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 Th41:
  Domains_Lattice Y = Closed_Domains_Lattice Y
proof
A1: D-Union Y = CLD-Union Y by Th40;
A2: D-Meet Y = CLD-Meet Y by Th40;
  Domains_of Y = Closed_Domains_of Y by Th39;
  hence
  Domains_Lattice Y = LattStr(#Closed_Domains_of Y,CLD-Union Y,CLD-Meet Y
  #) by A1,A2
    .= Closed_Domains_Lattice Y;
end;
