reserve X for TopSpace;

theorem Th15:
  for X be non empty TopSpace, A0 being non empty Subset of X st
  A0 is closed ex X0 being strict closed non empty SubSpace of X st A0 = the
  carrier of X0
proof
  let X be non empty TopSpace, A0 be non empty Subset of X such that
A1: A0 is closed;
  consider X0 being strict non empty SubSpace of X such that
A2: A0 = the carrier of X0 by Th10;
  reconsider Y0 = X0 as strict closed non empty SubSpace of X by A1,A2,Th11;
  take Y0;
  thus thesis by A2;
end;
