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

theorem Th28:
  for A,B,A1,B1 being Subset of FT holds A,B are_separated & A1 c=
  A & B1 c= B implies A1,B1 are_separated
proof
  let A,B,A1,B1 be Subset of FT;
  assume that
A1: A,B are_separated and
A2: A1 c= A and
A3: B1 c= B;
  A misses (B^b) by A1,FINTOPO4:def 1;
  then
A4: A/\(B^b)={};
  B1^b c= B^b by A3,FIN_TOPO:14;
  then A1 /\ (B1^b) = {}FT by A2,A4,XBOOLE_1:3,27;
  then
A5: A1 misses (B1^b);
  A^b misses B by A1,FINTOPO4:def 1;
  then
A6: A^b /\ B={};
  A1^b c= A^b by A2,FIN_TOPO:14;
  then (A1^b) /\ B1 = {}FT by A3,A6,XBOOLE_1:3,27;
  then (A1^b) misses B1;
  hence thesis by A5,FINTOPO4:def 1;
end;
