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;
reserve p for partition of b-'a, q for partition of b;
reserve J for set, m for bag of I;

theorem Th32:
  q is ordered & q = <*a*>^p & a.x > 0 & a.y > 0 & x <> y implies x ## y
  proof
    assume Z0: q is ordered;
    assume Z1: q = <*a*>^p;
    assume Z2: a.x > 0 & a.y > 0;
    a in {a} = rng <*a*> by TARSKI:def 1,FINSEQ_1:39;
    then a in rng <*a*> \/ rng p = rng q by Z1,XBOOLE_0:def 3,FINSEQ_1:31;
    hence thesis by Z0,Z2;
  end;
