SET007 Axioms: SET007+626.ax
%------------------------------------------------------------------------------
% File : SET007+626 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Continuous Lattices of Maps between T_0 Spaces
% 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 : waybel26 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 68 ( 1 unt; 0 def)
% Number of atoms : 765 ( 24 equ)
% Maximal formula atoms : 25 ( 11 avg)
% Number of connectives : 838 ( 141 ~; 0 |; 497 &)
% ( 7 <=>; 193 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 10 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 66 ( 65 usr; 0 prp; 1-4 aty)
% Number of functors : 43 ( 43 usr; 2 con; 0-5 aty)
% Number of variables : 209 ( 202 !; 7 ?)
% 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_waybel26,axiom,
! [A,B] :
( ( v1_yellow_1(B)
& v4_waybel_3(B)
& m1_pboole(B,A) )
=> ( ~ v3_struct_0(k5_yellow_1(A,B))
& v1_orders_2(k5_yellow_1(A,B))
& v1_monoid_0(k5_yellow_1(A,B)) ) ) ).
fof(fc2_waybel26,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ( ~ v3_struct_0(k1_waybel26(A,B))
& v1_orders_2(k1_waybel26(A,B))
& v2_orders_2(k1_waybel26(A,B))
& v3_orders_2(k1_waybel26(A,B))
& v1_monoid_0(k1_waybel26(A,B)) ) ) ).
fof(fc3_waybel26,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_t_0topsp(B)
& l1_pre_topc(B) )
=> ( ~ v3_struct_0(k1_waybel26(A,B))
& v1_orders_2(k1_waybel26(A,B))
& v2_orders_2(k1_waybel26(A,B))
& v3_orders_2(k1_waybel26(A,B))
& v4_orders_2(k1_waybel26(A,B))
& v1_monoid_0(k1_waybel26(A,B)) ) ) ).
fof(fc4_waybel26,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k1_waybel26(A,B)))) )
=> ~ v1_xboole_0(k5_card_3(C,D)) ) ).
fof(fc5_waybel26,axiom,
( ~ v3_struct_0(k9_waybel18)
& v1_pre_topc(k9_waybel18)
& v2_pre_topc(k9_waybel18)
& v2_t_0topsp(k9_waybel18)
& ~ v2_tdlat_3(k9_waybel18)
& ~ v1_urysohn1(k9_waybel18)
& ~ v3_realset2(k9_waybel18)
& v2_waybel18(k9_waybel18)
& v1_waybel25(k9_waybel18) ) ).
fof(rc1_waybel26,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& v2_pre_topc(A)
& v2_t_0topsp(A)
& ~ v2_tdlat_3(A)
& ~ v3_realset2(A)
& v2_waybel18(A)
& v1_waybel25(A) ) ).
fof(rc2_waybel26,axiom,
? [A] :
( l1_waybel_9(A)
& ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_yellow_0(A)
& v2_yellow_0(A)
& v3_yellow_0(A)
& v24_waybel_0(A)
& v25_waybel_0(A)
& v2_pre_topc(A)
& v4_waybel11(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v3_lattice3(A)
& v2_t_0topsp(A)
& ~ v2_tdlat_3(A)
& ~ v3_realset2(A) ) ).
fof(fc6_waybel26,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> v1_relat_1(k3_card_3(k2_card_3(A))) ) ).
fof(cc1_waybel26,axiom,
! [A,B] :
( ( v2_pre_topc(A)
& l1_pre_topc(A)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B))))
=> v1_relat_1(C) ) ) ).
fof(cc2_waybel26,axiom,
! [A,B] :
( ( v2_pre_topc(A)
& l1_pre_topc(A)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,u1_pre_topc(k6_borsuk_1(A,B)))
=> v1_relat_1(C) ) ) ).
fof(d1_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> k1_waybel26(A,B) = k3_waybel24(A,k1_waybel25(B)) ) ) ).
fof(t1_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_waybel26(A,B)))
<=> ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(k1_waybel25(B)))
& v5_pre_topc(C,A,k1_waybel25(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(k1_waybel25(B))) ) ) ) ) ).
fof(t2_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_waybel26(A,B)))
<=> ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) ) ) ) ) ).
