SET007 Axioms: SET007+473.ax
%------------------------------------------------------------------------------
% File : SET007+473 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Some Topological Properties of Cells in R^2
% 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 : gobrd11 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 60 ( 0 unt; 0 def)
% Number of atoms : 471 ( 60 equ)
% Maximal formula atoms : 15 ( 7 avg)
% Number of connectives : 462 ( 51 ~; 9 |; 256 &)
% ( 24 <=>; 122 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 9 avg)
% Maximal term depth : 5 ( 2 avg)
% Number of predicates : 29 ( 28 usr; 0 prp; 1-3 aty)
% Number of functors : 55 ( 55 usr; 6 con; 0-4 aty)
% Number of variables : 179 ( 131 !; 48 ?)
% 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_gobrd11,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)
& m1_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ~ v1_xboole_0(k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A))) ) ).
fof(t1_gobrd11,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))
=> ( ( r2_hidden(C,B)
& v2_connsp_1(B,A) )
=> r1_tarski(B,k1_connsp_1(A,C)) ) ) ) ) ).
fof(t2_gobrd11,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)))
=> ( ( r3_connsp_1(A,D)
& r1_tarski(B,D)
& v2_connsp_1(C,A) )
=> ( r1_xboole_0(k6_pre_topc(A,B),k6_pre_topc(A,C))
| r1_tarski(C,D) ) ) ) ) ) ) ).
fof(t3_gobrd11,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)))
=> ( ( r3_connsp_1(A,B)
& r3_connsp_1(A,C) )
=> m1_connsp_3(k4_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) ).
fof(t4_gobrd11,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)))
=> k4_connsp_3(A,k4_subset_1(u1_struct_0(A),B,C),D) = k4_subset_1(u1_struct_0(k3_pre_topc(A,D)),k4_connsp_3(A,B,D),k4_connsp_3(A,C,D)) ) ) ) ) ).
fof(t5_gobrd11,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)))
=> k4_connsp_3(A,k5_subset_1(u1_struct_0(A),B,C),D) = k5_subset_1(u1_struct_0(k3_pre_topc(A,D)),k4_connsp_3(A,B,D),k4_connsp_3(A,C,D)) ) ) ) ) ).
fof(t6_gobrd11,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))) )
=> k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)) != k1_xboole_0 ) ).
fof(t16_gobrd11,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)))) )
=> v4_pre_topc(k2_goboard5(B,A),k15_euclid(np__2)) ) ) ).
fof(t17_gobrd11,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)))) )
=> v4_pre_topc(k1_goboard5(B,A),k15_euclid(np__2)) ) ) ).
fof(t33_gobrd11,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> v4_pre_topc(k3_goboard5(C,A,B),k15_euclid(np__2)) ) ) ) ).
fof(t34_gobrd11,axiom,
! [A] :
( ( ~ v3_relat_1(A)
& v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r1_xreal_0(np__1,k3_finseq_1(A))
& r1_xreal_0(np__1,k1_matrix_1(A)) ) ) ).
fof(t35_gobrd11,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)) )
=> k3_goboard5(C,A,B) = k6_pre_topc(k15_euclid(np__2),k1_tops_1(k15_euclid(np__2),k3_goboard5(C,A,B))) ) ) ) ) ).
fof(t7_gobrd11,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))) )
=> u1_struct_0(k15_euclid(np__2)) = k3_tarski(a_1_0_gobrd11(A)) ) ).
fof(t8_gobrd11,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [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))))
=> ( ( B = a_1_1_gobrd11(A)
& C = a_1_2_gobrd11(A) )
=> B = k3_subset_1(u1_struct_0(k15_euclid(np__2)),C) ) ) ) ) ).
fof(t9_gobrd11,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [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))))
=> ( ( B = a_1_3_gobrd11(A)
& C = a_1_4_gobrd11(A) )
=> B = k3_subset_1(u1_struct_0(k15_euclid(np__2)),C) ) ) ) ) ).
