
theorem Th23:
  for I being non degenerated domRing-like commutative Ring for u,
  v,w being Element of Quot.I holds (quotmult(I)).((quotmult(I)).(u,v),w) = (
  quotmult(I)).(u,(quotmult(I)).(v,w))
proof
  let I be non degenerated domRing-like commutative Ring;
  let u,v,w be Element of Quot.I;
  (quotmult(I)).((quotmult(I)).(u,v),w) = (quotmult(I)).(qmult(u,v),w) by Def13
    .= qmult(qmult(u,v),w) by Def13
    .= qmult(u,qmult(v,w)) by Th13
    .= qmult(u,(quotmult(I)).(v,w)) by Def13
    .= (quotmult(I)).(u,(quotmult(I)).(v,w)) by Def13;
  hence thesis;
end;
