SET007 Axioms: SET007+613.ax
%------------------------------------------------------------------------------
% File : SET007+613 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Cages - the External Approximation of Jordan's Curve
% 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 : jordan9 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 37 ( 2 unt; 0 def)
% Number of atoms : 536 ( 40 equ)
% Maximal formula atoms : 36 ( 14 avg)
% Number of connectives : 583 ( 84 ~; 7 |; 333 &)
% ( 4 <=>; 155 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 13 avg)
% Maximal term depth : 5 ( 2 avg)
% Number of predicates : 36 ( 34 usr; 1 prp; 0-4 aty)
% Number of functors : 41 ( 41 usr; 5 con; 0-4 aty)
% Number of variables : 132 ( 128 !; 4 ?)
% 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_jordan9,axiom,
$true ).
fof(t2_jordan9,axiom,
$true ).
fof(t3_jordan9,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)
& r4_connsp_1(A,E,C)
& r4_connsp_1(A,E,D)
& r1_tarski(B,E) )
=> ( r1_xboole_0(B,C)
| r1_xboole_0(B,D)
| C = D ) ) ) ) ) ) ) ).
fof(t4_jordan9,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k16_finseq_1(A,B,D) = k16_finseq_1(A,C,D) )
=> B = C ) ) ) ) ).
fof(t5_jordan9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,B)
=> ~ ( r2_hidden(A,k4_finseq_1(C))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(D,k4_finseq_1(k4_finseq_5(B,C)))
& k1_nat_1(A,D) = k1_nat_1(k3_finseq_1(C),np__1)
& k4_finseq_4(k5_numbers,B,C,A) = k4_finseq_4(k5_numbers,B,k4_finseq_5(B,C),D) ) ) ) ) ) ) ).
fof(t6_jordan9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,B)
=> ~ ( r2_hidden(A,k4_finseq_1(k4_finseq_5(B,C)))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(D,k4_finseq_1(C))
& k1_nat_1(A,D) = k1_nat_1(k3_finseq_1(C),np__1)
& k4_finseq_4(k5_numbers,B,k4_finseq_5(B,C),A) = k4_finseq_4(k5_numbers,B,C,D) ) ) ) ) ) ) ).
fof(t7_jordan9,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(A)) )
=> ! [C] :
( m2_finseq_1(C,A)
=> ( r1_goboard1(A,C,B)
<=> r1_goboard1(A,k4_finseq_5(A,C),B) ) ) ) ) ).
fof(t8_jordan9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),C,B)
& r1_xreal_0(np__1,A)
& r1_xreal_0(A,k3_finseq_1(C)) )
=> r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,A),k2_gobrd13(B)) ) ) ) ) ).
fof(t9_jordan9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
~ ( r1_xreal_0(A,k3_finseq_1(B))
& r2_hidden(C,k5_topreal1(np__2,k1_rfinseq(u1_struct_0(k15_euclid(np__2)),B,A)))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(k1_nat_1(A,np__1),D)
& r1_xreal_0(k1_nat_1(D,np__1),k3_finseq_1(B))
& r2_hidden(C,k4_topreal1(np__2,B,D)) ) ) ) ) ) ).
fof(t10_jordan9,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)))) )
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),C,B)
& r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(C)) )
=> ( r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,A),k4_gobrd13(C,B,A))
& r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,A),k3_gobrd13(C,B,A)) ) ) ) ) ) ).
fof(t11_jordan9,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)))) )
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),C,B)
& r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(C)) )
=> ( k1_tops_1(k15_euclid(np__2),k4_gobrd13(C,B,A)) != k1_xboole_0
& k1_tops_1(k15_euclid(np__2),k3_gobrd13(C,B,A)) != k1_xboole_0 ) ) ) ) ) ).
fof(t12_jordan9,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)))) )
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),C,B)
& r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(C)) )
=> ( v2_connsp_1(k1_tops_1(k15_euclid(np__2),k4_gobrd13(C,B,A)),k15_euclid(np__2))
& v2_connsp_1(k1_tops_1(k15_euclid(np__2),k3_gobrd13(C,B,A)),k15_euclid(np__2)) ) ) ) ) ) ).
