
theorem Th15:
  for n being set, L being non empty addLoopStr, p,q being Series of n, L
  for x being bag of n holds (p+q).x = p.x + q.x
  proof
    let n be set;
    let L be non empty addLoopStr;
    let p, q be Series of n, L;
    let x be bag of n;
A1: dom p = Bags n & dom q = Bags n by FUNCT_2:def 1;
A2: x in Bags n by PRE_POLY:def 12;
    then
A3: p/.x = p.x & q/.x = q.x by A1,PARTFUN1:def 6;
A4: dom (p+q) = dom p /\ dom q by VFUNCT_1:def 1;
    hence (p+q).x = (p+q)/.x by A1,A2,PARTFUN1:def 6
    .= p.x + q.x by A1,A2,A3,A4,VFUNCT_1:def 1;
  end;
