reserve X for TopSpace;
reserve C for Subset of X;
reserve A, B for Subset of X;
reserve X for non empty TopSpace;
reserve Y for extremally_disconnected non empty TopSpace;

theorem
  for a, b being Element of Domains_Lattice Y for A, B being Element of
  Domains_of Y st a = A & b = B holds a "\/" b = A \/ B & a "/\" b = A /\ B
proof
  let a, b be Element of Domains_Lattice Y;
  let A, B be Element of Domains_of Y;
  assume that
A1: a = A and
A2: b = B;
  thus a "\/" b = (D-Union Y).(A,B) by A1,A2,LATTICES:def 1
    .= A \/ B by Th45;
  thus a "/\" b = (D-Meet Y).(A,B) by A1,A2,LATTICES:def 2
    .= A /\ B by Th45;
end;
