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
  L is modular implies latt (L,P) is modular
proof
  assume
A1: for a,b,c being Element of L st a [= c holds a"\/"(b"/\"c) = (a"\/"b
  )"/\"c;
  let a9,b9,c9 be Element of latt (L,P);
  reconsider a = a9, b = b9, c = c9, bc = b9"/\"c9, ab = a9"\/"b9 as Element
  of L by Th68;
  assume a9 [= c9;
  then
A2: a [= c by Th74;
  thus a9"\/"(b9"/\"c9) = a"\/"bc by Th73
    .= a"\/"(b"/\"c) by Th73
    .= (a"\/"b)"/\"c by A1,A2
    .= ab"/\"c by Th73
    .= (a9"\/"b9)"/\"c9 by Th73;
end;
