TPTP Problem File: LAT351+2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : LAT351+2 : TPTP v8.2.0. Released v3.4.0.
% Domain : Lattice Theory
% Problem : Representation Theorem for Free Continuous Lattices T18
% Version : [Urb08] axioms : Especial.
% English :
% Refs : [Rud96] Rudnicki (1998), Representation Theorem for Free Conti
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : t18_waybel22 [Urb08]
% Status : Theorem
% Rating : 0.83 v8.2.0, 0.89 v8.1.0, 0.92 v7.5.0, 0.94 v7.4.0, 0.93 v7.1.0, 0.96 v7.0.0, 0.97 v6.4.0, 0.96 v6.2.0, 1.00 v3.4.0
% Syntax : Number of formulae : 8976 (1784 unt; 0 def)
% Number of atoms : 51961 (6123 equ)
% Maximal formula atoms : 62 ( 5 avg)
% Number of connectives : 48508 (5523 ~; 316 |;24088 &)
% (1574 <=>;17007 =>; 0 <=; 0 <~>)
% Maximal formula depth : 36 ( 7 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 658 ( 656 usr; 1 prp; 0-6 aty)
% Number of functors : 1508 (1508 usr; 443 con; 0-10 aty)
% Number of variables : 22027 (20895 !;1132 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Bushy version: includes all articles that contribute axioms to the
% Normal version.
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : The problem encoding is based on set theory.
%------------------------------------------------------------------------------
include('Axioms/SET007/SET007+0.ax').
include('Axioms/SET007/SET007+1.ax').
include('Axioms/SET007/SET007+2.ax').
include('Axioms/SET007/SET007+3.ax').
include('Axioms/SET007/SET007+4.ax').
include('Axioms/SET007/SET007+6.ax').
include('Axioms/SET007/SET007+7.ax').
include('Axioms/SET007/SET007+8.ax').
include('Axioms/SET007/SET007+9.ax').
include('Axioms/SET007/SET007+10.ax').
include('Axioms/SET007/SET007+11.ax').
include('Axioms/SET007/SET007+13.ax').
include('Axioms/SET007/SET007+14.ax').
include('Axioms/SET007/SET007+15.ax').
include('Axioms/SET007/SET007+16.ax').
include('Axioms/SET007/SET007+17.ax').
include('Axioms/SET007/SET007+18.ax').
include('Axioms/SET007/SET007+19.ax').
include('Axioms/SET007/SET007+20.ax').
include('Axioms/SET007/SET007+21.ax').
include('Axioms/SET007/SET007+23.ax').
include('Axioms/SET007/SET007+24.ax').
include('Axioms/SET007/SET007+25.ax').
include('Axioms/SET007/SET007+26.ax').
include('Axioms/SET007/SET007+31.ax').
include('Axioms/SET007/SET007+34.ax').
include('Axioms/SET007/SET007+35.ax').
include('Axioms/SET007/SET007+40.ax').
include('Axioms/SET007/SET007+48.ax').
include('Axioms/SET007/SET007+51.ax').
include('Axioms/SET007/SET007+54.ax').
include('Axioms/SET007/SET007+55.ax').
include('Axioms/SET007/SET007+59.ax').
include('Axioms/SET007/SET007+60.ax').
include('Axioms/SET007/SET007+61.ax').
include('Axioms/SET007/SET007+64.ax').
include('Axioms/SET007/SET007+67.ax').
include('Axioms/SET007/SET007+68.ax').
include('Axioms/SET007/SET007+76.ax').
include('Axioms/SET007/SET007+77.ax').
include('Axioms/SET007/SET007+79.ax').
include('Axioms/SET007/SET007+80.ax').
include('Axioms/SET007/SET007+86.ax').
include('Axioms/SET007/SET007+91.ax').
include('Axioms/SET007/SET007+117.ax').
include('Axioms/SET007/SET007+125.ax').
include('Axioms/SET007/SET007+159.ax').
include('Axioms/SET007/SET007+182.ax').
include('Axioms/SET007/SET007+188.ax').
include('Axioms/SET007/SET007+200.ax').
include('Axioms/SET007/SET007+202.ax').
include('Axioms/SET007/SET007+205.ax').
include('Axioms/SET007/SET007+206.ax').
include('Axioms/SET007/SET007+210.ax').
include('Axioms/SET007/SET007+212.ax').
include('Axioms/SET007/SET007+213.ax').
include('Axioms/SET007/SET007+218.ax').
include('Axioms/SET007/SET007+225.ax').