fof(t13_jordan9,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)))) )
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),C,B)
& r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(C)) )
=> ( k6_pre_topc(k15_euclid(np__2),k1_tops_1(k15_euclid(np__2),k4_gobrd13(C,B,A))) = k4_gobrd13(C,B,A)
& k6_pre_topc(k15_euclid(np__2),k1_tops_1(k15_euclid(np__2),k3_gobrd13(C,B,A))) = k3_gobrd13(C,B,A) ) ) ) ) ) ).
fof(t14_jordan9,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)))) )
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),C,B)
& v1_sppol_1(k4_topreal1(np__2,C,A))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,D)
& r1_xreal_0(D,k1_matrix_1(B))
& ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(E,k4_topreal1(np__2,C,A))
=> k22_euclid(E) = k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),B,np__1,D)) ) ) ) ) ) ) ) ) ).
fof(t15_jordan9,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)))) )
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),C,B)
& v2_sppol_1(k4_topreal1(np__2,C,A))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,D)
& r1_xreal_0(D,k3_finseq_1(B))
& ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(E,k4_topreal1(np__2,C,A))
=> k21_euclid(E) = k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),B,D,np__1)) ) ) ) ) ) ) ) ) ).
fof(t16_jordan9,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)))) )
=> ! [D] :
( m2_finseq_1(D,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),D,C)
& v1_topreal1(D)
& r1_xreal_0(A,k3_finseq_1(C))
& r1_xreal_0(B,k1_matrix_1(C)) )
=> r1_xboole_0(k1_tops_1(k15_euclid(np__2),k3_goboard5(C,A,B)),k5_topreal1(np__2,D)) ) ) ) ) ) ).
fof(t17_jordan9,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)))) )
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),C,B)
& v1_topreal1(C)
& r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(C)) )
=> ( r1_xboole_0(k1_tops_1(k15_euclid(np__2),k4_gobrd13(C,B,A)),k5_topreal1(np__2,C))
& r1_xboole_0(k1_tops_1(k15_euclid(np__2),k3_gobrd13(C,B,A)),k5_topreal1(np__2,C)) ) ) ) ) ) ).
fof(t18_jordan9,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(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(C))
& r1_xreal_0(np__1,B)
& r1_xreal_0(k1_nat_1(B,np__1),k1_matrix_1(C)) )
=> ( k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,A,B)) = k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,A,k1_nat_1(B,np__1)))
& k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,A,B)) = k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,k1_nat_1(A,np__1),B))
& k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,k1_nat_1(A,np__1),k1_nat_1(B,np__1))) = k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,k1_nat_1(A,np__1),B))
& k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,k1_nat_1(A,np__1),k1_nat_1(B,np__1))) = k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,A,k1_nat_1(B,np__1))) ) ) ) ) ) ).
fof(t19_jordan9,axiom,
! [A] :
( ( ~ v3_relat_1(A)
& v1_matrix_1(A)
& v3_goboard1(A)
& v4_goboard1(A)
& v5_goboard1(A)
& v6_goboard1(A)
& m2_finseq_1(A,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [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),k1_matrix_1(A)) )
=> ( r2_hidden(B,k3_goboard5(A,C,D))
<=> ( r1_xreal_0(k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),A,C,D)),k21_euclid(B))
& r1_xreal_0(k21_euclid(B),k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),A,k1_nat_1(C,np__1),D)))
& r1_xreal_0(k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),A,C,D)),k22_euclid(B))
& r1_xreal_0(k22_euclid(B),k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),A,C,k1_nat_1(D,np__1)))) ) ) ) ) ) ) ) ).
fof(t21_jordan9,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)))) )
=> ! [D] :
( m1_subset_1(D,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))
& r1_xreal_0(np__1,B)
& r1_xreal_0(k1_nat_1(B,np__1),k1_matrix_1(C))
& r2_hidden(D,k2_gobrd13(C))
& r2_hidden(D,k3_goboard5(C,A,B))
& D != k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,A,B)
& D != k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,A,k1_nat_1(B,np__1))
& D != k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,k1_nat_1(A,np__1),k1_nat_1(B,np__1))
& D != k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,k1_nat_1(A,np__1),B) ) ) ) ) ) ).