fof(t3_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_waybel26(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k1_waybel26(A,B)))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(k1_waybel25(B)))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(k1_waybel25(B))) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(A),u1_struct_0(k1_waybel25(B)))
& m2_relset_1(F,u1_struct_0(A),u1_struct_0(k1_waybel25(B))) )
=> ( ( C = E
& D = F )
=> ( r3_orders_2(k1_waybel26(A,B),C,D)
<=> r1_yellow_2(u1_struct_0(A),k1_waybel25(B),E,F) ) ) ) ) ) ) ) ) ).
fof(t4_waybel26,axiom,
m1_yellow_9(k1_waybel25(k9_waybel18),k3_yellow_1(np__1)) ).
fof(t5_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k2_yellow_1(u1_pre_topc(A))),u1_struct_0(k1_waybel26(A,k9_waybel18)))
& m2_relset_1(B,u1_struct_0(k2_yellow_1(u1_pre_topc(A))),u1_struct_0(k1_waybel26(A,k9_waybel18)))
& v23_waybel_0(B,k2_yellow_1(u1_pre_topc(A)),k1_waybel26(A,k9_waybel18))
& ! [C] :
( ( v3_pre_topc(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> k1_funct_1(B,C) = k5_funct_3(C,u1_struct_0(A)) ) ) ) ).
fof(t6_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> r5_waybel_1(k2_yellow_1(u1_pre_topc(A)),k1_waybel26(A,k9_waybel18)) ) ).
fof(d2_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(C))
& v5_pre_topc(D,B,C)
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k1_waybel26(A,B)),u1_struct_0(k1_waybel26(A,C)))
& m2_relset_1(E,u1_struct_0(k1_waybel26(A,B)),u1_struct_0(k1_waybel26(A,C))) )
=> ( E = k3_waybel26(A,B,C,D)
<=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(F,A,B)
& m2_relset_1(F,u1_struct_0(A),u1_struct_0(B)) )
=> k1_funct_1(E,F) = k4_borsuk_1(A,B,C,F,D) ) ) ) ) ) ) ) ).
fof(d3_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(C))
& v5_pre_topc(D,B,C)
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k1_waybel26(C,A)),u1_struct_0(k1_waybel26(B,A)))
& m2_relset_1(E,u1_struct_0(k1_waybel26(C,A)),u1_struct_0(k1_waybel26(B,A))) )
=> ( E = k4_waybel26(A,B,C,D)
<=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(C),u1_struct_0(A))
& v5_pre_topc(F,C,A)
& m2_relset_1(F,u1_struct_0(C),u1_struct_0(A)) )
=> k1_funct_1(E,F) = k4_borsuk_1(B,C,A,D,F) ) ) ) ) ) ) ) ).
fof(t7_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_t_0topsp(B)
& v1_waybel25(B)
& l1_pre_topc(B) )
=> ( v4_waybel_0(k1_waybel26(A,B),k6_yellow_1(u1_struct_0(A),k1_waybel25(B)))
& m1_yellow_0(k1_waybel26(A,B),k6_yellow_1(u1_struct_0(A),k1_waybel25(B))) ) ) ) ).
fof(t8_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_t_0topsp(B)
& v1_waybel25(B)
& l1_pre_topc(B) )
=> v24_waybel_0(k1_waybel26(A,B)) ) ) ).
fof(t9_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(C))
& v5_pre_topc(D,B,C)
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
=> v5_orders_3(k3_waybel26(A,B,C,D),k1_waybel26(A,B),k1_waybel26(A,C)) ) ) ) ) ).
fof(t10_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(B),u1_struct_0(B))
& v5_pre_topc(C,B,B)
& m2_relset_1(C,u1_struct_0(B),u1_struct_0(B)) )
=> ( v11_quantal1(C)
=> v11_quantal1(k3_waybel26(A,B,B,C)) ) ) ) ) ).
fof(t11_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(C))
& v5_pre_topc(D,B,C)
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
=> v5_orders_3(k4_waybel26(A,B,C,D),k1_waybel26(C,A),k1_waybel26(B,A)) ) ) ) ) ).
fof(t12_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(B),u1_struct_0(B))
& v5_pre_topc(C,B,B)
& m2_relset_1(C,u1_struct_0(B),u1_struct_0(B)) )
=> ( v11_quantal1(C)
=> v11_quantal1(k4_waybel26(A,B,B,C)) ) ) ) ) ).