include('Axioms/SET007/SET007+237.ax').
include('Axioms/SET007/SET007+242.ax').
include('Axioms/SET007/SET007+295.ax').
include('Axioms/SET007/SET007+311.ax').
include('Axioms/SET007/SET007+335.ax').
include('Axioms/SET007/SET007+363.ax').
include('Axioms/SET007/SET007+393.ax').
include('Axioms/SET007/SET007+396.ax').
include('Axioms/SET007/SET007+412.ax').
include('Axioms/SET007/SET007+427.ax').
include('Axioms/SET007/SET007+449.ax').
include('Axioms/SET007/SET007+466.ax').
include('Axioms/SET007/SET007+480.ax').
include('Axioms/SET007/SET007+481.ax').
include('Axioms/SET007/SET007+483.ax').
include('Axioms/SET007/SET007+484.ax').
include('Axioms/SET007/SET007+485.ax').
include('Axioms/SET007/SET007+486.ax').
include('Axioms/SET007/SET007+489.ax').
include('Axioms/SET007/SET007+490.ax').
include('Axioms/SET007/SET007+494.ax').
include('Axioms/SET007/SET007+495.ax').
include('Axioms/SET007/SET007+498.ax').
include('Axioms/SET007/SET007+505.ax').
include('Axioms/SET007/SET007+509.ax').
include('Axioms/SET007/SET007+520.ax').
include('Axioms/SET007/SET007+527.ax').
include('Axioms/SET007/SET007+537.ax').
include('Axioms/SET007/SET007+538.ax').
include('Axioms/SET007/SET007+559.ax').
%------------------------------------------------------------------------------
fof(fraenkel_a_1_0_waybel22,axiom,
! [A,B] :
( r2_hidden(A,a_1_0_waybel22(B))
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(k3_yellow_1(B)))
& A = k7_waybel_0(k3_yellow_1(B),C)
& ? [D] :
( m1_subset_1(D,B)
& C = k1_tarski(D) ) ) ) ).
fof(dt_k1_waybel22,axiom,
! [A] : m1_subset_1(k1_waybel22(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k3_yellow_1(A))))) ).
fof(dt_k2_waybel22,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v3_lattice3(B)
& v3_waybel_3(B)
& l1_orders_2(B)
& v1_funct_1(C)
& v1_funct_2(C,k1_waybel22(A),u1_struct_0(B))
& m1_relset_1(C,k1_waybel22(A),u1_struct_0(B)) )
=> ( v1_funct_1(k2_waybel22(A,B,C))
& v1_funct_2(k2_waybel22(A,B,C),u1_struct_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A)))),u1_struct_0(B))
& m2_relset_1(k2_waybel22(A,B,C),u1_struct_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A)))),u1_struct_0(B)) ) ) ).
fof(l1_waybel22,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v3_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( r1_tarski(B,k1_zfmisc_1(u1_struct_0(A)))
=> k2_yellow_0(A,k3_tarski(B)) = k2_yellow_0(A,a_2_2_waybel22(A,B)) ) ) ).
fof(l2_waybel22,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v3_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( r1_tarski(B,k1_zfmisc_1(u1_struct_0(A)))
=> k1_yellow_0(A,k3_tarski(B)) = k1_yellow_0(A,a_2_4_waybel22(A,B)) ) ) ).
fof(t1_waybel22,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_yellow_0(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_waybel_0(B,k2_yellow_1(k9_waybel_0(A)))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k2_yellow_1(k9_waybel_0(A))))) )
=> k1_yellow_0(k2_yellow_1(k9_waybel_0(A)),B) = k3_tarski(B) ) ) ).
fof(t2_waybel22,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v3_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v3_lattice3(B)
& l1_orders_2(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_orders_2(C)
& v3_orders_2(C)
& v4_orders_2(C)
& v3_lattice3(C)
& l1_orders_2(C) )
=> ! [D] :
( m1_waybel16(D,A,B)
=> ! [E] :
( m1_waybel16(E,B,C)
=> m1_waybel16(k7_funct_2(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),D,E),A,C) ) ) ) ) ) ).
fof(t3_waybel22,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> v17_waybel_0(k7_grcat_1(A),A,A) ) ).
fof(t4_waybel22,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> v22_waybel_0(k7_grcat_1(A),A,A) ) ).
fof(t5_waybel22,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v3_lattice3(A)
& l1_orders_2(A) )
=> m1_waybel16(k7_grcat_1(A),A,A) ) ).
