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 Th5:
  for X9 being SubSpace of GX, P1,Q1 being Subset of GX, P,Q being
Subset of X9 st P=P1 & Q=Q1 holds P,Q are_separated implies P1,Q1 are_separated
proof
  let X9 be SubSpace of GX, P1,Q1 be Subset of GX, P,Q be Subset of X9 such
  that
A1: P = P1 and
A2: Q = Q1;
  reconsider P2 = P, Q2 = Q as Subset of GX by PRE_TOPC:11;
  assume that
A3: (Cl P) /\ Q = {} and
A4: P /\ Cl Q = {};
  P /\ Cl Q = P /\ (([#] X9) /\ Cl Q2) by PRE_TOPC:17
    .= P /\ [#] X9 /\ Cl Q2 by XBOOLE_1:16
    .= P2 /\ Cl Q2 by XBOOLE_1:28;
  then
A5: P2 misses Cl Q2 by A4;
  (Cl P) /\ Q = ((Cl P2) /\ ([#](X9))) /\ Q by PRE_TOPC:17
    .= (Cl P2) /\ (Q /\ [#] X9) by XBOOLE_1:16
    .= (Cl P2) /\ Q2 by XBOOLE_1:28;
  then (Cl P2) misses Q2 by A3;
  hence thesis by A1,A2,A5;
end;
