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
  for F being Subset-Family of T st F is open-domains-family for X being
Subset of Domains_Lattice T st X = F holds "\/"(X,Domains_Lattice T) = Int Cl(
  union F)
proof
  let F be Subset-Family of T;
A1: Int(union F) c= Int Cl(union F) by PRE_TOPC:18,TOPS_1:19;
  assume
A2: F is open-domains-family;
  then union F is open by Th80,TOPS_2:19;
  then union F c= Int Cl(union F) by A1,TOPS_1:23;
  then
A3: (union F) \/ Int Cl(union F) = Int Cl(union F) by XBOOLE_1:12;
  let X be Subset of Domains_Lattice T;
  assume X = F;
  hence thesis by A2,A3,Th79,Th91;
end;
