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

theorem
  for Y0 being T_0 non empty SubSpace of X ex X0 being strict SubSpace
  of X st Y0 is SubSpace of X0 & X0 is maximal_T_0
proof
  let Y0 be T_0 non empty SubSpace of X;
  reconsider A = the carrier of Y0 as Subset of X by Lm1;
  A is T_0 by TSP_1:13;
  then consider M being Subset of X such that
A1: A c= M and
A2: M is maximal_T_0 by Th9;
  M is non empty by A2,Th4;
  then consider X0 being strict non empty SubSpace of X such that
A3: X0 is maximal_T_0 and
A4: M = the carrier of X0 by A2,Th12;
  take X0;
  thus thesis by A1,A3,A4,TSEP_1:4;
end;
