reserve FT for non empty RelStr;
reserve x, y, z for Element of FT;
reserve A for Subset of FT;
reserve F for Subset of FT;

theorem
  A^s <> {} & card A <> 1 implies A is not connected
proof
  assume that
A1: A^s <> {} and
A2: card A <> 1;
  consider z being Element of FT such that
A3: z in A^s by A1,SUBSET_1:4;
  set C = {z};
  set B = A \ {z};
  card {z} = 1 & A <> {} by A3,Th9,CARD_1:30;
  then
A4: B <> {} by A2,ZFMISC_1:58;
  z in A by A3,Th9;
  then {z} is Subset of A by SUBSET_1:41;
  then
A5: A = B \/ C by XBOOLE_1:45;
A6: (U_FT z \ {z}) misses A by A3,Th9;
A7: B^b misses C
  proof
    assume B^b meets C;
    then consider x being object such that
A8: x in B^b and
A9: x in C by XBOOLE_0:3;
    reconsider x as Element of FT by A8;
A10: x = z by A9,TARSKI:def 1;
    U_FT x meets B by A8,Th8;
    then consider y being object such that
A11: y in U_FT x and
A12: y in B by XBOOLE_0:3;
    not x in B by A9,XBOOLE_0:def 5;
    then y in U_FT x \ {x} by A11,A12,ZFMISC_1:56;
    then (U_FT z \ {z}) meets B by A12,A10,XBOOLE_0:3;
    then
A13: ex w being object st w in (U_FT z \ {z}) /\ B by XBOOLE_0:4;
    (U_FT z \ {z}) /\ B c= (U_FT z \ {z}) /\ A by XBOOLE_1:26,36;
    hence contradiction by A6,A13;
  end;
  B misses C by XBOOLE_1:79;
  hence thesis by A5,A4,A7;
end;