fof(t10_gobrd11,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [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))))
=> ( ( B = a_1_5_gobrd11(A)
& C = a_1_6_gobrd11(A) )
=> B = k3_subset_1(u1_struct_0(k15_euclid(np__2)),C) ) ) ) ) ).
fof(t11_gobrd11,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [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))))
=> ( ( B = a_1_7_gobrd11(A)
& C = a_1_8_gobrd11(A) )
=> B = k3_subset_1(u1_struct_0(k15_euclid(np__2)),C) ) ) ) ) ).
fof(t12_gobrd11,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( A = a_1_1_gobrd11(B)
=> v4_pre_topc(A,k15_euclid(np__2)) ) ) ) ).
fof(t13_gobrd11,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( A = a_1_3_gobrd11(B)
=> v4_pre_topc(A,k15_euclid(np__2)) ) ) ) ).
fof(t14_gobrd11,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( A = a_1_7_gobrd11(B)
=> v4_pre_topc(A,k15_euclid(np__2)) ) ) ) ).
fof(t15_gobrd11,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( A = a_1_5_gobrd11(B)
=> v4_pre_topc(A,k15_euclid(np__2)) ) ) ) ).
fof(t18_gobrd11,axiom,
! [A] :
( ( ~ v3_relat_1(A)
& v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( v3_goboard1(A)
=> k1_goboard5(A,np__0) = a_1_9_gobrd11(A) ) ) ).
fof(t19_gobrd11,axiom,
! [A] :
( ( ~ v3_relat_1(A)
& v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( v3_goboard1(A)
=> k1_goboard5(A,k3_finseq_1(A)) = a_1_10_gobrd11(A) ) ) ).
fof(t20_gobrd11,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_relat_1(B)
& v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( ( v3_goboard1(B)
& r1_xreal_0(np__1,A) )
=> ( r1_xreal_0(k3_finseq_1(B),A)
| k1_goboard5(B,A) = a_2_0_gobrd11(A,B) ) ) ) ) ).
fof(t21_gobrd11,axiom,
! [A] :
( ( ~ v3_relat_1(A)
& v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( v4_goboard1(A)
=> k2_goboard5(A,np__0) = a_1_11_gobrd11(A) ) ) ).
fof(t22_gobrd11,axiom,
! [A] :
( ( ~ v3_relat_1(A)
& v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( v4_goboard1(A)
=> k2_goboard5(A,k1_matrix_1(A)) = a_1_12_gobrd11(A) ) ) ).
fof(t23_gobrd11,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_relat_1(B)
& v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( ( v4_goboard1(B)
& r1_xreal_0(np__1,A) )
=> ( r1_xreal_0(k1_matrix_1(B),A)
| k2_goboard5(B,A) = a_2_1_gobrd11(A,B) ) ) ) ) ).
fof(t24_gobrd11,axiom,
! [A] :
( ( ~ v3_relat_1(A)
& v1_matrix_1(A)
& v3_goboard1(A)
& v4_goboard1(A)
& m2_finseq_1(A,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> k3_goboard5(A,np__0,np__0) = a_1_13_gobrd11(A) ) ).
fof(t25_gobrd11,axiom,
! [A] :
( ( ~ v3_relat_1(A)
& v1_matrix_1(A)
& v3_goboard1(A)
& v4_goboard1(A)
& m2_finseq_1(A,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> k3_goboard5(A,np__0,k1_matrix_1(A)) = a_1_14_gobrd11(A) ) ).
fof(t26_gobrd11,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)
& m2_finseq_1(B,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r1_xreal_0(np__1,A)
=> ( r1_xreal_0(k1_matrix_1(B),A)
| k3_goboard5(B,np__0,A) = a_2_2_gobrd11(A,B) ) ) ) ) ).
fof(t27_gobrd11,axiom,
! [A] :
( ( ~ v3_relat_1(A)
& v1_matrix_1(A)
& v3_goboard1(A)
& v4_goboard1(A)
& m2_finseq_1(A,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> k3_goboard5(A,k3_finseq_1(A),np__0) = a_1_15_gobrd11(A) ) ).