fof(t22_jordan9,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(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(C))
& r1_xreal_0(np__1,B)
& r1_xreal_0(k1_nat_1(B,np__1),k1_matrix_1(C)) )
=> ( r2_hidden(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,A,B),k3_goboard5(C,A,B))
& r2_hidden(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,A,k1_nat_1(B,np__1)),k3_goboard5(C,A,B))
& r2_hidden(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,k1_nat_1(A,np__1),k1_nat_1(B,np__1)),k3_goboard5(C,A,B))
& r2_hidden(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,k1_nat_1(A,np__1),B),k3_goboard5(C,A,B)) ) ) ) ) ) ).
fof(t23_jordan9,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)))) )
=> ! [D] :
( m1_subset_1(D,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))
& r1_xreal_0(np__1,B)
& r1_xreal_0(k1_nat_1(B,np__1),k1_matrix_1(C))
& r2_hidden(D,k2_gobrd13(C))
& r2_hidden(D,k3_goboard5(C,A,B)) )
=> r1_sppol_1(np__2,D,k3_goboard5(C,A,B)) ) ) ) ) ) ).
fof(t24_jordan9,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)))) )
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r1_xreal_0(np__2,k3_finseq_1(B))
& r1_xreal_0(np__2,k1_matrix_1(B))
& r1_goboard1(u1_struct_0(k15_euclid(np__2)),C,B)
& r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(C))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,D)
& r1_xreal_0(k1_nat_1(D,np__1),k3_finseq_1(B))
& r1_xreal_0(np__1,E)
& r1_xreal_0(k1_nat_1(E,np__1),k1_matrix_1(B))
& r1_tarski(k4_topreal1(np__2,C,A),k3_goboard5(B,D,E)) ) ) ) ) ) ) ) ).
fof(t25_jordan9,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)))) )
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r1_xreal_0(np__2,k3_finseq_1(B))
& r1_xreal_0(np__2,k1_matrix_1(B))
& r1_goboard1(u1_struct_0(k15_euclid(np__2)),C,B)
& r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(C))
& r2_hidden(D,k2_gobrd13(B))
& r2_hidden(D,k4_topreal1(np__2,C,A))
& D != k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,A)
& D != k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k1_nat_1(A,np__1)) ) ) ) ) ) ).
fof(t26_jordan9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( ( ~ v3_relat_1(D)
& v1_matrix_1(D)
& v3_goboard1(D)
& v4_goboard1(D)
& v5_goboard1(D)
& v6_goboard1(D)
& m2_finseq_1(D,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( ( r2_hidden(k4_tarski(A,B),k2_matrix_1(D))
& r1_xreal_0(np__1,C)
& r1_xreal_0(C,k1_matrix_1(D)) )
=> r1_xreal_0(k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),D,A,B)),k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),D,k3_finseq_1(D),C))) ) ) ) ) ) ).
fof(t27_jordan9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( ( ~ v3_relat_1(D)
& v1_matrix_1(D)
& v3_goboard1(D)
& v4_goboard1(D)
& v5_goboard1(D)
& v6_goboard1(D)
& m2_finseq_1(D,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( ( r2_hidden(k4_tarski(A,B),k2_matrix_1(D))
& r1_xreal_0(np__1,C)
& r1_xreal_0(C,k3_finseq_1(D)) )
=> r1_xreal_0(k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),D,A,B)),k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),D,C,k1_matrix_1(D)))) ) ) ) ) ) ).
fof(t28_jordan9,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)))) )
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m2_finseq_1(D,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),C,B)
& v1_topreal1(C)
& r1_tarski(k5_topreal1(np__2,D),k5_topreal1(np__2,C))
& r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(C)) )
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( ( E = k6_subset_1(u1_struct_0(k15_euclid(np__2)),k3_gobrd13(C,B,A),k5_topreal1(np__2,D))
| E = k6_subset_1(u1_struct_0(k15_euclid(np__2)),k4_gobrd13(C,B,A),k5_topreal1(np__2,D)) )
=> v2_connsp_1(E,k15_euclid(np__2)) ) ) ) ) ) ) ) ).
fof(t29_jordan9,axiom,
! [A] :
( ( ~ v3_relat_1(A)
& v1_matrix_1(A)
& v3_goboard1(A)
& v4_goboard1(A)
& v5_goboard1(A)
& v6_goboard1(A)
& m2_finseq_1(A,k3_finseq_2(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))) )
=> ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),B,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)) )
=> ( r1_tarski(k6_subset_1(u1_struct_0(k15_euclid(np__2)),k3_gobrd13(B,A,C),k5_topreal1(np__2,B)),k3_goboard9(B))
& r1_tarski(k6_subset_1(u1_struct_0(k15_euclid(np__2)),k4_gobrd13(B,A,C),k5_topreal1(np__2,B)),k2_goboard9(B)) ) ) ) ) ) ) ).
