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

theorem Th29:
  A,B are_separated & A,C are_separated implies A,B \/ C are_separated
proof
  assume that
A1: A,B are_separated and
A2: A,C are_separated;
A3: (A^b) misses C by A2,FINTOPO4:def 1;
  (A^b) misses B by A1,FINTOPO4:def 1;
  then
A4: (A^b) /\ B = {};
  (A^b) /\ (B \/ C) = ((A^b) /\ B) \/ ((A^b) /\ C) by XBOOLE_1:23
    .= {} by A3,A4;
  then
A5: (A^b) misses (B \/ C);
  A misses (B^b) by A1,FINTOPO4:def 1;
  then
A6: A /\ (B^b) = {};
A7: A misses (C^b) by A2,FINTOPO4:def 1;
  A /\ ((B \/ C)^b) = A /\ ((B^b) \/ (C^b)) by FINTOPO3:8
    .= (A /\ (B^b)) \/ (A /\ (C^b)) by XBOOLE_1:23
    .= {} by A7,A6;
  then A misses ((B \/ C)^b);
  hence thesis by A5,FINTOPO4:def 1;
end;