fof(t28_gobrd11,axiom,
! [A] :
( ( ~ v3_relat_1(A)
& v1_matrix_1(A)
& v3_goboard1(A)
& v4_goboard1(A)
& m2_finseq_1(A,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> k3_goboard5(A,k3_finseq_1(A),k1_matrix_1(A)) = a_1_16_gobrd11(A) ) ).
fof(t29_gobrd11,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)
& m2_finseq_1(B,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r1_xreal_0(np__1,A)
=> ( r1_xreal_0(k1_matrix_1(B),A)
| k3_goboard5(B,k3_finseq_1(B),A) = a_2_3_gobrd11(A,B) ) ) ) ) ).
fof(t30_gobrd11,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)
& m2_finseq_1(B,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r1_xreal_0(np__1,A)
=> ( r1_xreal_0(k3_finseq_1(B),A)
| k3_goboard5(B,A,np__0) = a_2_4_gobrd11(A,B) ) ) ) ) ).
fof(t31_gobrd11,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)
& m2_finseq_1(B,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r1_xreal_0(np__1,A)
=> ( r1_xreal_0(k3_finseq_1(B),A)
| k3_goboard5(B,A,k1_matrix_1(B)) = a_2_5_gobrd11(A,B) ) ) ) ) ).
fof(t32_gobrd11,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)
& m2_finseq_1(C,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( ( r1_xreal_0(np__1,A)
& r1_xreal_0(np__1,B) )
=> ( r1_xreal_0(k3_finseq_1(C),A)
| r1_xreal_0(k1_matrix_1(C),B)
| k3_goboard5(C,A,B) = a_3_0_gobrd11(A,B,C) ) ) ) ) ) ).
fof(fraenkel_a_1_0_gobrd11,axiom,
! [A,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))) )
=> ( r2_hidden(A,a_1_0_gobrd11(B))
<=> ? [C,D] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& m2_subset_1(D,k1_numbers,k5_numbers)
& A = k3_goboard5(k3_goboard2(B),C,D)
& r1_xreal_0(C,k3_finseq_1(k3_goboard2(B)))
& r1_xreal_0(D,k1_matrix_1(k3_goboard2(B))) ) ) ) ).
fof(fraenkel_a_1_1_gobrd11,axiom,
! [A,B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_hidden(A,a_1_1_gobrd11(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_2_gobrd11,axiom,
! [A,B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_hidden(A,a_1_2_gobrd11(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_gobrd11,axiom,
! [A,B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_hidden(A,a_1_3_gobrd11(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_1_4_gobrd11,axiom,
! [A,B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_hidden(A,a_1_4_gobrd11(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_1_5_gobrd11,axiom,
! [A,B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_hidden(A,a_1_5_gobrd11(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_6_gobrd11,axiom,
! [A,B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_hidden(A,a_1_6_gobrd11(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_7_gobrd11,axiom,
! [A,B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_hidden(A,a_1_7_gobrd11(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_8_gobrd11,axiom,
! [A,B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_hidden(A,a_1_8_gobrd11(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_9_gobrd11,axiom,
! [A,B] :
( ( ~ v3_relat_1(B)
& v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r2_hidden(A,a_1_9_gobrd11(B))
<=> ? [C,D] :
( m1_subset_1(C,k1_numbers)
& m1_subset_1(D,k1_numbers)
& A = k23_euclid(C,D)
& r1_xreal_0(C,k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),B,np__1,np__1))) ) ) ) ).
fof(fraenkel_a_1_10_gobrd11,axiom,
! [A,B] :
( ( ~ v3_relat_1(B)
& v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r2_hidden(A,a_1_10_gobrd11(B))
<=> ? [C,D] :
( m1_subset_1(C,k1_numbers)
& m1_subset_1(D,k1_numbers)
& A = k23_euclid(C,D)
& r1_xreal_0(k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B),np__1)),C) ) ) ) ).
