reserve X for TopSpace;

theorem Th4:
  for X1, X2 being SubSpace of X holds the carrier of X1 c= the
  carrier of X2 iff X1 is SubSpace of X2
proof
  let X1, X2 be SubSpace of X;
  set A1 = the carrier of X1, A2 = the carrier of X2;
A1: A1 = [#]X1;
A2: A2 = [#]X2;
  thus the carrier of X1 c= the carrier of X2 implies X1 is SubSpace of X2
  proof
    assume
A3: A1 c= A2;
    for P being Subset of X1 holds P in the topology of X1 iff ex Q being
    Subset of X2 st Q in the topology of X2 & P = Q /\ A1
    proof
      let P be Subset of X1;
      thus P in the topology of X1 implies ex Q being Subset of X2 st Q in the
      topology of X2 & P = Q /\ A1
      proof
        assume P in the topology of X1;
        then consider R being Subset of X such that
A4:     R in the topology of X and
A5:     P = R /\ A1 by A1,PRE_TOPC:def 4;
        reconsider Q = R /\ A2 as Subset of X2 by XBOOLE_1:17;
        take Q;
        thus Q in the topology of X2 by A2,A4,PRE_TOPC:def 4;
        Q /\ A1 = R /\ (A2 /\ A1) by XBOOLE_1:16
          .= R /\ A1 by A3,XBOOLE_1:28;
        hence thesis by A5;
      end;
      given Q being Subset of X2 such that
A6:   Q in the topology of X2 and
A7:   P = Q /\ A1;
      consider R being Subset of X such that
A8:   R in the topology of X and
A9:   Q = R /\ [#]X2 by A6,PRE_TOPC:def 4;
      P = R /\ (A2 /\ A1) by A7,A9,XBOOLE_1:16
        .= R /\ A1 by A3,XBOOLE_1:28;
      hence thesis by A1,A8,PRE_TOPC:def 4;
    end;
    hence thesis by A1,A2,A3,PRE_TOPC:def 4;
  end;
  thus thesis by A1,A2,PRE_TOPC:def 4;
end;