fof(t6_waybel22,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_yellow_0(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ( ~ v3_struct_0(k2_yellow_1(k9_waybel_0(A)))
& v4_yellow_0(k2_yellow_1(k9_waybel_0(A)),k3_yellow_1(u1_struct_0(A)))
& v7_yellow_0(k2_yellow_1(k9_waybel_0(A)),k3_yellow_1(u1_struct_0(A)))
& v4_waybel_0(k2_yellow_1(k9_waybel_0(A)),k3_yellow_1(u1_struct_0(A)))
& m1_yellow_0(k2_yellow_1(k9_waybel_0(A)),k3_yellow_1(u1_struct_0(A))) ) ) ).
fof(fc1_waybel22,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_yellow_0(A)
& v2_lattice3(A)
& l1_orders_2(A) )
=> ( ~ v3_struct_0(k2_yellow_1(k9_waybel_0(A)))
& v1_orders_2(k2_yellow_1(k9_waybel_0(A)))
& v2_orders_2(k2_yellow_1(k9_waybel_0(A)))
& v3_orders_2(k2_yellow_1(k9_waybel_0(A)))
& v4_orders_2(k2_yellow_1(k9_waybel_0(A)))
& v1_yellow_0(k2_yellow_1(k9_waybel_0(A)))
& v2_yellow_0(k2_yellow_1(k9_waybel_0(A)))
& v3_yellow_0(k2_yellow_1(k9_waybel_0(A)))
& v24_waybel_0(k2_yellow_1(k9_waybel_0(A)))
& v25_waybel_0(k2_yellow_1(k9_waybel_0(A)))
& v1_lattice3(k2_yellow_1(k9_waybel_0(A)))
& v2_lattice3(k2_yellow_1(k9_waybel_0(A)))
& v3_lattice3(k2_yellow_1(k9_waybel_0(A)))
& v3_waybel_3(k2_yellow_1(k9_waybel_0(A))) ) ) ).
fof(cc1_waybel22,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_yellow_0(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k2_yellow_1(k9_waybel_0(A))))
=> ~ v1_xboole_0(B) ) ) ).
fof(d1_waybel22,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v3_lattice3(A)
& v3_waybel_3(A)
& l1_orders_2(A) )
=> ! [B] :
( r1_waybel22(A,B)
<=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_orders_2(C)
& v3_orders_2(C)
& v4_orders_2(C)
& v3_lattice3(C)
& v3_waybel_3(C)
& l1_orders_2(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,u1_struct_0(C))
& m2_relset_1(D,B,u1_struct_0(C)) )
=> ? [E] :
( m1_waybel16(E,A,C)
& k7_relat_1(E,B) = D
& ! [F] :
( m1_waybel16(F,A,C)
=> ( k7_relat_1(F,B) = D
=> F = E ) ) ) ) ) ) ) ).
fof(t7_waybel22,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v3_lattice3(A)
& v3_waybel_3(A)
& l1_orders_2(A) )
=> ! [B] :
( r1_waybel22(A,B)
=> m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(t8_waybel22,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v3_lattice3(A)
& v3_waybel_3(A)
& l1_orders_2(A) )
=> ! [B] :
( r1_waybel22(A,B)
=> ! [C] :
( m1_waybel16(C,A,A)
=> ( k7_relat_1(C,B) = k1_pralg_3(B)
=> C = k7_grcat_1(A) ) ) ) ) ).
fof(d2_waybel22,axiom,
! [A] : k1_waybel22(A) = a_1_0_waybel22(A) ).
fof(t9_waybel22,axiom,
! [A] : r1_tarski(k1_waybel22(A),k9_waybel_0(k3_yellow_1(A))) ).
fof(t10_waybel22,axiom,
! [A] : k1_card_1(k1_waybel22(A)) = k1_card_1(A) ).
fof(t11_waybel22,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v2_waybel_0(B,k3_yellow_1(A))
& v13_waybel_0(B,k3_yellow_1(A))
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k3_yellow_1(A)))) )
=> B = k1_yellow_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),a_2_0_waybel22(A,B)) ) ).
fof(d3_waybel22,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v3_lattice3(B)
& v3_waybel_3(B)
& l1_orders_2(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_waybel22(A),u1_struct_0(B))
& m2_relset_1(C,k1_waybel22(A),u1_struct_0(B)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A)))),u1_struct_0(B))
& m2_relset_1(D,u1_struct_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A)))),u1_struct_0(B)) )
=> ( D = k2_waybel22(A,B,C)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A)))))
=> k1_waybel_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B,D,E) = k1_yellow_0(B,a_4_0_waybel22(A,B,C,E)) ) ) ) ) ) ).
