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 F being Subset-Family of Y st F is domains-family for S being
  Subset of Domains_Lattice Y st S = F holds (S <> {} implies "/\"(S,
Domains_Lattice Y) = Int(meet F)) & (S = {} implies "/\"(S,Domains_Lattice Y) =
  [#]Y)
proof
  let F be Subset-Family of Y;
  assume
A1: F is domains-family;
  then F c= Domains_of Y by TDLAT_2:65;
  then F c= Open_Domains_of Y by Th42;
  then
A2: F is open-domains-family by TDLAT_2:79;
  let S be Subset of Domains_Lattice Y;
  assume
A3: S = F;
  Domains_Lattice Y = Open_Domains_Lattice Y by Th44;
  hence S <> {} implies "/\"(S,Domains_Lattice Y) = Int(meet F) by A2,A3,
TDLAT_2:110;
  assume S = {};
  hence thesis by A1,A3,TDLAT_2:93;
end;
