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;
