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

theorem Th32:
  FT is filled & A is connected & A c= B \/ C & B,C are_separated
  implies A c= B or A c= C
proof
  assume that
A1: FT is filled and
A2: A is connected and
A3: A c= B \/ C and
A4: B,C are_separated;
A5: (A /\ B) \/ (A /\ C) = A /\ (B \/ C) by XBOOLE_1:23
    .= A by A3,XBOOLE_1:28;
  assume that
A6: not A c= B and
A7: not A c= C;
  A meets B by A3,A7,XBOOLE_1:73;
  then
A8: A /\ B <> {};
  A meets C by A3,A6,XBOOLE_1:73;
  then
A9: A /\ C <> {};
A10: A /\ B c= B & A /\ C c= C by XBOOLE_1:17;
  then {}FT={} & (A /\ B) misses (A /\ C) by A1,A4,Th28,FINTOPO4:6;
  hence contradiction by A2,A4,A8,A9,A10,A5,Th3,Th28;
end;
