1(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,D)),F)
& r2_hidden(E,F)
& r2_hidden(C,F) ) ) ) ) ) ) ) ) ) ).
fof(t32_jordan12,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& ~ v5_seqm_3(A)
& v1_topreal1(A)
& v2_topreal1(A)
& v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(A)
& v2_goboard5(A)
& m2_finseq_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)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ( k5_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A),k3_topreal1(np__2,B,C)) = k1_tarski(D)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ~ ( ~ r2_hidden(B,k5_topreal1(np__2,A))
& ~ r2_hidden(C,k5_topreal1(np__2,A))
& ( ( k21_euclid(B) = k21_euclid(C)
& k21_euclid(B) = k21_euclid(E)
& k21_euclid(E) = k21_euclid(F) )
| ( k22_euclid(B) = k22_euclid(C)
& k22_euclid(B) = k22_euclid(E)
& k22_euclid(E) = k22_euclid(F) ) )
& ? [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
& r1_xreal_0(np__1,G)
& r1_xreal_0(k1_nat_1(G,np__1),k3_finseq_1(A))
& r2_hidden(E,k4_gobrd13(A,k3_goboard2(A),G))
& r2_hidden(F,k3_gobrd13(A,k3_goboard2(A),G))
& r2_hidden(D,k4_topreal1(np__2,A,G)) )
& ~ r2_hidden(E,k5_topreal1(np__2,A))
& ~ r2_hidden(F,k5_topreal1(np__2,A))
& ? [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& r4_connsp_1(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)),G)
& r2_hidden(B,G)
& r2_hidden(C,G) ) ) ) ) ) ) ) ) ) ).
fof(t33_jordan12,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( ( ~ v1_xboole_0(B)
& ~ v5_seqm_3(B)
& v1_topreal1(B)
& v2_topreal1(B)
& v1_finseq_6(B,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(B)
& v2_goboard5(B)
& m2_finseq_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)))
=> ( ( k5_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,B),k3_topreal1(np__2,C,D)) = k1_tarski(A)
& r1_xboole_0(k2_relat_1(B),k3_topreal1(np__2,C,D)) )
=> ( ( k21_euclid(C) != k21_euclid(D)
& k22_euclid(C) != k22_euclid(D) )
| r2_hidden(C,k5_topreal1(np__2,B))
| r2_hidden(D,k5_topreal1(np__2,B))
| ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ~ ( r4_connsp_1(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,B)),E)
& r2_hidden(C,E)
& r2_hidden(D,E) ) ) ) ) ) ) ) ) ).
fof(t34_jordan12,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& ~ v5_seqm_3(A)
& v1_topreal1(A)
& v2_topreal1(A)
& v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(A)
& v2_goboard5(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ! [B] :
( ( v1_topreal1(B)
& m2_finseq_1(B,u1_struct_0(k15_euclid(np__2))) )
=> ( r2_jordan12(np__2,A,B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,C)
& r1_xreal_0(k1_nat_1(C,np__1),k3_finseq_1(B)) )
=> ( ( v1_abian(k1_card_1(k5_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A),k4_topreal1(np__2,B,C))))
& m2_subset_1(k1_card_1(k5_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A),k4_topreal1(np__2,B,C))),k1_numbers,k5_numbers) )
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& r4_connsp_1(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)),D)
& r2_hidden(k1_funct_1(B,C),D)
& r2_hidden(k1_funct_1(B,k1_nat_1(C,np__1)),D) ) ) ) ) ) ) ) ).
fof(t35_jordan12,axiom,
! [A] :
( ( v1_topreal1(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ! [B] :
( ( v1_topreal1(B)
& m2_finseq_1(B,u1_struct_0(k15_euclid(np__2))) )
=> ! [C] :
( ( v1_topreal1(C)
& m2_finseq_1(C,u1_struct_0(k15_euclid(np__2))) )
=> ( ( ~ v1_xboole_0(k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B))
& ~ v5_seqm_3(k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B))
& v1_topreal1(k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B))
& v2_topreal1(k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B))
& v1_finseq_6(k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B),u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B))
& v2_goboard5(k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B))
& m2_finseq_1(k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B),u1_struct_0(k15_euclid(np__2)))
& r2_jordan12(np__2,k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B),C)
& r1_xreal_0(np__2,k3_finseq_1(C))
& v2_topreal1(C)
& v3_topreal1(C) )
=> ( ( v1_abian(k1_card_1(k5_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B)),k5_topreal1(np__2,C))))
& m2_subset_1(k1_card_1(k5_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B)),k5_topreal1(np__2,C))),k1_numbers,k5_numbers) )
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& r4_connsp_1(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B))),D)
& r2_hidden(k1_funct_1(C,np__1),D)
& r2_hidden(k1_funct_1(C,k3_finseq_1(C)),D) ) ) ) ) ) ) ).
