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

theorem
  for A being Subset of X st A is open or A is closed holds A is
  maximal_discrete implies A is not proper
proof
  let A be Subset of X;
  assume
A1: A is open or A is closed;
  then A is closed by TDLAT_3:21;
  then
A2: A = Cl A by PRE_TOPC:22;
  assume
A3: A is maximal_discrete;
  A is open by A1,TDLAT_3:22;
  then A is dense by A3,Th41;
  then A = the carrier of X by A2,TOPS_3:def 2;
  hence thesis;
end;
