reserve L,L1,L2 for Lattice,
  F1,F2 for Filter of L,
  p,q,r,s for Element of L,
  p1,q1,r1,s1 for Element of L1,
  p2,q2,r2,s2 for Element of L2,
  X,x,x1,x2,y,y1,y2 for set,
  D,D1,D2 for non empty set,
  R for Relation,
  RD for Equivalence_Relation of D,
  a,b,d for Element of D,
  a1,b1,c1 for Element of D1,
  a2,b2,c2 for Element of D2,
  B for B_Lattice,
  FB for Filter of B,
  I for I_Lattice,
  FI for Filter of I ,
  i,i1,i2,j,j1,j2,k for Element of I,
  f1,g1 for BinOp of D1,
  f2,g2 for BinOp of D2;
reserve F,G for BinOp of D,RD;
reserve B for B_Lattice,
  a,b,c,d for Element of B;

theorem Th60:
  c"\/"(c <=>d) in Class(equivalence_wrt <.d.),c) & for b st b in
  Class(equivalence_wrt <.d.),c) holds b [= c"\/"(c <=>d)
proof
  set A = Class(equivalence_wrt <.d.),c);
A1: c in A by EQREL_1:20;
A2: (c <=>d)<=>c = c <=>(c <=>d);
A3: d in <.d.);
  c <=>(c <=>d) = d by Th53;
  then c <=>d in A by A3,A2,Lm4;
  hence (c"\/"(c <=>d)) in A by A1,Th59;
  let b;
  assume b in A;
  then b<=>c in <.d.) by Lm4;
  then
A4: d [= b<=>c by FILTER_0:15;
  (b<=>c)` = (b"/\"c`)"\/"(b`"/\"c) by Th51;
  then (b"/\"c`)"\/"(b`"/\"c) [= d` by A4,LATTICES:26;
  then
A5: ((b"/\"c`)"\/"(b`"/\"c))"/\"c` [= d`"/\"c` by LATTICES:9;
A6: ((b"/\"c`)"\/"(b`"/\"c))"/\"c` = ((b"/\"c`)"/\"c`)"\/"((b`"/\"c)"/\"c`)
  by LATTICES:def 11;
A7: (b`"/\"c)"/\"c`= b`"/\"(c"/\"c`) by LATTICES:def 7;
A8: (c`"/\"d`)"\/"(b"/\"c) [= (c`"/\"d`)"\/"c by FILTER_0:1,LATTICES:6;
A9: (b"/\"c`)"\/"(b"/\" c) = b "/\"(c`"\/"c) by LATTICES:def 11;
A10: c"\/"(c"/\"d)"\/"(c`"/\"d`) = c"\/"((c"/\"d)"\/"(c`"/\"d`)) by
LATTICES:def 5;
A11: c = c"\/"(c"/\"d) by LATTICES:def 8;
A12: (c"/\"d)"\/"(c`"/\" d`) = c <=>d by Th50;
A13: c`"\/"c = Top B by LATTICES:21;
A14: Bottom B = c"/\"c` by LATTICES:20;
  (b"/\"c`)"/\"c`= b"/\"(c `"/\" c`) by LATTICES:def 7;
  then (b"/\"c`)"\/"(b"/\"c) [= (c`"/\"d`)"\/"(b"/\"c) by A5,A6,A7,A14,
FILTER_0:1;
  hence thesis by A9,A13,A8,A11,A12,A10,LATTICES:7;
end;