fof(t12_waybel22,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v3_lattice3(B)
& v3_waybel_3(B)
& l1_orders_2(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_waybel22(A),u1_struct_0(B))
& m2_relset_1(C,k1_waybel22(A),u1_struct_0(B)) )
=> v5_orders_3(k2_waybel22(A,B,C),k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B) ) ) ).
fof(t13_waybel22,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v3_lattice3(B)
& v3_waybel_3(B)
& l1_orders_2(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_waybel22(A),u1_struct_0(B))
& m2_relset_1(C,k1_waybel22(A),u1_struct_0(B)) )
=> k1_waybel_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B,k2_waybel22(A,B,C),k4_yellow_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A))))) = k4_yellow_0(B) ) ) ).
fof(fc2_waybel22,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v3_lattice3(B)
& v3_waybel_3(B)
& l1_orders_2(B)
& v1_funct_1(C)
& v1_funct_2(C,k1_waybel22(A),u1_struct_0(B))
& m1_relset_1(C,k1_waybel22(A),u1_struct_0(B)) )
=> ( v1_relat_1(k2_waybel22(A,B,C))
& v1_funct_1(k2_waybel22(A,B,C))
& v1_funct_2(k2_waybel22(A,B,C),u1_struct_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A)))),u1_struct_0(B))
& v22_waybel_0(k2_waybel22(A,B,C),k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B) ) ) ).
fof(l19_waybel22,axiom,
! [A] :
( v1_setfam_1(A)
=> ( v2_relat_1(k1_pralg_3(A))
& m1_pboole(k1_pralg_3(A),A) ) ) ).
fof(fc3_waybel22,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v3_lattice3(B)
& v3_waybel_3(B)
& l1_orders_2(B)
& v1_funct_1(C)
& v1_funct_2(C,k1_waybel22(A),u1_struct_0(B))
& m1_relset_1(C,k1_waybel22(A),u1_struct_0(B)) )
=> ( v1_relat_1(k2_waybel22(A,B,C))
& v1_funct_1(k2_waybel22(A,B,C))
& v1_funct_2(k2_waybel22(A,B,C),u1_struct_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A)))),u1_struct_0(B))
& v17_waybel_0(k2_waybel22(A,B,C),k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B)
& v19_waybel_0(k2_waybel22(A,B,C),k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B)
& v21_waybel_0(k2_waybel22(A,B,C),k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B)
& v22_waybel_0(k2_waybel22(A,B,C),k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B) ) ) ).
fof(t14_waybel22,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v3_lattice3(B)
& v3_waybel_3(B)
& l1_orders_2(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_waybel22(A),u1_struct_0(B))
& m2_relset_1(C,k1_waybel22(A),u1_struct_0(B)) )
=> k7_relat_1(k2_waybel22(A,B,C),k1_waybel22(A)) = C ) ) ).
fof(t15_waybel22,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v3_lattice3(B)
& v3_waybel_3(B)
& l1_orders_2(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_waybel22(A),u1_struct_0(B))
& m2_relset_1(C,k1_waybel22(A),u1_struct_0(B)) )
=> ! [D] :
( m1_waybel16(D,k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B)
=> ( k7_relat_1(D,k1_waybel22(A)) = C
=> D = k2_waybel22(A,B,C) ) ) ) ) ).
fof(t16_waybel22,axiom,
! [A] : r1_waybel22(k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),k1_waybel22(A)) ).
fof(t17_waybel22,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v3_lattice3(A)
& v3_waybel_3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v1_lattice3(B)
& v2_lattice3(B)
& v3_lattice3(B)
& v3_waybel_3(B)
& l1_orders_2(B) )
=> ! [C,D] :
( ( r1_waybel22(A,C)
& r1_waybel22(B,D)
& k1_card_1(C) = k1_card_1(D) )
=> r5_waybel_1(A,B) ) ) ) ).
fof(t18_waybel22,conjecture,
! [A,B] :
( ( v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v1_lattice3(B)
& v2_lattice3(B)
& v3_lattice3(B)
& v3_waybel_3(B)
& l1_orders_2(B) )
=> ! [C] :
( ( r1_waybel22(B,C)
& k1_card_1(C) = k1_card_1(A) )
=> r5_waybel_1(B,k2_yellow_1(k9_waybel_0(k3_yellow_1(A)))) ) ) ).
%------------------------------------------------------------------------------