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,Domains_Lattice Y) = Cl(union
  F)
proof
  let F be Subset-Family of Y;
  assume F is domains-family;
  then F c= Domains_of Y by TDLAT_2:65;
  then F c= Closed_Domains_of Y by Th39;
  then
A1: F is closed-domains-family by TDLAT_2:72;
  let S be Subset of Domains_Lattice Y;
  reconsider P = S as Subset of Closed_Domains_Lattice Y by Th41;
  assume
A2: S = F;
  thus "\/"(S,Domains_Lattice Y) = "\/"(P,Closed_Domains_Lattice Y) by Th41
    .= Cl(union F) by A1,A2,TDLAT_2:100;
end;
