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 Lem10:
  for p being Bags I-valued FinSequence st Sum p = EmptyBag I &
  for a being bag of I st a in rng p holds a <> EmptyBag I
  holds p = {}
  proof
    let p be Bags I-valued FinSequence;
    assume Z0: Sum p = EmptyBag I;
    assume Z1: for a being bag of I st a in rng p holds a <> EmptyBag I;
    assume p <> {};
    then consider a being object such that
A1: a in rng p by XBOOLE_0:7;
    rng p c= Bags I by RELAT_1:def 19;
    then reconsider a as Element of Bags I by A1;
    consider x being object such that
A2: x in I & a.x <> (EmptyBag I).x by A1,Z1,PBOOLE:def 10;
A4: (EmptyBag I).x = 0 by A2,FUNCOP_1:7;
    then
A3: a.x > 0 by A2;
    thus thesis by A4,A3,Z0,A1,Th26,PRE_POLY:def 11;
  end;
