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;

theorem Th86:
  for a, b being Element of Domains_Lattice T for A, B being
Element of Domains_of T st a = A & b = B holds a "\/" b = Int(Cl(A \/ B)) \/ (A
  \/ B) & a "/\" b = Cl(Int(A /\ B)) /\ (A /\ B)
proof
  let a, b be Element of Domains_Lattice T;
  let A, B be Element of Domains_of T;
  assume that
A1: a = A and
A2: b = B;
A3: Domains_Lattice T = LattStr(#Domains_of T,D-Union T,D-Meet T#) by
TDLAT_1:def 4;
  hence a "\/" b = (D-Union T).(A,B) by A1,A2,LATTICES:def 1
    .= Int(Cl(A \/ B)) \/ (A \/ B) by TDLAT_1:def 2;
  thus a "/\" b = (D-Meet T).(A,B) by A3,A1,A2,LATTICES:def 2
    .= Cl(Int(A /\ B)) /\ (A /\ B) by TDLAT_1:def 3;
end;
