reserve X for TopSpace;

theorem Th7:
  for X1 being SubSpace of X for X2 being SubSpace of X1 holds X2
  is SubSpace of X
proof
  let X1 be SubSpace of X;
  let X2 be SubSpace of X1;
A1: [#]X2 c= [#]X1 by PRE_TOPC:def 4;
  [#]X1 c= [#]X by PRE_TOPC:def 4;
  hence [#](X2) c= [#](X) by A1,XBOOLE_1:1;
  thus for P being Subset of X2 holds P in the topology of X2 iff ex Q being
  Subset of X st Q in the topology of X & P = Q /\ [#]X2
  proof
    let P be Subset of X2;
    reconsider P1 = P as Subset of X2;
    thus P in the topology of X2 implies ex Q being Subset of X st Q in the
    topology of X & P = Q /\ [#]X2
    proof
      assume P in the topology of X2;
      then consider R being Subset of X1 such that
A2:   R in the topology of X1 and
A3:   P = R /\ [#]X2 by PRE_TOPC:def 4;
      consider Q being Subset of X such that
A4:   Q in the topology of X and
A5:   R = Q /\ [#]X1 by A2,PRE_TOPC:def 4;
      Q /\ [#]X2 = Q /\ ([#]X1 /\ [#]X2) by A1,XBOOLE_1:28
        .= P by A3,A5,XBOOLE_1:16;
      hence thesis by A4;
    end;
    given Q being Subset of X such that
A6: Q in the topology of X and
A7: P = Q /\ [#]X2;
    reconsider R = Q /\ [#]X1 as Subset of X1;
    reconsider Q1 = Q as Subset of X;
    Q1 is open by A6;
    then
A8: R is open by TOPS_2:24;
    R /\ [#]X2 = Q /\ ([#]X1 /\ [#]X2) by XBOOLE_1:16
      .= P by A1,A7,XBOOLE_1:28;
    then P1 is open by A8,TOPS_2:24;
    hence thesis;
  end;
end;
