
theorem Th20:
  for I being non degenerated domRing-like commutative Ring for u,
  v,w being Element of Quot.I holds (quotadd(I)).((quotadd(I)).(u,v),w) = (
  quotadd(I)).(u,(quotadd(I)).(v,w))
proof
  let I be non degenerated domRing-like commutative Ring;
  let u,v,w be Element of Quot.I;
  (quotadd(I)).((quotadd(I)).(u,v),w) = (quotadd(I)).(qadd(u,v),w) by Def12
    .= qadd(qadd(u,v),w) by Def12
    .= qadd(u,qadd(v,w)) by Th11
    .= qadd(u,(quotadd(I)).(v,w)) by Def12
    .= (quotadd(I)).(u,(quotadd(I)).(v,w)) by Def12;
  hence thesis;
end;
