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 Lem11:
  for a,b being bag of I st a <> EmptyBag I holds a+b <> EmptyBag I
  proof let a,b be bag of I;
    given i being object such that
A1: i in I & a.i <> (EmptyBag I).i;
    take i; thus i in I by A1;
    (EmptyBag I).i = 0 by A1,FUNCOP_1:7;
    then a.i+b.i <> (EmptyBag I).i by A1;
    hence thesis by PRE_POLY:def 5;
  end;