fof(t30_jordan9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v6_compts_1(B,k15_euclid(np__2))
& ~ v1_sppol_1(B)
& ~ v2_sppol_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ? [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(k1_jordan8(B,A)))
& r2_hidden(k32_pscomp_1(B),k3_goboard5(k1_jordan8(B,A),C,k5_binarith(k1_matrix_1(k1_jordan8(B,A)),np__1)))
& k32_pscomp_1(B) != k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(B,A),C,k5_binarith(k1_matrix_1(k1_jordan8(B,A)),np__1)) ) ) ) ).
fof(t31_jordan9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v6_compts_1(B,k15_euclid(np__2))
& ~ v1_sppol_1(B)
& ~ v2_sppol_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [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(k1_jordan8(B,A)))
& r2_hidden(k32_pscomp_1(B),k3_goboard5(k1_jordan8(B,A),C,k5_binarith(k1_matrix_1(k1_jordan8(B,A)),np__1)))
& r1_xreal_0(np__1,D)
& r1_xreal_0(k1_nat_1(D,np__1),k3_finseq_1(k1_jordan8(B,A)))
& r2_hidden(k32_pscomp_1(B),k3_goboard5(k1_jordan8(B,A),D,k5_binarith(k1_matrix_1(k1_jordan8(B,A)),np__1))) )
=> ( k32_pscomp_1(B) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(B,A),C,k5_binarith(k1_matrix_1(k1_jordan8(B,A)),np__1))
| k32_pscomp_1(B) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(B,A),D,k5_binarith(k1_matrix_1(k1_jordan8(B,A)),np__1))
| C = D ) ) ) ) ) ) ).
fof(t32_jordan9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v6_compts_1(B,k15_euclid(np__2))
& ~ v1_sppol_1(B)
& ~ v2_sppol_1(B)
& m1_subset_1(B,k1_zfmisc_1(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))) )
=> ( ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),C,k1_jordan8(B,A))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,D)
& r1_xreal_0(k1_nat_1(D,np__1),k3_finseq_1(C)) )
=> ( r1_xboole_0(k4_gobrd13(C,k1_jordan8(B,A),D),B)
& ~ r1_xboole_0(k3_gobrd13(C,k1_jordan8(B,A),D),B) ) ) ) )
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,D)
& r1_xreal_0(k1_nat_1(D,np__1),k3_finseq_1(k1_jordan8(B,A)))
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(B,A),D,k1_matrix_1(k1_jordan8(B,A)))
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__2) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(B,A),k1_nat_1(D,np__1),k1_matrix_1(k1_jordan8(B,A)))
& r2_hidden(k32_pscomp_1(B),k3_goboard5(k1_jordan8(B,A),D,k5_binarith(k1_matrix_1(k1_jordan8(B,A)),np__1)))
& k32_pscomp_1(B) != k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(B,A),D,k5_binarith(k1_matrix_1(k1_jordan8(B,A)),np__1)) ) )
| k32_pscomp_1(k5_topreal1(np__2,C)) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1) ) ) ) ) ) ).
