reserve a,b for object, I,J for set;

theorem Th7:
  for m,n being bag of I holds n = (m-'(m-'n))+(n-'m)
  proof
    let m,n be bag of I;
    let a; assume a in I;
    thus n.a = (m.a-'(m.a-'n.a))+(n.a-'m.a) by Th6
    .= (m.a-'(m-'n).a)+(n.a-'m.a) by PRE_POLY:def 6
    .= (m-'(m-'n)).a+(n.a-'m.a) by PRE_POLY:def 6
    .= (m-'(m-'n)).a+(n-'m).a by PRE_POLY:def 6
    .= ((m-'(m-'n))+(n-'m)).a by PRE_POLY:def 5;
  end;
