
theorem Th27:
  for I being non degenerated domRing-like commutative Ring for u,
  v,w being Element of Quot.I holds (quotmult(I)).(u,(quotadd(I)).(v,w)) = (
  quotadd(I)).((quotmult(I)).(u,v),(quotmult(I)).(u,w))
proof
  let I be non degenerated domRing-like commutative Ring;
  let u,v,w be Element of Quot.I;
  (quotmult(I)).(u,(quotadd(I)).(v,w)) = (quotmult(I)).(u,qadd(v,w)) by Def12
    .= qmult(u,qadd(v,w)) by Def13
    .= qadd(qmult(u,v),qmult(u,w)) by Th16
    .= qadd((quotmult(I)).(u,v),qmult(u,w)) by Def13
    .= qadd((quotmult(I)).(u,v),(quotmult(I)).(u,w)) by Def13
    .= (quotadd(I)).((quotmult(I)).(u,v),(quotmult(I)).(u,w)) by Def12;
  hence thesis;
end;
