reserve T for non empty RelStr,
  a for Element of T;
reserve a for set;

theorem Th5:
  for DP be discrete non empty Poset st (ex a,b be Element of DP st
  a <> b) holds DP is disconnected
proof
  let DP be discrete non empty Poset;
  given a,b be Element of DP such that
A1: a <> b;
  not b in {a} by A1,TARSKI:def 1;
  then reconsider
A = ( (the carrier of DP) \ {a} ) as non empty Subset of DP by XBOOLE_0:def 5;
  reconsider B = {a} as non empty Subset of DP;
A2: the carrier of DP = ( (the carrier of DP) \ {a} ) \/ {a} & ( (the
  carrier of DP) \ {a} ) misses {a} by XBOOLE_1:45,79;
  the InternalRel of DP c= ([:A,A:] \/ [:B,B:])
  proof
    let x be object;
    assume
A3: x in the InternalRel of DP;
    then consider x1,x2 be object such that
A4: x = [x1,x2] by RELAT_1:def 1;
A5: x in id (the carrier of DP) by A3,Def1;
    then
A6: x1 = x2 by A4,RELAT_1:def 10;
    per cases;
    suppose
A7:   x1 in A;
A8:   [:A,A:] c= ([:A,A:] \/ [:B,B:]) by XBOOLE_1:7;
      [x1,x1] in [:A,A:] by A7,ZFMISC_1:87;
      hence thesis by A4,A6,A8;
    end;
    suppose
A9:   not x1 in A;
      x1 in the carrier of DP by A5,A4,RELAT_1:def 10;
      then x1 in (the carrier of DP) \ A by A9,XBOOLE_0:def 5;
      then x1 in (the carrier of DP) /\ B by XBOOLE_1:48;
      then x1 in B by XBOOLE_1:28;
      then
A10:  [x1,x1] in [:B,B:] by ZFMISC_1:87;
      [:B,B:] c= ([:A,A:] \/ [:B,B:]) by XBOOLE_1:7;
      hence thesis by A4,A6,A10;
    end;
  end;
  then
A11: the InternalRel of DP = ((the InternalRel of DP) /\ ([:A,A:] \/ [:B,B:]
  )) by XBOOLE_1:28;
  (the InternalRel of DP) |_2 A = ((the InternalRel of DP) /\ [:A,A:]) & (
the InternalRel of DP) |_2 B = ((the InternalRel of DP) /\ [:B,B:]) by
WELLORD1:def 6;
  then the InternalRel of DP = (the InternalRel of DP) |_2 A \/ (the
  InternalRel of DP) |_2 B by A11,XBOOLE_1:23;
  then [#] DP is disconnected by A2;
  hence thesis;
end;
