theorem Th51:
  (a => b)` = a "/\" b` & (a <=> b)` = (a "/\" b`) "\/" (a` "/\" b
  ) & (a <=> b)` = a <=> b` & (a <=> b)` = a` <=> b
proof
A1: now
    let a,b;
    thus (a => b)` = (a` "\/" b)` by FILTER_0:42
      .= a`` "/\" b` by LATTICES:24
      .= a "/\" b`;
  end;
  hence (a => b)` = a "/\" b`;
  thus (a <=> b)` = (a=>b)`"\/"(b=>a)` by LATTICES:23
    .= (a"/\"b`)"\/"(b=>a)` by A1
    .= (a"/\"b`)"\/"(a`"/\"b) by A1;
  hence (a <=> b)` = (a"/\"b`)"\/"(a`"/\"b``)
    .= a <=> b` by Th50;
  hence (a <=> b)` = (a"/\"b`)"\/"(a`"/\"b``) by Th50
    .= (a``"/\"b`)"\/"(a`"/\"b``)
    .= (a`"/\"b)"\/"(a``"/\"b`)
    .= a` <=> b by Th50;
end;
