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
  for c being bag of the carrier of R,
  r being Bags the carrier of R-valued FinSequence
  st q is ordered & q = <*a,c*>^r & c.y > 0
  ex x st a.x > 0 & y <= x
  proof
    let c be bag of the carrier of R;
    let r be Bags the carrier of R-valued FinSequence;
    assume Z0: q is ordered;
    assume Z1: q = <*a,c*>^r;
    assume Z2: c.y > 0;
    len <*a,c*> = 2 by FINSEQ_1:44;
    then
A1: 1 in dom <*a,c*> & 2 in dom <*a,c*> & dom <*a,c*> c= dom q
    by Z1,FINSEQ_1:26,FINSEQ_3:25;
    q/.1 = q.1 = <*a,c*>.1 = a & q/.(1+1) = q.2 = <*a,c*>.2 = c
    by Z1,A1,PARTFUN1:def 6,FINSEQ_1:def 7;
    hence thesis by Z0,Z2,A1;
  end;