fof(fraenkel_a_2_0_gobrd11,axiom,
! [A,B,C] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& ~ v3_relat_1(C)
& v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r2_hidden(A,a_2_0_gobrd11(B,C))
<=> ? [D,E] :
( m1_subset_1(D,k1_numbers)
& m1_subset_1(E,k1_numbers)
& A = k23_euclid(D,E)
& r1_xreal_0(k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,B,np__1)),D)
& r1_xreal_0(D,k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,k1_nat_1(B,np__1),np__1))) ) ) ) ).
fof(fraenkel_a_1_11_gobrd11,axiom,
! [A,B] :
( ( ~ v3_relat_1(B)
& v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r2_hidden(A,a_1_11_gobrd11(B))
<=> ? [C,D] :
( m1_subset_1(C,k1_numbers)
& m1_subset_1(D,k1_numbers)
& A = k23_euclid(C,D)
& r1_xreal_0(D,k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),B,np__1,np__1))) ) ) ) ).
fof(fraenkel_a_1_12_gobrd11,axiom,
! [A,B] :
( ( ~ v3_relat_1(B)
& v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r2_hidden(A,a_1_12_gobrd11(B))
<=> ? [C,D] :
( m1_subset_1(C,k1_numbers)
& m1_subset_1(D,k1_numbers)
& A = k23_euclid(C,D)
& r1_xreal_0(k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),B,np__1,k1_matrix_1(B))),D) ) ) ) ).
fof(fraenkel_a_2_1_gobrd11,axiom,
! [A,B,C] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& ~ v3_relat_1(C)
& v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r2_hidden(A,a_2_1_gobrd11(B,C))
<=> ? [D,E] :
( m1_subset_1(D,k1_numbers)
& m1_subset_1(E,k1_numbers)
& A = k23_euclid(D,E)
& r1_xreal_0(k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,np__1,B)),E)
& r1_xreal_0(E,k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,np__1,k1_nat_1(B,np__1)))) ) ) ) ).
fof(fraenkel_a_1_13_gobrd11,axiom,
! [A,B] :
( ( ~ v3_relat_1(B)
& v1_matrix_1(B)
& v3_goboard1(B)
& v4_goboard1(B)
& m2_finseq_1(B,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r2_hidden(A,a_1_13_gobrd11(B))
<=> ? [C,D] :
( m1_subset_1(C,k1_numbers)
& m1_subset_1(D,k1_numbers)
& A = k23_euclid(C,D)
& r1_xreal_0(C,k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),B,np__1,np__1)))
& r1_xreal_0(D,k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),B,np__1,np__1))) ) ) ) ).
fof(fraenkel_a_1_14_gobrd11,axiom,
! [A,B] :
( ( ~ v3_relat_1(B)
& v1_matrix_1(B)
& v3_goboard1(B)
& v4_goboard1(B)
& m2_finseq_1(B,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r2_hidden(A,a_1_14_gobrd11(B))
<=> ? [C,D] :
( m1_subset_1(C,k1_numbers)
& m1_subset_1(D,k1_numbers)
& A = k23_euclid(C,D)
& r1_xreal_0(C,k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),B,np__1,np__1)))
& r1_xreal_0(k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),B,np__1,k1_matrix_1(B))),D) ) ) ) ).
fof(fraenkel_a_2_2_gobrd11,axiom,
! [A,B,C] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& ~ v3_relat_1(C)
& v1_matrix_1(C)
& v3_goboard1(C)
& v4_goboard1(C)
& m2_finseq_1(C,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r2_hidden(A,a_2_2_gobrd11(B,C))
<=> ? [D,E] :
( m1_subset_1(D,k1_numbers)
& m1_subset_1(E,k1_numbers)
& A = k23_euclid(D,E)
& r1_xreal_0(D,k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,np__1,np__1)))
& r1_xreal_0(k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,np__1,B)),E)
& r1_xreal_0(E,k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,np__1,k1_nat_1(B,np__1)))) ) ) ) ).