fof(t13_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(C))
& v5_pre_topc(D,B,C)
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k1_waybel26(A,B))))
=> k2_waybel26(A,C,E,k4_pre_topc(k1_waybel26(A,B),k1_waybel26(A,C),k3_waybel26(A,B,C,D),F)) = k4_pre_topc(B,C,D,k2_waybel26(A,B,E,F)) ) ) ) ) ) ) ).
fof(t14_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_t_0topsp(B)
& v1_waybel25(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_pre_topc(C)
& v2_t_0topsp(C)
& v1_waybel25(C)
& l1_pre_topc(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(C))
& v5_pre_topc(D,B,C)
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
=> v22_waybel_0(k3_waybel26(A,B,C,D),k1_waybel26(A,B),k1_waybel26(A,C)) ) ) ) ) ).
fof(t15_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(C))
& v5_pre_topc(D,B,C)
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k1_waybel26(C,A))))
=> k2_waybel26(B,A,E,k4_pre_topc(k1_waybel26(C,A),k1_waybel26(B,A),k4_waybel26(A,B,C,D),F)) = k2_waybel26(C,A,k7_yellow_2(u1_struct_0(B),C,D,E),F) ) ) ) ) ) ) ).
fof(t16_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_pre_topc(C)
& v2_t_0topsp(C)
& v1_waybel25(C)
& l1_pre_topc(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(D,A,B)
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
=> v22_waybel_0(k4_waybel26(C,A,B,D),k1_waybel26(B,C),k1_waybel26(A,C)) ) ) ) ) ).
fof(t17_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m2_tsp_1(C,B) )
=> ( v4_yellow_0(k1_waybel26(A,C),k1_waybel26(A,B))
& m1_yellow_0(k1_waybel26(A,C),k1_waybel26(A,B)) ) ) ) ) ).
fof(t18_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v2_t_0topsp(A)
& v1_waybel25(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m2_tsp_1(B,A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v3_borsuk_1(C,A,B)
=> ( v4_waybel_0(k1_waybel25(B),k1_waybel25(A))
& m1_yellow_0(k1_waybel25(B),k1_waybel25(A)) ) ) ) ) ) ).
fof(t19_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_t_0topsp(B)
& v1_waybel25(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m2_tsp_1(C,B) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(C))
& v5_pre_topc(D,B,C)
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
=> ( v3_borsuk_1(D,B,C)
=> r1_yellow16(k1_waybel26(A,C),k1_waybel26(A,B),k3_waybel26(A,B,C,D)) ) ) ) ) ) ).
fof(t20_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_t_0topsp(B)
& v1_waybel25(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m2_tsp_1(C,B) )
=> ( r1_borsuk_1(B,C)
=> r3_yellow16(k1_waybel26(A,C),k1_waybel26(A,B)) ) ) ) ) ).
fof(t21_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(C))
& v5_pre_topc(D,B,C)
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
=> ( v3_tops_2(D,B,C)
=> v23_waybel_0(k3_waybel26(A,B,C,D),k1_waybel26(A,B),k1_waybel26(A,C)) ) ) ) ) ) ).
fof(t22_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ( r1_t_0topsp(B,C)
=> r5_waybel_1(k1_waybel26(A,B),k1_waybel26(A,C)) ) ) ) ) ).
fof(t23_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_t_0topsp(B)
& v1_waybel25(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m2_tsp_1(C,B) )
=> ( ( r1_borsuk_1(B,C)
& v3_lattice3(k1_waybel26(A,B))
& v3_waybel_3(k1_waybel26(A,B)) )
=> ( v3_lattice3(k1_waybel26(A,C))
& v3_waybel_3(k1_waybel26(A,C)) ) ) ) ) ) ).
fof(t24_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_t_0topsp(B)
& v1_waybel25(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_pre_topc(C)
& v2_t_0topsp(C)
& v1_waybel25(C)
& l1_pre_topc(C) )
=> ( ( r1_waybel18(B,C)
& v3_lattice3(k1_waybel26(A,C))
& v3_waybel_3(k1_waybel26(A,C)) )
=> ( v3_lattice3(k1_waybel26(A,B))
& v3_waybel_3(k1_waybel26(A,B)) ) ) ) ) ) ).
fof(t25_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v2_t_0topsp(A)
& ~ v3_realset2(A)
& l1_pre_topc(A) )
=> ( ~ v1_urysohn1(A)
=> r1_waybel18(k9_waybel18,A) ) ) ).
