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 Th6:
  for X9 being SubSpace of GX, P,Q being Subset of GX, P1,Q1 being
Subset of X9 st P=P1 & Q=Q1 & P \/ Q c= [#](X9) holds P,Q are_separated implies
  P1,Q1 are_separated
proof
  let X9 be SubSpace of GX, P,Q be Subset of GX, P1,Q1 be Subset of X9 such
  that
A1: P = P1 and
A2: Q = Q1 and
A3: P \/ Q c= [#](X9);
A4: Q c= P \/ Q by XBOOLE_1:7;
  P c= P \/ Q by XBOOLE_1:7;
  then reconsider P2 = P, Q2 = Q as Subset of X9 by A3,A4,XBOOLE_1:1;
  assume that
A5: (Cl P) /\ Q = {} and
A6: P /\ Cl Q = {};
  P2 /\ Cl Q2 = P2 /\ (([#] X9) /\ Cl Q) by PRE_TOPC:17
    .= (P2 /\ [#] X9) /\ Cl Q by XBOOLE_1:16
    .= P /\ Cl Q by XBOOLE_1:28;
  then
A7: P2 misses Cl Q2 by A6;
  Cl P2 = (Cl P) /\ [#] X9 by PRE_TOPC:17;
  then (Cl P2) /\ Q2 = (Cl P) /\ (Q2 /\ [#] X9) by XBOOLE_1:16
    .= (Cl P) /\ Q by XBOOLE_1:28;
  then (Cl P2) misses Q2 by A5;
  hence thesis by A1,A2,A7;
end;
