SET007 Axioms: SET007+602.ax
%------------------------------------------------------------------------------
% File : SET007+602 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Properties of Left and Right Components
% 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 : gobrd14 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 61 ( 0 unt; 0 def)
% Number of atoms : 597 ( 43 equ)
% Maximal formula atoms : 23 ( 9 avg)
% Number of connectives : 655 ( 119 ~; 4 |; 375 &)
% ( 4 <=>; 153 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 11 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of predicates : 42 ( 41 usr; 0 prp; 1-3 aty)
% Number of functors : 58 ( 58 usr; 9 con; 0-5 aty)
% Number of variables : 140 ( 138 !; 2 ?)
% 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_gobrd14,axiom,
! [A] :
( ( v4_pre_topc(A,k15_euclid(np__2))
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> v3_pre_topc(k1_jordan2c(np__2,A),k15_euclid(np__2)) ) ).
fof(fc2_gobrd14,axiom,
! [A] :
( ( v4_pre_topc(A,k15_euclid(np__2))
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> v3_pre_topc(k2_jordan2c(np__2,A),k15_euclid(np__2)) ) ).
fof(fc3_gobrd14,axiom,
! [A] :
( ( v6_compts_1(A,k15_euclid(np__2))
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> v2_connsp_1(k2_jordan2c(np__2,A),k15_euclid(np__2)) ) ).
fof(fc4_gobrd14,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(k5_topreal1(np__2,A))
& v6_compts_1(k5_topreal1(np__2,A),k15_euclid(np__2))
& v2_connsp_1(k5_topreal1(np__2,A),k15_euclid(np__2))
& v2_jordan1(k5_topreal1(np__2,A))
& v1_jordan2c(k5_topreal1(np__2,A),np__2)
& ~ v1_sppol_1(k5_topreal1(np__2,A))
& ~ v2_sppol_1(k5_topreal1(np__2,A))
& v1_topreal2(k5_topreal1(np__2,A))
& v2_tops_1(k5_topreal1(np__2,A),k15_euclid(np__2)) ) ) ).
fof(t1_gobrd14,axiom,
! [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)))
=> ( r4_connsp_1(A,B,C)
=> v2_connsp_1(C,A) ) ) ) ) ).
fof(t2_gobrd14,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( r1_jordan2c(A,B,C)
=> v2_connsp_1(C,k15_euclid(A)) ) ) ) ) ).
fof(t3_gobrd14,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( r2_jordan2c(A,B,C)
=> v2_connsp_1(C,k15_euclid(A)) ) ) ) ) ).
fof(t4_gobrd14,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( r4_connsp_1(k15_euclid(A),k3_subset_1(u1_struct_0(k15_euclid(A)),B),C)
=> r1_xboole_0(B,C) ) ) ) ) ).
fof(t5_gobrd14,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [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))))
=> ( ( r2_jordan2c(np__2,B,A)
& r1_jordan2c(np__2,B,C) )
=> r1_xboole_0(A,C) ) ) ) ) ).
fof(t6_gobrd14,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__2,A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ( ( v1_jordan2c(D,A)
& r2_jordan2c(A,D,B)
& r2_jordan2c(A,D,C) )
=> B = C ) ) ) ) ) ) ).
fof(t7_gobrd14,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,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k14_euclid(np__2)))
=> ~ ( C = k16_euclid(np__2)
& r2_jordan2c(np__2,k5_topreal1(np__2,A),B)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& r1_tarski(k3_subset_1(u1_struct_0(k14_euclid(np__2)),k9_metric_1(k14_euclid(np__2),C,D)),B) ) ) ) ) ) ) ).
fof(t8_gobrd14,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(A)))
=> ( k5_topreal1(A,B) != k1_xboole_0
=> r1_xreal_0(np__2,k3_finseq_1(B)) ) ) ) ).