fof(t26_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_t_0topsp(B)
& ~ v3_realset2(B)
& l1_pre_topc(B) )
=> ~ ( v1_lattice3(k1_waybel26(A,B))
& v1_urysohn1(B) ) ) ) ).
fof(t27_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_t_0topsp(B)
& ~ v3_realset2(B)
& v1_waybel25(B)
& l1_pre_topc(B) )
=> ( ( v3_lattice3(k1_waybel26(A,B))
& v3_waybel_3(k1_waybel26(A,B)) )
=> v3_waybel_3(k2_yellow_1(u1_pre_topc(A))) ) ) ) ).
fof(t28_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_pre_topc(C)
& v2_t_0topsp(C)
& v1_waybel25(C)
& l1_pre_topc(C) )
=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k1_waybel26(A,C)),u1_struct_0(k1_waybel26(A,C)))
& v22_waybel_0(D,k1_waybel26(A,C),k1_waybel26(A,C))
& v6_waybel_1(D,k1_waybel26(A,C))
& m2_relset_1(D,u1_struct_0(k1_waybel26(A,C)),u1_struct_0(k1_waybel26(A,C)))
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(C))
& v5_pre_topc(E,A,C)
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(C)) )
=> k1_funct_1(D,E) = k3_borsuk_1(A,C,k7_yellow_2(u1_struct_0(A),C,E,B)) )
& ? [E] :
( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(A))
& v5_pre_topc(E,A,A)
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(A))
& E = k3_borsuk_1(A,A,B)
& D = k4_waybel26(C,A,A,E) ) ) ) ) ) ).
fof(t29_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_t_0topsp(B)
& v1_waybel25(B)
& l1_pre_topc(B) )
=> ( ( v3_lattice3(k1_waybel26(A,B))
& v3_waybel_3(k1_waybel26(A,B)) )
=> ( v3_lattice3(k1_waybel25(B))
& v3_waybel_3(k1_waybel25(B)) ) ) ) ) ).
fof(t30_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v4_waybel_3(C)
& v1_waybel18(C)
& m1_pboole(C,B) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(k3_waybel18(B,C)))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(k3_waybel18(B,C))) )
=> ! [E] :
( m1_subset_1(E,B)
=> k1_funct_1(k10_funct_6(D),E) = k7_funct_2(u1_struct_0(A),u1_struct_0(k3_waybel18(B,C)),u1_struct_0(k4_waybel18(B,C,E)),D,k6_waybel18(B,C,E)) ) ) ) ) ) ).
fof(t31_waybel26,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] : r6_pboole(B,k12_pralg_1(B,k2_pre_circ(B,A)),k2_pre_circ(B,u1_struct_0(A))) ) ).
fof(t32_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(k3_waybel18(C,k2_pre_circ(C,B))))
& v5_pre_topc(D,A,k3_waybel18(C,k2_pre_circ(C,B)))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(k3_waybel18(C,k2_pre_circ(C,B)))) )
=> ( v1_funct_1(k10_funct_6(D))
& v1_funct_2(k10_funct_6(D),C,u1_struct_0(k1_waybel26(A,B)))
& m2_relset_1(k10_funct_6(D),C,u1_struct_0(k1_waybel26(A,B))) ) ) ) ) ) ).
fof(t33_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> r1_tarski(u1_struct_0(k1_waybel26(A,B)),k1_funct_2(u1_struct_0(A),u1_struct_0(B))) ) ) ).
fof(t34_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,C,u1_struct_0(k1_waybel26(A,B)))
& m2_relset_1(D,C,u1_struct_0(k1_waybel26(A,B))) )
=> ( v1_funct_1(k10_funct_6(D))
& v1_funct_2(k10_funct_6(D),u1_struct_0(A),u1_struct_0(k3_waybel18(C,k2_pre_circ(C,B))))
& v5_pre_topc(k10_funct_6(D),A,k3_waybel18(C,k2_pre_circ(C,B)))
& m2_relset_1(k10_funct_6(D),u1_struct_0(A),u1_struct_0(k3_waybel18(C,k2_pre_circ(C,B)))) ) ) ) ) ) ).