fof(d1_jordan9,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v6_compts_1(A,k15_euclid(np__2))
& ~ v1_sppol_1(A)
& ~ v2_sppol_1(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( v2_connsp_1(A,k15_euclid(np__2))
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_sprect_2(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))) )
=> ( C = k1_jordan9(A,B)
<=> ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),C,k1_jordan8(A,B))
& ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& r1_xreal_0(np__1,D)
& r1_xreal_0(k1_nat_1(D,np__1),k3_finseq_1(k1_jordan8(A,B)))
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(A,B),D,k1_matrix_1(k1_jordan8(A,B)))
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__2) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(A,B),k1_nat_1(D,np__1),k1_matrix_1(k1_jordan8(A,B)))
& r2_hidden(k32_pscomp_1(A),k3_goboard5(k1_jordan8(A,B),D,k5_binarith(k1_matrix_1(k1_jordan8(A,B)),np__1)))
& k32_pscomp_1(A) != k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(A,B),D,k5_binarith(k1_matrix_1(k1_jordan8(A,B)),np__1)) )
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,D)
& r1_xreal_0(k1_nat_1(D,np__2),k3_finseq_1(C)) )
=> ( ( ( r1_xboole_0(k6_gobrd13(C,k1_jordan8(A,B),D),A)
& r1_xboole_0(k5_gobrd13(C,k1_jordan8(A,B),D),A) )
=> r1_gobrd13(u1_struct_0(k15_euclid(np__2)),C,k1_jordan8(A,B),D) )
& ( r1_xboole_0(k6_gobrd13(C,k1_jordan8(A,B),D),A)
=> ( r1_xboole_0(k5_gobrd13(C,k1_jordan8(A,B),D),A)
| r3_gobrd13(u1_struct_0(k15_euclid(np__2)),C,k1_jordan8(A,B),D) ) )
& ( ~ r1_xboole_0(k6_gobrd13(C,k1_jordan8(A,B),D),A)
=> r2_gobrd13(u1_struct_0(k15_euclid(np__2)),C,k1_jordan8(A,B),D) ) ) ) ) ) ) ) ) ) ) ).
fof(t33_jordan9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v6_compts_1(C,k15_euclid(np__2))
& ~ v1_sppol_1(C)
& ~ v2_sppol_1(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( ( v2_connsp_1(C,k15_euclid(np__2))
& r1_xreal_0(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(k1_jordan9(C,B))) )
=> ( r1_xboole_0(k4_gobrd13(k1_jordan9(C,B),k1_jordan8(C,B),A),C)
& ~ r1_xboole_0(k3_gobrd13(k1_jordan9(C,B),k1_jordan8(C,B),A),C) ) ) ) ) ) ).
fof(t34_jordan9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v6_compts_1(B,k15_euclid(np__2))
& ~ v1_sppol_1(B)
& ~ v2_sppol_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( v2_connsp_1(B,k15_euclid(np__2))
=> k32_pscomp_1(k5_topreal1(np__2,k1_jordan9(B,A))) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),k1_jordan9(B,A),np__1) ) ) ) ).
fof(dt_k1_jordan9,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v6_compts_1(A,k15_euclid(np__2))
& ~ v1_sppol_1(A)
& ~ v2_sppol_1(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
& m1_subset_1(B,k5_numbers) )
=> ( ~ v1_xboole_0(k1_jordan9(A,B))
& v1_sprect_2(k1_jordan9(A,B))
& ~ v5_seqm_3(k1_jordan9(A,B))
& v1_topreal1(k1_jordan9(A,B))
& v2_topreal1(k1_jordan9(A,B))
& v1_finseq_6(k1_jordan9(A,B),u1_struct_0(k15_euclid(np__2)))
& v1_goboard5(k1_jordan9(A,B))
& v2_goboard5(k1_jordan9(A,B))
& m2_finseq_1(k1_jordan9(A,B),u1_struct_0(k15_euclid(np__2))) ) ) ).
fof(t20_jordan9,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(np__1,A)
& r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(C))
& r1_xreal_0(np__1,B)
& r1_xreal_0(k1_nat_1(B,np__1),k1_matrix_1(C)) )
=> k3_goboard5(C,A,B) = a_3_0_jordan9(A,B,C) ) ) ) ) ).
fof(fraenkel_a_3_0_jordan9,axiom,
! [A,B,C,D] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& m2_subset_1(C,k1_numbers,k5_numbers)
& ~ v3_relat_1(D)
& v1_matrix_1(D)
& v3_goboard1(D)
& v4_goboard1(D)
& v5_goboard1(D)
& v6_goboard1(D)
& m2_finseq_1(D,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r2_hidden(A,a_3_0_jordan9(B,C,D))
<=> ? [E,F] :
( m1_subset_1(E,k1_numbers)
& m1_subset_1(F,k1_numbers)
& A = k23_euclid(E,F)
& r1_xreal_0(k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),D,B,C)),E)
& r1_xreal_0(E,k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),D,k1_nat_1(B,np__1),C)))
& r1_xreal_0(k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),D,B,C)),F)
& r1_xreal_0(F,k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),D,B,k1_nat_1(C,np__1)))) ) ) ) ).
%------------------------------------------------------------------------------