reserve GX for TopSpace;
reserve A, B, C for Subset of GX;
reserve TS for TopStruct;
reserve K, K1, L, L1 for Subset of TS;

theorem Th23:
  for X9 being SubSpace of GX, A being Subset of GX, B being
  Subset of X9 st A = B holds A is connected iff B is connected
proof
  let X9 be SubSpace of GX, A be Subset of GX, B be Subset of X9;
  assume
A1: A = B;
  reconsider GX9 = GX, X = X9 as TopSpace;
A2: [#](GX|A) = A by PRE_TOPC:def 5;
  reconsider B9 = B as Subset of X;
  reconsider A9 = A as Subset of GX9;
A3: [#](X9|B) = B by PRE_TOPC:def 5;
A4: now
    assume not A is connected;
    then not GX9|A9 is connected;
    then consider P,Q being Subset of GX|A such that
A5: [#](GX|A) = P \/ Q and
A6: P <> {}(GX|A) and
A7: Q <> {}(GX|A) and
A8: P is closed and
A9: Q is closed and
A10: P misses Q by Th10;
    consider P1 being Subset of GX such that
A11: P1 is closed and
A12: P1 /\ ([#](GX|A)) = P by A8,PRE_TOPC:13;
    reconsider P11 = P1 /\ ([#](X9)) as Subset of X9;
A13: P11 is closed by A11,PRE_TOPC:13;
    reconsider PP = P, QQ=Q as Subset of X9|B by A1,A2,A3;
A14: P c= [#](X9) by A1,A2,XBOOLE_1:1;
    P1 /\ A c= P1 by XBOOLE_1:17;
    then P c= P1 /\ ([#](X9)) by A2,A12,A14,XBOOLE_1:19;
    then
A15: P c= P1 /\ [#](X9) /\ A by A2,XBOOLE_1:19;
    P1 /\ ([#](X9)) c= P1 by XBOOLE_1:17;
    then P1 /\ [#](X9) /\ A c= P by A2,A12,XBOOLE_1:27;
    then P11 /\ [#](X9|B) = PP by A1,A3,A15;
    then
A16: PP is closed by A13,PRE_TOPC:13;
    consider Q1 being Subset of GX such that
A17: Q1 is closed and
A18: Q1 /\ ([#](GX|A)) = Q by A9,PRE_TOPC:13;
    reconsider Q11 = Q1 /\ ([#](X9)) as Subset of X9;
A19: Q c= [#](X9) by A1,A2,XBOOLE_1:1;
    Q1 /\ A c= Q1 by XBOOLE_1:17;
    then Q c= Q1 /\ ([#](X9)) by A2,A18,A19,XBOOLE_1:19;
    then
A20: Q c= (Q1 /\ ([#](X9))) /\ A by A2,XBOOLE_1:19;
    Q1 /\ ([#](X9)) c= Q1 by XBOOLE_1:17;
    then (Q1 /\ ([#](X9))) /\ A c= Q by A2,A18,XBOOLE_1:27;
    then
A21: (Q1 /\ ([#](X9))) /\ A = Q by A20;
    Q11 is closed by A17,PRE_TOPC:13;
    then QQ is closed by A1,A3,A21,PRE_TOPC:13;
    then not X|B9 is connected by A1,A2,A3,A5,A6,A7,A10,A16,Th10;
    hence not B is connected;
  end;
  now
    assume not B is connected;
    then not X9|B is connected;
    then consider P,Q being Subset of X9|B such that
A22: [#](X9|B) = P \/ Q and
A23: P <> {}(X9|B) and
A24: Q <> {}(X9|B) and
A25: P is closed and
A26: Q is closed and
A27: P misses Q by Th10;
    reconsider QQ = Q as Subset of GX|A by A1,A2,A3;
    reconsider PP = P as Subset of GX|A by A1,A2,A3;
    consider P1 being Subset of X9 such that
A28: P1 is closed and
A29: P1 /\ ([#](X9|B)) = P by A25,PRE_TOPC:13;
    consider Q1 being Subset of X9 such that
A30: Q1 is closed and
A31: Q1 /\ ([#](X9|B)) = Q by A26,PRE_TOPC:13;
    consider Q2 being Subset of GX such that
A32: Q2 is closed and
A33: Q2 /\ ([#](X9)) = Q1 by A30,PRE_TOPC:13;
    Q2 /\ ([#](GX|A)) = Q2 /\ (([#](X9)) /\ B) by A1,A2,XBOOLE_1:28
      .= QQ by A3,A31,A33,XBOOLE_1:16;
    then
A34: QQ is closed by A32,PRE_TOPC:13;
    consider P2 being Subset of GX such that
A35: P2 is closed and
A36: P2 /\ ([#](X9)) = P1 by A28,PRE_TOPC:13;
    P2 /\ ([#](GX|A)) = P2 /\ (([#](X9)) /\ B) by A1,A2,XBOOLE_1:28
      .= PP by A3,A29,A36,XBOOLE_1:16;
    then PP is closed by A35,PRE_TOPC:13;
    then not GX9|A9 is connected by A1,A2,A3,A22,A23,A24,A27,A34,Th10;
    hence not A is connected;
  end;
  hence thesis by A4;
end;