fof(d1_gobrd14,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( D = k1_gobrd14(A,B,C)
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(k14_euclid(A)))
& ? [F] :
( m1_subset_1(F,u1_struct_0(k14_euclid(A)))
& E = B
& F = C
& D = k4_metric_1(k14_euclid(A),E,F) ) ) ) ) ) ) ) ).
fof(t9_gobrd14,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)))
=> k1_gobrd14(np__2,A,B) = k9_square_1(k3_real_1(k7_square_1(k5_real_1(k21_euclid(A),k21_euclid(B))),k7_square_1(k5_real_1(k22_euclid(A),k22_euclid(B))))) ) ) ).
fof(t10_gobrd14,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> k1_gobrd14(A,B,B) = np__0 ) ) ).
fof(t11_gobrd14,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> r1_xreal_0(k1_gobrd14(A,B,D),k3_real_1(k1_gobrd14(A,B,C),k1_gobrd14(A,C,D))) ) ) ) ) ).
fof(t12_gobrd14,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(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)))
=> ( ( r1_xreal_0(A,k21_euclid(E))
& r1_xreal_0(k21_euclid(E),B)
& r1_xreal_0(C,k22_euclid(E))
& r1_xreal_0(k22_euclid(E),D)
& r1_xreal_0(A,k21_euclid(F))
& r1_xreal_0(k21_euclid(F),B)
& r1_xreal_0(C,k22_euclid(F))
& r1_xreal_0(k22_euclid(F),D) )
=> r1_xreal_0(k1_gobrd14(np__2,E,F),k2_xcmplx_0(k6_xcmplx_0(B,A),k6_xcmplx_0(D,C))) ) ) ) ) ) ) ) ).
fof(t13_gobrd14,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(np__1,B) )
=> ( r1_xreal_0(k3_finseq_1(C),A)
| r1_xreal_0(k1_matrix_1(C),B)
| k3_goboard5(C,A,B) = k4_card_3(k5_funct_4(k1_zfmisc_1(k1_numbers),np__1,np__2,k1_rcomp_1(k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,A,np__1)),k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,k1_nat_1(A,np__1),np__1))),k1_rcomp_1(k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,np__1,B)),k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,np__1,k1_nat_1(B,np__1)))))) ) ) ) ) ) ).
fof(t14_gobrd14,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(np__1,B) )
=> ( r1_xreal_0(k3_finseq_1(C),A)
| r1_xreal_0(k1_matrix_1(C),B)
| v6_compts_1(k3_goboard5(C,A,B),k15_euclid(np__2)) ) ) ) ) ) ).
fof(t15_gobrd14,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))
& r2_hidden(k4_tarski(k1_nat_1(A,C),B),k2_matrix_1(D)) )
=> k1_gobrd14(np__2,k3_matrix_1(u1_struct_0(k15_euclid(np__2)),D,A,B),k3_matrix_1(u1_struct_0(k15_euclid(np__2)),D,k1_nat_1(A,C),B)) = k5_real_1(k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),D,k1_nat_1(A,C),B)),k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),D,A,B))) ) ) ) ) ) ).
fof(t16_gobrd14,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))
& r2_hidden(k4_tarski(A,k1_nat_1(B,C)),k2_matrix_1(D)) )
=> k1_gobrd14(np__2,k3_matrix_1(u1_struct_0(k15_euclid(np__2)),D,A,B),k3_matrix_1(u1_struct_0(k15_euclid(np__2)),D,A,k1_nat_1(B,C))) = k5_real_1(k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),D,A,k1_nat_1(B,C))),k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),D,A,B))) ) ) ) ) ) ).
fof(t17_gobrd14,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [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)))) )
=> r1_xreal_0(np__3,k5_binarith(k3_finseq_1(k1_jordan8(B,A)),np__1)) ) ) ).
