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

theorem
  for Y0 being non empty SubSpace of Y, Y1 being T_0 non empty SubSpace
  of Y st Y0 is SubSpace of Y1 holds Y0 is T_0
proof
  let Y0 be non empty SubSpace of Y, Y1 be T_0 non empty SubSpace of Y;
  reconsider A1 = the carrier of Y1, A0 = the carrier of Y0 as non empty
  Subset of Y by Lm3;
  assume
A1: Y0 is SubSpace of Y1;
A2: A1 is T_0 by Th13;
  [#]Y0 = A0 & [#]Y1 = A1;
  then A0 c= A1 by A1,PRE_TOPC:def 4;
  then A0 is T_0 by A2;
  hence thesis;
end;
