reserve X for TopSpace;
reserve C for Subset of X;
reserve A, B for Subset of X;
reserve X for non empty TopSpace;

theorem Th36:
  X is extremally_disconnected iff 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
  thus X is extremally_disconnected implies 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
    assume
A1: X is extremally_disconnected;
    let A be Subset of X;
    thus A is open_condensed implies A is closed_condensed
    proof
      assume A is open_condensed;
      then A is condensed by TOPS_1:67;
      hence thesis by A1,Th33;
    end;
    thus A is closed_condensed implies A is open_condensed
    proof
      assume A is closed_condensed;
      then A is condensed by TOPS_1:66;
      hence thesis by A1,Th33;
    end;
  end;
  assume
A2: 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);
  for A being Subset of X st A is condensed holds Int Cl A = Cl Int A
  proof
    let A be Subset of X;
    assume
A3: A is condensed;
    then
A4: A c= Cl Int A by TOPS_1:def 6;
    Cl Int A is closed_condensed by TDLAT_1:22;
    then Cl Int A is open_condensed by A2;
    then Cl Int A = Int Cl Cl Int A by TOPS_1:def 8;
    then
A5: Cl Int A c= Int Cl A by TDLAT_2:1;
    Int Cl A c= A by A3,TOPS_1:def 6;
    then Int Cl A c= Cl Int A by A4;
    hence thesis by A5;
  end;
  hence thesis by Th34;
end;
