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 closed-domains-family for X
  being Subset of Domains_Lattice T st X = F holds (X <> {} implies "/\"(X,
Domains_Lattice T) = Cl(Int(meet F))) & (X = {} implies "/\"(X,Domains_Lattice
  T) = [#]T)
proof
  let F be Subset-Family of T;
A1: Cl Int(meet F) c= Cl(meet F) by PRE_TOPC:19,TOPS_1:16;
  assume
A2: F is closed-domains-family;
  then
A3: F is domains-family by Th72;
  let X be Subset of Domains_Lattice T;
  assume
A4: X = F;
  meet F is closed by A2,Th73,TOPS_2:22;
  then Cl Int(meet F) c= meet F by A1,PRE_TOPC:22;
  then (meet F) /\ (Cl Int(meet F)) = Cl Int(meet F) by XBOOLE_1:28;
  hence X <> {} implies "/\"(X,Domains_Lattice T) = Cl Int(meet F) by A3,A4
,Th92;
  thus thesis by A3,A4,Th92;
end;