fof(t36_jordan12,axiom,
! [A] :
( ( v1_topreal1(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ! [B] :
( ( v1_topreal1(B)
& m2_finseq_1(B,u1_struct_0(k15_euclid(np__2))) )
=> ! [C] :
( ( v1_topreal1(C)
& m2_finseq_1(C,u1_struct_0(k15_euclid(np__2))) )
=> ! [D] :
( ( v1_topreal1(D)
& m2_finseq_1(D,u1_struct_0(k15_euclid(np__2))) )
=> ( ( ~ v1_xboole_0(k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B))
& ~ v5_seqm_3(k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B))
& v1_topreal1(k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B))
& v2_topreal1(k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B))
& v1_finseq_6(k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B),u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B))
& v2_goboard5(k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B))
& m2_finseq_1(k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B),u1_struct_0(k15_euclid(np__2)))
& ~ v1_xboole_0(k4_graph_2(u1_struct_0(k15_euclid(np__2)),C,D))
& ~ v5_seqm_3(k4_graph_2(u1_struct_0(k15_euclid(np__2)),C,D))
& v1_topreal1(k4_graph_2(u1_struct_0(k15_euclid(np__2)),C,D))
& v2_topreal1(k4_graph_2(u1_struct_0(k15_euclid(np__2)),C,D))
& v1_finseq_6(k4_graph_2(u1_struct_0(k15_euclid(np__2)),C,D),u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(k4_graph_2(u1_struct_0(k15_euclid(np__2)),C,D))
& v2_goboard5(k4_graph_2(u1_struct_0(k15_euclid(np__2)),C,D))
& m2_finseq_1(k4_graph_2(u1_struct_0(k15_euclid(np__2)),C,D),u1_struct_0(k15_euclid(np__2)))
& r1_xboole_0(k5_topreal1(np__2,A),k5_topreal1(np__2,D))
& r1_xboole_0(k5_topreal1(np__2,B),k5_topreal1(np__2,C))
& r2_jordan12(np__2,k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B),k4_graph_2(u1_struct_0(k15_euclid(np__2)),C,D)) )
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(k15_euclid(np__2)))
=> ~ ( k1_funct_1(A,np__1) = E
& k1_funct_1(A,k3_finseq_1(A)) = F
& k1_funct_1(C,np__1) = G
& k1_funct_1(C,k3_finseq_1(C)) = H
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A)) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1)
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),D,np__1)
& r2_hidden(E,k5_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A),k5_topreal1(np__2,B)))
& r2_hidden(G,k5_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,C),k5_topreal1(np__2,D)))
& ? [I] :
( m1_subset_1(I,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& r4_connsp_1(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k4_graph_2(u1_struct_0(k15_euclid(np__2)),A,B))),I)
& r2_hidden(G,I)
& r2_hidden(H,I) )
& ! [I] :
( m1_subset_1(I,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ~ ( r4_connsp_1(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k4_graph_2(u1_struct_0(k15_euclid(np__2)),C,D))),I)
& r2_hidden(E,I)
& r2_hidden(F,I) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(symmetry_r2_jordan12,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_finseq_1(B,u1_struct_0(k15_euclid(A)))
& m1_finseq_1(C,u1_struct_0(k15_euclid(A))) )
=> ( r2_jordan12(A,B,C)
=> r2_jordan12(A,C,B) ) ) ).
fof(t11_jordan12,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> v1_finset_1(k3_setfam_1(a_1_0_jordan12(A),a_1_0_jordan12(B))) ) ) ).
fof(fraenkel_a_1_0_jordan12,axiom,
! [A,B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(A,a_1_0_jordan12(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = k4_topreal1(np__2,B,C)
& r1_xreal_0(np__1,C)
& r1_xreal_0(k1_nat_1(C,np__1),k3_finseq_1(B)) ) ) ) ).
%------------------------------------------------------------------------------