fof(t18_gobrd14,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [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)))) )
=> ( r1_xreal_0(A,B)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__2,D)
& r1_xreal_0(D,k5_real_1(k3_finseq_1(k1_jordan8(C,A)),np__1))
& r1_xreal_0(np__2,E)
& r1_xreal_0(E,k5_real_1(k3_finseq_1(k1_jordan8(C,A)),np__1))
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__2,F)
& r1_xreal_0(F,k5_real_1(k3_finseq_1(k1_jordan8(C,B)),np__1))
& r1_xreal_0(np__2,G)
& r1_xreal_0(G,k5_real_1(k3_finseq_1(k1_jordan8(C,B)),np__1))
& r2_hidden(k4_tarski(F,G),k2_matrix_1(k1_jordan8(C,B)))
& k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(C,A),D,E) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(C,B),F,G)
& F = k1_nat_1(np__2,k2_nat_1(k1_card_4(np__2,k5_binarith(B,A)),k5_binarith(D,np__2)))
& G = k1_nat_1(np__2,k2_nat_1(k1_card_4(np__2,k5_binarith(B,A)),k5_binarith(E,np__2))) ) ) ) ) ) ) ) ) ) ) ).
fof(t19_gobrd14,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] :
( ( v6_compts_1(D,k15_euclid(np__2))
& ~ v1_sppol_1(D)
& ~ v2_sppol_1(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( ( r2_hidden(k4_tarski(A,B),k2_matrix_1(k1_jordan8(D,C)))
& r2_hidden(k4_tarski(A,k1_nat_1(B,np__1)),k2_matrix_1(k1_jordan8(D,C))) )
=> k1_gobrd14(np__2,k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(D,C),A,B),k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(D,C),A,k1_nat_1(B,np__1))) = k6_real_1(k5_real_1(k19_pscomp_1(D),k21_pscomp_1(D)),k1_card_4(np__2,C)) ) ) ) ) ) ).
fof(t20_gobrd14,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] :
( ( v6_compts_1(D,k15_euclid(np__2))
& ~ v1_sppol_1(D)
& ~ v2_sppol_1(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( ( r2_hidden(k4_tarski(A,B),k2_matrix_1(k1_jordan8(D,C)))
& r2_hidden(k4_tarski(k1_nat_1(A,np__1),B),k2_matrix_1(k1_jordan8(D,C))) )
=> k1_gobrd14(np__2,k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(D,C),A,B),k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(D,C),k1_nat_1(A,np__1),B)) = k6_real_1(k5_real_1(k20_pscomp_1(D),k18_pscomp_1(D)),k1_card_4(np__2,C)) ) ) ) ) ) ).
fof(t21_gobrd14,axiom,
! [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] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(B,np__0)
& ~ r1_xreal_0(C,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(D,np__1)
& ~ r1_xreal_0(B,k1_gobrd14(np__2,k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(A,D),np__1,np__1),k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(A,D),np__1,np__2)))
& ~ r1_xreal_0(C,k1_gobrd14(np__2,k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(A,D),np__1,np__1),k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(A,D),np__2,np__1))) ) ) ) ) ) ) ).
fof(t22_gobrd14,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,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A))))))
=> ~ ( r3_connsp_1(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A))),B)
& B != k3_goboard9(A)
& B != k2_goboard9(A) ) ) ) ).
fof(t23_gobrd14,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,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)),k5_topreal1(np__2,A)) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),B,C)
& r1_xboole_0(B,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)),k5_topreal1(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)),k5_topreal1(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)),k5_topreal1(np__2,A))),D)
& r3_connsp_1(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A))),E) ) ) ) )
& ~ ( B = k3_goboard9(A)
& C = k2_goboard9(A) )
& ~ ( B = k2_goboard9(A)
& C = k3_goboard9(A) ) ) ) ) ) ).
fof(t24_gobrd14,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))) )
=> r1_subset_1(k2_goboard9(A),k3_goboard9(A)) ) ).
fof(t25_gobrd14,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))) )
=> k4_subset_1(u1_struct_0(k15_euclid(np__2)),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A),k3_goboard9(A)),k2_goboard9(A)) = u1_struct_0(k15_euclid(np__2)) ) ).