fof(t35_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k1_waybel26(A,k3_waybel18(B,k2_pre_circ(B,k9_waybel18)))),u1_struct_0(k5_yellow_1(B,k2_pre_circ(B,k1_waybel26(A,k9_waybel18)))))
& m2_relset_1(C,u1_struct_0(k1_waybel26(A,k3_waybel18(B,k2_pre_circ(B,k9_waybel18)))),u1_struct_0(k5_yellow_1(B,k2_pre_circ(B,k1_waybel26(A,k9_waybel18)))))
& v23_waybel_0(C,k1_waybel26(A,k3_waybel18(B,k2_pre_circ(B,k9_waybel18))),k5_yellow_1(B,k2_pre_circ(B,k1_waybel26(A,k9_waybel18))))
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(k3_waybel18(B,k2_pre_circ(B,k9_waybel18))))
& v5_pre_topc(D,A,k3_waybel18(B,k2_pre_circ(B,k9_waybel18)))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(k3_waybel18(B,k2_pre_circ(B,k9_waybel18)))) )
=> k1_funct_1(C,D) = k10_funct_6(D) ) ) ) ) ).
fof(t36_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> r5_waybel_1(k1_waybel26(A,k3_waybel18(B,k2_pre_circ(B,k9_waybel18))),k5_yellow_1(B,k2_pre_circ(B,k1_waybel26(A,k9_waybel18)))) ) ) ).
fof(t37_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v3_waybel_3(k2_yellow_1(u1_pre_topc(A)))
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v2_t_0topsp(B)
& v2_waybel18(B)
& l1_pre_topc(B) )
=> ( v3_lattice3(k1_waybel26(A,B))
& v3_waybel_3(k1_waybel26(A,B)) ) ) ) ) ).
fof(t38_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v2_pre_topc(B)
& v4_waybel11(B)
& v1_lattice3(B)
& v2_lattice3(B)
& v3_lattice3(B)
& ~ v3_realset2(B)
& l1_waybel_9(B) )
=> ( ( ( v3_lattice3(k1_waybel26(A,B))
& v3_waybel_3(k1_waybel26(A,B)) )
=> ( v3_waybel_3(k2_yellow_1(u1_pre_topc(A)))
& v3_waybel_3(B) ) )
& ( ( v3_waybel_3(k2_yellow_1(u1_pre_topc(A)))
& v3_waybel_3(B) )
=> ( v3_lattice3(k1_waybel26(A,B))
& v3_waybel_3(k1_waybel26(A,B)) ) ) ) ) ) ).
fof(d4_waybel26,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> k5_waybel26(A) = k4_relat_1(k3_card_3(k2_card_3(A))) ) ).
fof(t39_waybel26,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(k4_tarski(A,B),k5_waybel26(C))
<=> ( r2_hidden(A,k1_relat_1(C))
& r2_hidden(B,k1_funct_1(C,A)) ) ) ) ).
fof(t40_waybel26,axiom,
! [A] :
( v1_finset_1(A)
=> ( v1_finset_1(k1_funct_5(A))
& v1_finset_1(k2_funct_5(A)) ) ) ).
fof(t41_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v4_waybel11(C)
& m1_yellow_9(C,k2_yellow_1(u1_pre_topc(B))) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(C))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(C)) )
=> ( ( v3_pre_topc(k5_waybel26(D),k6_borsuk_1(A,B))
& m1_subset_1(k5_waybel26(D),k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))) )
=> v5_pre_topc(D,A,C) ) ) ) ) ) ).
fof(d5_waybel26,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( C = k6_waybel26(A,B)
<=> ( k1_relat_1(C) = B
& ! [D] :
( r2_hidden(D,B)
=> k1_funct_1(C,D) = k9_relat_1(A,k1_tarski(D)) ) ) ) ) ) ).
fof(t42_waybel26,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r1_tarski(k1_relat_1(A),B)
=> k5_waybel26(k6_waybel26(A,B)) = A ) ) ).
fof(t43_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v3_pre_topc(C,k6_borsuk_1(A,B))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( v3_pre_topc(k9_relat_1(C,k1_struct_0(A,D)),B)
& m1_subset_1(k9_relat_1(C,k1_struct_0(A,D)),k1_zfmisc_1(u1_struct_0(B))) ) ) ) ) ) ).
