theorem Th54:
  L is distributive implies latt F is distributive
proof
  assume
A1: for p,q,r holds p"/\"(q"\/"r) = (p"/\"q)"\/"(p"/\"r);
  let p9,q9,r9 be Element of latt F;
  reconsider p = p9, q = q9, r = r9, qr = q9"\/"r9 as Element of F by Th49;
A2: p9"/\"q9 = p"/\"q & p9"/\"r9 = p"/\"r by Th50;
  thus p9"/\"(q9"\/"r9) = p"/\"qr by Th50
    .= p"/\"(q"\/"r) by Th50
    .= (p"/\"q)"\/"(p"/\"r) by A1
    .= (p9"/\"q9)"\/"(p9"/\"r9) by A2,Th50;
end;
