
theorem
  for FT being filled non empty RelStr, B being Subset of FT st FT is
symmetric holds B is connected iff not (ex C being Subset of FT st C<>{} & B\C
  <>{} & C c= B & (C^b) misses (B\C))
proof
  let FT be filled non empty RelStr, B be Subset of FT;
  assume
A1: FT is symmetric;
  thus B is connected implies not (ex C being Subset of FT st C<>{} & B\C <>{}
  & C c= B & (C^b) misses (B\C))
  proof
    assume
A2: B is connected;
    for C being Subset of FT st C c= B & (C^b) misses (B\C) holds C={} or
    B\C={}
    proof
      let C be Subset of FT;
      assume that
A3:   C c= B and
A4:   (C^b) misses (B\C);
      C misses ((B\C)^b) by A1,A4,Th8;
      then
A5:   C,B\C are_separated by A4;
      C \/ (B\C)=C \/ B by XBOOLE_1:39
        .=B by A3,XBOOLE_1:12;
      then C = B or B\C = B by A1,A2,A5,Th12;
      hence thesis by A3,XBOOLE_1:37,38;
    end;
    hence thesis;
  end;
  thus not (ex C being Subset of FT st C<>{} & B\C <>{} & C c= B & (C^b)
  misses (B\C)) implies B is connected
  proof
    assume
A6: not (ex C being Subset of FT st C<>{} & B\C <>{} & C c= B & (C^b)
    misses (B\C));
    for A, B2 being Subset of FT st B = A \/ B2 & A,B2 are_separated
    holds A = B or B2 = B
    proof
      let A, B2 be Subset of FT;
      assume that
A7:   B = A \/ B2 and
A8:   A,B2 are_separated;
A9:   (A \/ B2) \A =B2\A by XBOOLE_1:40;
      A^b misses B2 by A8;
      then A^b misses (B\A) by A7,A9,XBOOLE_1:36,63;
      then A={} or B\A={} by A6,A7,XBOOLE_1:7;
      then
A10:  B=B2 or B c= A by A7,XBOOLE_1:37;
      A c= B by A7,XBOOLE_1:7;
      hence thesis by A10,XBOOLE_0:def 10;
    end;
    hence thesis by A1,Th12;
  end;
end;
