
theorem
  for n being set, L being non empty addLoopStr, p,q,r being Series of n, L
  st for x being bag of n holds r.x = p.x + q.x holds
  r = p + q
  proof
    let n be set;
    let L be non empty addLoopStr;
    let p,q,r be Series of n, L;
    assume
A1: for x being bag of n holds r.x = p.x + q.x;
    let x be Element of Bags n;
A2: dom (p+q) = Bags n by FUNCT_2:def 1;
A3: (p+q)/.x = (p+q).x;
A4: p/.x = p.x & q/.x = q.x;
    thus r.x = p.x + q.x by A1
    .= (p + q).x by A2,A3,A4,VFUNCT_1:def 1;
  end;