fof(fraenkel_a_1_15_gobrd11,axiom,
! [A,B] :
( ( ~ v3_relat_1(B)
& v1_matrix_1(B)
& v3_goboard1(B)
& v4_goboard1(B)
& m2_finseq_1(B,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r2_hidden(A,a_1_15_gobrd11(B))
<=> ? [C,D] :
( m1_subset_1(C,k1_numbers)
& m1_subset_1(D,k1_numbers)
& A = k23_euclid(C,D)
& r1_xreal_0(k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B),np__1)),C)
& r1_xreal_0(D,k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),B,np__1,np__1))) ) ) ) ).
fof(fraenkel_a_1_16_gobrd11,axiom,
! [A,B] :
( ( ~ v3_relat_1(B)
& v1_matrix_1(B)
& v3_goboard1(B)
& v4_goboard1(B)
& m2_finseq_1(B,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r2_hidden(A,a_1_16_gobrd11(B))
<=> ? [C,D] :
( m1_subset_1(C,k1_numbers)
& m1_subset_1(D,k1_numbers)
& A = k23_euclid(C,D)
& r1_xreal_0(k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B),np__1)),C)
& r1_xreal_0(k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),B,np__1,k1_matrix_1(B))),D) ) ) ) ).
fof(fraenkel_a_2_3_gobrd11,axiom,
! [A,B,C] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& ~ v3_relat_1(C)
& v1_matrix_1(C)
& v3_goboard1(C)
& v4_goboard1(C)
& m2_finseq_1(C,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r2_hidden(A,a_2_3_gobrd11(B,C))
<=> ? [D,E] :
( m1_subset_1(D,k1_numbers)
& m1_subset_1(E,k1_numbers)
& A = k23_euclid(D,E)
& r1_xreal_0(k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C),np__1)),D)
& r1_xreal_0(k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,np__1,B)),E)
& r1_xreal_0(E,k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,np__1,k1_nat_1(B,np__1)))) ) ) ) ).
fof(fraenkel_a_2_4_gobrd11,axiom,
! [A,B,C] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& ~ v3_relat_1(C)
& v1_matrix_1(C)
& v3_goboard1(C)
& v4_goboard1(C)
& m2_finseq_1(C,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r2_hidden(A,a_2_4_gobrd11(B,C))
<=> ? [D,E] :
( m1_subset_1(D,k1_numbers)
& m1_subset_1(E,k1_numbers)
& A = k23_euclid(D,E)
& r1_xreal_0(k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,B,np__1)),D)
& r1_xreal_0(D,k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,k1_nat_1(B,np__1),np__1)))
& r1_xreal_0(E,k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,np__1,np__1))) ) ) ) ).
fof(fraenkel_a_2_5_gobrd11,axiom,
! [A,B,C] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& ~ v3_relat_1(C)
& v1_matrix_1(C)
& v3_goboard1(C)
& v4_goboard1(C)
& m2_finseq_1(C,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r2_hidden(A,a_2_5_gobrd11(B,C))
<=> ? [D,E] :
( m1_subset_1(D,k1_numbers)
& m1_subset_1(E,k1_numbers)
& A = k23_euclid(D,E)
& r1_xreal_0(k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,B,np__1)),D)
& r1_xreal_0(D,k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,k1_nat_1(B,np__1),np__1)))
& r1_xreal_0(k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,np__1,k1_matrix_1(C))),E) ) ) ) ).
fof(fraenkel_a_3_0_gobrd11,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)
& m2_finseq_1(D,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ( r2_hidden(A,a_3_0_gobrd11(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,np__1)),E)
& r1_xreal_0(E,k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),D,k1_nat_1(B,np__1),np__1)))
& r1_xreal_0(k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),D,np__1,C)),F)
& r1_xreal_0(F,k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),D,np__1,k1_nat_1(C,np__1)))) ) ) ) ).
%------------------------------------------------------------------------------