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

theorem Th44:
  for X being anti-discrete non empty TopSpace, A being non empty
  Subset of X holds A is maximal_discrete iff A is trivial
proof
  let X be anti-discrete non empty TopSpace, A be non empty Subset of X;
  thus A is maximal_discrete implies A is trivial by Th38;
  hereby
A1: now
      let D be Subset of X;
      assume
A2:   D is discrete;
      assume
A3:   A c= D;
      then reconsider E = D as non empty Subset of X;
      E is trivial by A2,Th38;
      hence A = D by A3,Th1;
    end;
    assume A is trivial;
    then A is discrete by Th38;
    hence A is maximal_discrete by A1;
  end;
end;
