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

theorem Th33:
  X is extremally_disconnected iff for A being Subset of X st A is
  condensed holds A is closed_condensed & A is open_condensed
proof
  thus X is extremally_disconnected implies for A being Subset of X st A is
  condensed holds A is closed_condensed & A is open_condensed
  by Th32,TOPS_1:66,TOPS_1:67;
  assume
A1: for A being Subset of X st A is condensed holds A is
  closed_condensed & A is open_condensed;
  now
    let A be Subset of X;
    assume
A2: A is condensed;
    then
A3: A is open_condensed by A1;
    A is closed_condensed by A1,A2;
    hence A is closed & A is open by A3,TOPS_1:66,67;
  end;
  hence thesis by Th32;
end;
