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

theorem Th43:
  for X being discrete non empty TopSpace, A being Subset of X
  holds A is maximal_discrete iff A is non proper
proof
  let X be discrete non empty TopSpace, A be Subset of X;
  hereby
    X is SubSpace of X by TSEP_1:2;
    then reconsider C = the carrier of X as Subset of X by TSEP_1:1;
    assume
A1: A is maximal_discrete;
    C is discrete by Th21;
    then A = C by A1;
    hence A is non proper;
  end;
  thus thesis by Th21,XBOOLE_0:def 10;
end;
