reserve X for non empty TopSpace;
reserve X for non empty TopSpace;

theorem Th12:
  for A0 being non empty Subset of X st A0 is maximal_T_0 ex X0
being strict non empty SubSpace of X st X0 is maximal_T_0 & A0 = the carrier of
  X0
proof
  let A0 be non empty Subset of X;
  assume
A1: A0 is maximal_T_0;
  consider X0 being strict non empty SubSpace of X such that
A2: A0 = the carrier of X0 by TSEP_1:10;
  take X0;
  thus thesis by A1,A2;
end;
