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;
reserve X for non empty TopSpace;

theorem
  X is extremally_disconnected iff Domains_Lattice X is B_Lattice
proof
  thus X is extremally_disconnected implies Domains_Lattice X is B_Lattice
  proof
    assume X is extremally_disconnected;
    then Domains_Lattice X = Open_Domains_Lattice X by Th44;
    hence thesis;
  end;
  assume Domains_Lattice X is B_Lattice;
  hence thesis by Th49;
end;