fof(t26_gobrd14,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)))
=> ( r2_hidden(B,k5_topreal1(np__2,A))
<=> ( ~ r2_hidden(B,k2_goboard9(A))
& ~ r2_hidden(B,k3_goboard9(A)) ) ) ) ) ).
fof(t27_gobrd14,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)))
=> ( r2_hidden(B,k2_goboard9(A))
<=> ( ~ r2_hidden(B,k5_topreal1(np__2,A))
& ~ r2_hidden(B,k3_goboard9(A)) ) ) ) ) ).
fof(t28_gobrd14,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)))
=> ( r2_hidden(B,k3_goboard9(A))
<=> ( ~ r2_hidden(B,k5_topreal1(np__2,A))
& ~ r2_hidden(B,k2_goboard9(A)) ) ) ) ) ).
fof(t29_gobrd14,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))) )
=> k5_topreal1(np__2,A) = k6_subset_1(u1_struct_0(k15_euclid(np__2)),k6_pre_topc(k15_euclid(np__2),k3_goboard9(A)),k3_goboard9(A)) ) ).
fof(t30_gobrd14,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))) )
=> k5_topreal1(np__2,A) = k6_subset_1(u1_struct_0(k15_euclid(np__2)),k6_pre_topc(k15_euclid(np__2),k2_goboard9(A)),k2_goboard9(A)) ) ).
fof(t31_gobrd14,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))) )
=> k6_pre_topc(k15_euclid(np__2),k3_goboard9(A)) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k3_goboard9(A),k5_topreal1(np__2,A)) ) ).
fof(t32_gobrd14,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))) )
=> k6_pre_topc(k15_euclid(np__2),k2_goboard9(A)) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k2_goboard9(A),k5_topreal1(np__2,A)) ) ).
fof(t33_gobrd14,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)
& v1_sprect_2(B)
& m2_finseq_1(B,u1_struct_0(k15_euclid(np__2))) )
=> ~ ( r2_hidden(A,k3_goboard9(B))
& r1_xreal_0(k21_euclid(A),k18_pscomp_1(k5_topreal1(np__2,B))) ) ) ) ).
fof(t34_gobrd14,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)
& v1_sprect_2(B)
& m2_finseq_1(B,u1_struct_0(k15_euclid(np__2))) )
=> ~ ( r2_hidden(A,k3_goboard9(B))
& r1_xreal_0(k20_pscomp_1(k5_topreal1(np__2,B)),k21_euclid(A)) ) ) ) ).
fof(t35_gobrd14,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)
& v1_sprect_2(B)
& m2_finseq_1(B,u1_struct_0(k15_euclid(np__2))) )
=> ~ ( r2_hidden(A,k3_goboard9(B))
& r1_xreal_0(k19_pscomp_1(k5_topreal1(np__2,B)),k22_euclid(A)) ) ) ) ).
fof(t36_gobrd14,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)
& v1_sprect_2(B)
& m2_finseq_1(B,u1_struct_0(k15_euclid(np__2))) )
=> ~ ( r2_hidden(A,k3_goboard9(B))
& r1_xreal_0(k22_euclid(A),k21_pscomp_1(k5_topreal1(np__2,B))) ) ) ) ).
fof(t37_gobrd14,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)
& v1_sprect_2(C)
& m2_finseq_1(C,u1_struct_0(k15_euclid(np__2))) )
=> ~ ( r2_hidden(A,k3_goboard9(C))
& r2_hidden(B,k2_goboard9(C))
& r2_subset_1(k3_topreal1(np__2,A,B),k5_topreal1(np__2,C)) ) ) ) ) ).
fof(t38_gobrd14,axiom,
! [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)))) )
=> k6_pre_topc(k15_euclid(np__2),k3_goboard9(k1_sprect_1(A))) = k4_card_3(k5_funct_4(k1_zfmisc_1(k1_numbers),np__1,np__2,k1_rcomp_1(k18_pscomp_1(k5_topreal1(np__2,k1_sprect_1(A))),k20_pscomp_1(k5_topreal1(np__2,k1_sprect_1(A)))),k1_rcomp_1(k21_pscomp_1(k5_topreal1(np__2,k1_sprect_1(A))),k19_pscomp_1(k5_topreal1(np__2,k1_sprect_1(A)))))) ) ).