fof(t44_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v4_waybel11(C)
& m1_yellow_9(C,k2_yellow_1(u1_pre_topc(B))) )
=> ! [D] :
( ( v3_pre_topc(D,k6_borsuk_1(A,B))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))) )
=> ( v1_funct_1(k6_waybel26(D,u1_struct_0(A)))
& v1_funct_2(k6_waybel26(D,u1_struct_0(A)),u1_struct_0(A),u1_struct_0(C))
& v5_pre_topc(k6_waybel26(D,u1_struct_0(A)),A,C)
& m2_relset_1(k6_waybel26(D,u1_struct_0(A)),u1_struct_0(A),u1_struct_0(C)) ) ) ) ) ) ).
fof(t45_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v4_waybel11(C)
& m1_yellow_9(C,k2_yellow_1(u1_pre_topc(B))) )
=> ! [D] :
( ( v3_pre_topc(D,k6_borsuk_1(A,B))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))) )
=> ! [E] :
( ( v3_pre_topc(E,k6_borsuk_1(A,B))
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))) )
=> ( r1_tarski(D,E)
=> ! [F] :
( m1_subset_1(F,u1_struct_0(k1_waybel26(A,C)))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(k1_waybel26(A,C)))
=> ( ( F = k6_waybel26(D,u1_struct_0(A))
& G = k6_waybel26(E,u1_struct_0(A)) )
=> r3_orders_2(k1_waybel26(A,C),F,G) ) ) ) ) ) ) ) ) ) ).
fof(t46_waybel26,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( ( v4_waybel11(C)
& m1_yellow_9(C,k2_yellow_1(u1_pre_topc(B))) )
=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k2_yellow_1(u1_pre_topc(k6_borsuk_1(A,B)))),u1_struct_0(k1_waybel26(A,C)))
& m2_relset_1(D,u1_struct_0(k2_yellow_1(u1_pre_topc(k6_borsuk_1(A,B)))),u1_struct_0(k1_waybel26(A,C)))
& v5_orders_3(D,k2_yellow_1(u1_pre_topc(k6_borsuk_1(A,B))),k1_waybel26(A,C))
& ! [E] :
( ( v3_pre_topc(E,k6_borsuk_1(A,B))
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))) )
=> k1_funct_1(D,E) = k6_waybel26(E,u1_struct_0(A)) ) ) ) ) ) ).
fof(dt_k1_waybel26,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ( ~ v3_struct_0(k1_waybel26(A,B))
& v1_orders_2(k1_waybel26(A,B))
& l1_orders_2(k1_waybel26(A,B)) ) ) ).
fof(dt_k2_waybel26,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k1_waybel26(A,B)))) )
=> m1_subset_1(k2_waybel26(A,B,C,D),k1_zfmisc_1(u1_struct_0(k1_waybel25(B)))) ) ).
fof(redefinition_k2_waybel26,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k1_waybel26(A,B)))) )
=> k2_waybel26(A,B,C,D) = k5_card_3(C,D) ) ).
fof(dt_k3_waybel26,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C)
& v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(C))
& v5_pre_topc(D,B,C)
& m1_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
=> ( v1_funct_1(k3_waybel26(A,B,C,D))
& v1_funct_2(k3_waybel26(A,B,C,D),u1_struct_0(k1_waybel26(A,B)),u1_struct_0(k1_waybel26(A,C)))
& m2_relset_1(k3_waybel26(A,B,C,D),u1_struct_0(k1_waybel26(A,B)),u1_struct_0(k1_waybel26(A,C))) ) ) ).
fof(dt_k4_waybel26,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C)
& v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(C))
& v5_pre_topc(D,B,C)
& m1_relset_1(D,u1_struct_0(B),u1_struct_0(C)) )
=> ( v1_funct_1(k4_waybel26(A,B,C,D))
& v1_funct_2(k4_waybel26(A,B,C,D),u1_struct_0(k1_waybel26(C,A)),u1_struct_0(k1_waybel26(B,A)))
& m2_relset_1(k4_waybel26(A,B,C,D),u1_struct_0(k1_waybel26(C,A)),u1_struct_0(k1_waybel26(B,A))) ) ) ).
fof(dt_k5_waybel26,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> v1_relat_1(k5_waybel26(A)) ) ).
fof(dt_k6_waybel26,axiom,
! [A,B] :
( v1_relat_1(A)
=> ( v1_relat_1(k6_waybel26(A,B))
& v1_funct_1(k6_waybel26(A,B)) ) ) ).
%------------------------------------------------------------------------------