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 Th73:
  for p,q for p9,q9 being Element of latt (L,P) st p = p9 & q = q9
  holds p"\/"q = p9"\/"q9 & p"/\"q = p9"/\"q9
proof
  let p,q;
  let p9,q9 be Element of latt (L,P);
  assume
A1: p = p9 & q = q9;
  consider o1, o2 such that
A2: o1 = join(L)||P and
A3: o2 = met(L)||P and
A4: latt (L,P) = LattStr (#P, o1, o2#) by Def14;
A5: [p9,q9] in [:P,P:] by A4;
  dom o1 = [:P,P:] by FUNCT_2:def 1;
  hence p"\/"q = p9"\/"q9 by A1,A2,A4,A5,FUNCT_1:47;
  dom o2 = [:P,P:] by FUNCT_2:def 1;
  hence thesis by A1,A3,A4,A5,FUNCT_1:47;
end;
