theorem Th77:
  L is distributive implies latt (L,P) is distributive
proof
  assume
A1: for a,b,c being Element of L holds a"/\"(b"\/"c) = (a"/\"b)"\/"(a "/\"c);
  let a9,b9,c9 be Element of latt (L,P);
  reconsider a = a9, b = b9, c = c9, bc = b9"\/"c9, ab = a9"/\"b9, ac = a9"/\"
  c9 as Element of L by Th68;
  thus a9"/\"(b9"\/"c9) = a"/\"bc by Th73
    .= a"/\"(b"\/"c) by Th73
    .= (a"/\"b)"\/"(a"/\"c) by A1
    .= ab"\/"(a"/\"c) by Th73
    .= ab"\/" ac by Th73
    .= (a9"/\"b9)"\/"(a9"/\"c9) by Th73;
end;
