reserve FT for non empty RelStr,
  A,B,C for Subset of FT;

theorem
  for X9 being non empty SubSpace of FT, P1,Q1 being Subset of FT, 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 non empty SubSpace of FT, P1,Q1 be Subset of FT, P,Q be Subset of
  X9 such that
A1: P = P1 & Q = Q1;
  reconsider P2 = P, Q2 = Q as Subset of FT by Th9;
  assume
A2: P,Q are_separated;
  then P^b misses Q by FINTOPO4:def 1;
  then
A3: (P^b) /\ Q = {};
  P misses (Q^b) by A2,FINTOPO4:def 1;
  then
A4: P /\ (Q^b) = {};
  P /\ (Q^b) = P /\ (([#] X9) /\ (Q2^b)) by Th12
    .= P /\ [#] X9 /\ (Q2^b) by XBOOLE_1:16
    .= P2 /\ (Q2^b) by XBOOLE_1:28;
  then
A5: P2 misses Q2^b by A4;
  (P^b) /\ Q = ((P2^b) /\ ([#](X9))) /\ Q by Th12
    .= (P2^b) /\ (Q /\ [#] X9) by XBOOLE_1:16
    .= (P2^b) /\ Q2 by XBOOLE_1:28;
  then (P2^b) misses Q2 by A3;
  hence thesis by A1,A5,FINTOPO4:def 1;
end;
