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

theorem
  for FT being filled non empty RelStr, A being non empty Subset of FT
st FT is symmetric holds A is connected iff for P,Q being Subset of FT st A = P
  \/ Q & P misses Q & P,Q are_separated holds P = {}FT or Q = {}FT
proof
  let FT be filled non empty RelStr, A be non empty Subset of FT;
  assume
A1: FT is symmetric;
  now
    assume not A is connected;
    then not FT|A is connected by Th15;
    then not [#](FT|A) is connected;
    then consider P,Q being Subset of FT|A such that
A2: [#](FT|A) = P \/ Q and
A3: P <> {} & Q <> {} and
A4: P misses Q and
A5: P^b misses Q;
    reconsider P1 = P, Q1 = Q as Subset of FT by Th9;
    take P1,Q1;
    thus A = P1 \/ Q1 & P1 misses Q1 by A2,A4,Def3;
A6: P^b=P1^b /\ [#](FT|A) by Th12;
    ([#](FT|A))/\ Q1=Q1 by XBOOLE_1:28;
    then P1^b /\ Q1=P1^b /\ ([#](FT|A))/\ Q by XBOOLE_1:16
      .={} by A5,A6;
    then P1^b misses Q1;
    hence P1,Q1 are_separated & P1 <> {}FT & Q1 <> {}FT by A1,A3,FINTOPO4:10;
  end;
  hence thesis by Th3;
end;
