reserve Y for TopStruct;
reserve X for non empty TopSpace;

theorem
  for X being discrete non empty TopSpace, A being Subset of X holds A
  is discrete
proof
  let X be discrete non empty TopSpace, A be Subset of X;
  hereby
    per cases;
    suppose
      A is empty;
      hence thesis by Th29;
    end;
    suppose
      A is non empty;
      then
      ex X0 being strict non empty SubSpace of X st A = the carrier of X0
      by TSEP_1:10;
      hence thesis by Th20;
    end;
  end;
end;
