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
  Closed_Domains_Lattice X = Open_Domains_Lattice X implies
    X is extremally_disconnected
proof
  assume Closed_Domains_Lattice X = Open_Domains_Lattice X; then
A1: Closed_Domains_of X = Open_Domains_of X;
  for A being Subset of X holds (A is open_condensed implies A is
  closed_condensed) & (A is closed_condensed implies A is open_condensed)
  proof
    let A be Subset of X;
    thus A is open_condensed implies A is closed_condensed
    proof
      assume A is open_condensed;
      then A in {E where E is Subset of X : E is open_condensed}; then
      A in Closed_Domains_of X by A1;
      then A in {E where E is Subset of X : E is closed_condensed};
      then ex D being Subset of X st D = A & D is closed_condensed;
      hence thesis;
    end;
    assume A is closed_condensed;
    then A in {E where E is Subset of X : E is closed_condensed}; then
    A in Open_Domains_of X by A1;
    then A in {E where E is Subset of X : E is open_condensed };
    then ex D being Subset of X st D = A & D is open_condensed;
    hence thesis;
  end;
  hence thesis by Th36;
end;
