theorem Th53:
  a <=> (a <=> b) = b
proof
A1: a"/\"((a"/\"b)"\/"(a`"/\"b`)) = (a"/\"(a"/\"b))"\/"(a"/\"(a`"/\"b`)) by
LATTICES:def 11;
A2: a`"/\"((a"/\"b`)"\/"(a`"/\"b)) = (a`"/\"(a"/\"b`))"\/"(a`"/\"(a`"/\"b))
  by LATTICES:def 11;
A3: a"\/"a` = Top B by LATTICES:21;
A4: (a"/\"b)"\/"(a`"/\"b) = (a"\/"a`)"/\"b by LATTICES:def 11;
A5: (a<=>b)` = (a"/\"b`)"\/"(a`"/\"b) by Th51;
A6: a`"/\"(a"/\"b`) = a`"/\"a"/\"b` by LATTICES:def 7;
A7: a"/\"a` = Bottom B by LATTICES:20;
A8: a<=>b = (a"/\"b)"\/"(a`"/\"b`) by Th50;
A9: a"/\"(a"/\"b) = a"/\"a"/\"b by LATTICES:def 7;
A10: a"/\"(a`"/\"b`) = a"/\"a`"/\"b` by LATTICES:def 7;
A11: a`"/\"(a`"/\"b) = a`"/\"a`"/\"b by LATTICES:def 7;
  a<=>(a<=>b) = (a"/\"(a<=>b))"\/"(a`"/\"(a<=>b)`) by Th50;
  hence thesis by A8,A5,A1,A2,A9,A6,A11,A10,A7,A4,A3;
end;
