reserve L for Lattice,
  p,q,r for Element of L,
  p9,q9,r9 for Element of L.:,
  x, y for set;
reserve I,J for Ideal of L,
  F for Filter of L;
reserve D for non empty Subset of L,
  D9 for non empty Subset of L.:;
reserve D1,D2 for non empty Subset of L,
  D19,D29 for non empty Subset of L.:;
reserve B for B_Lattice,
  IB,JB for Ideal of B,
  a,b for Element of B;
reserve a9 for Element of (B qua Lattice).:;
reserve P for non empty ClosedSubset of L,
  o1,o2 for BinOp of P;

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;