fof(t39_gobrd14,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)
& v1_sprect_2(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k16_pscomp_1,k6_pre_topc(k15_euclid(np__2),k3_goboard9(A))) = k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k16_pscomp_1,k5_topreal1(np__2,A)) ) ).
fof(t40_gobrd14,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)
& v1_sprect_2(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k17_pscomp_1,k6_pre_topc(k15_euclid(np__2),k3_goboard9(A))) = k2_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k17_pscomp_1,k5_topreal1(np__2,A)) ) ).
fof(t41_gobrd14,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)
& v1_sprect_2(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> r1_tarski(k3_goboard9(A),k3_goboard9(k1_sprect_1(k5_topreal1(np__2,A)))) ) ).
fof(t42_gobrd14,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)
& v1_sprect_2(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> v6_compts_1(k6_pre_topc(k15_euclid(np__2),k3_goboard9(A)),k15_euclid(np__2)) ) ).
fof(t43_gobrd14,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)
& v1_sprect_2(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ~ v1_jordan2c(k2_goboard9(A),np__2) ) ).
fof(t44_gobrd14,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)
& v1_sprect_2(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> r2_jordan2c(np__2,k5_topreal1(np__2,A),k2_goboard9(A)) ) ).
fof(t45_gobrd14,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)
& v1_sprect_2(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> r1_jordan2c(np__2,k5_topreal1(np__2,A),k3_goboard9(A)) ) ).
fof(t46_gobrd14,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)
& v1_sprect_2(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> k2_jordan2c(np__2,k5_topreal1(np__2,A)) = k2_goboard9(A) ) ).
fof(t47_gobrd14,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)
& v1_sprect_2(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> k1_jordan2c(np__2,k5_topreal1(np__2,A)) = k3_goboard9(A) ) ).
fof(t48_gobrd14,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)
& v1_sprect_2(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( r2_jordan2c(np__2,k5_topreal1(np__2,A),B)
=> B = k2_goboard9(A) ) ) ) ).
fof(t49_gobrd14,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)
& v1_sprect_2(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ~ ( r1_jordan2c(np__2,k5_topreal1(np__2,A),B)
& r1_xboole_0(B,k3_goboard9(A)) ) ) ) ).
fof(t50_gobrd14,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)
& v1_sprect_2(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ( r1_jordan2c(np__2,k5_topreal1(np__2,A),B)
=> B = k1_jordan2c(np__2,k5_topreal1(np__2,A)) ) ) ) ).
fof(t51_gobrd14,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)
& v1_sprect_2(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> k18_pscomp_1(k5_topreal1(np__2,A)) = k18_pscomp_1(k3_goboard9(A)) ) ).
fof(t52_gobrd14,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)
& v1_sprect_2(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> k20_pscomp_1(k5_topreal1(np__2,A)) = k20_pscomp_1(k3_goboard9(A)) ) ).
fof(t53_gobrd14,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)
& v1_sprect_2(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> k19_pscomp_1(k5_topreal1(np__2,A)) = k19_pscomp_1(k3_goboard9(A)) ) ).
fof(t54_gobrd14,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)
& v1_sprect_2(A)
& m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
=> k21_pscomp_1(k5_topreal1(np__2,A)) = k21_pscomp_1(k3_goboard9(A)) ) ).
fof(dt_k1_gobrd14,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& m1_subset_1(C,u1_struct_0(k15_euclid(A))) )
=> m1_subset_1(k1_gobrd14(A,B,C),k1_numbers) ) ).
fof(commutativity_k1_gobrd14,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& m1_subset_1(C,u1_struct_0(k15_euclid(A))) )
=> k1_gobrd14(A,B,C) = k1_gobrd14(A,C,B) ) ).
%------------------------------------------------------------------------------