reserve T for TopSpace;
reserve T for non empty TopSpace;
reserve F for Subset-Family of T;
reserve T for non empty TopSpace;
reserve T for non empty TopSpace;
reserve T for non empty TopSpace;

theorem Th103:
  for a, b being Element of Open_Domains_Lattice T for A, B being
Element of Open_Domains_of T st a = A & b = B holds a "\/" b = Int(Cl(A \/ B))
  & a "/\" b = A /\ B
proof
  let a, b be Element of Open_Domains_Lattice T;
  let A, B be Element of Open_Domains_of T;
  assume that
A1: a = A and
A2: b = B;
A3: Open_Domains_Lattice T = LattStr(#Open_Domains_of T,OPD-Union T,OPD-Meet
    T#) by TDLAT_1:def 12;
  hence a "\/" b = (OPD-Union T).(A,B) by A1,A2,LATTICES:def 1
    .= Int(Cl(A \/ B)) by TDLAT_1:def 10;
  thus a "/\" b = (OPD-Meet T).(A,B) by A3,A1,A2,LATTICES:def 2
    .= A /\ B by TDLAT_1:def 11;
end;
