reserve a,b for object, I,J for set;
reserve b for bag of I;
reserve R for asymmetric transitive non empty RelStr,
  a,b,c for bag of the carrier of R,
  x,y,z for Element of R;

theorem
  <*a*> is ordered iff a <> EmptyBag the carrier of R &
  for x,y st a.x > 0 & a.y > 0 & x <> y holds x ## y
  proof
    hereby
      assume Z0: <*a*> is ordered;
      a in {a} = rng <*a*> by TARSKI:def 1,FINSEQ_1:39;
      hence a <> EmptyBag the carrier of R &
      for x,y st a.x > 0 & a.y > 0 & x <> y holds x ## y by Z0;
    end;
    assume Z3: a <> EmptyBag the carrier of R;
    assume Z4: for x,y st a.x > 0 & a.y > 0 & x <> y holds x ## y;
    set p = <*a*>;
    hereby let m be bag of the carrier of R;
      assume m in rng p;
      then m in {a} by FINSEQ_1:39;
      hence for x being Element of R st m.x > 0 holds m.x = a.x
      by TARSKI:def 1;
    end;
    hereby let m be bag of the carrier of R;
      assume m in rng p;
      then m in {a} by FINSEQ_1:39;
      then m = a by TARSKI:def 1;
      hence for x,y being Element of R st m.x > 0 & m.y > 0 & x <> y
      holds x ## y by Z4;
    end;
    hereby let m be bag of the carrier of R;
      assume m in rng p;
      then m in {a} by FINSEQ_1:39;
      hence m <> EmptyBag the carrier of R by Z3,TARSKI:def 1;
    end;
    let i be Nat;
    assume i in dom p & i+1 in dom p;
    then 1 <= i < i+1 <= len p = 1 by FINSEQ_1:40,FINSEQ_3:25,NAT_1:13;
    hence thesis by XXREAL_0:2;
  end;
