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

theorem Th27:
  FT is filled & [#]FT = A \/ B & A,B are_separated implies A is open closed
proof
  assume that
A1: FT is filled and
A2: [#]FT = A \/ B and
A3: A,B are_separated;
    B c= B^b by A1,FIN_TOPO:13;
   then A misses (B^b) implies B^b = B by A2,XBOOLE_1:73;
  then
A4: B is closed by A3,FINTOPO4:def 1;
    A c= A^b by A1,FIN_TOPO:13;
  then
A5: A^b misses B implies A^b = A by A2,XBOOLE_1:73;
  B`= A by A1,A2,A3,FINTOPO4:6,PRE_TOPC:5;
  hence thesis by A3,A5,A4,FINTOPO4:def 1,FIN_TOPO:24;
end;
