
theorem Th26:
  for I being non degenerated domRing-like commutative Ring for u,
  v,w being Element of Quot.I holds (quotmult(I)).((quotadd(I)).(u,v),w) = (
  quotadd(I)).((quotmult(I)).(u,w),(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)).((quotadd(I)).(u,v),w) = (quotmult(I)).(qadd(u,v),w) by Def12
    .= qmult(qadd(u,v),w) by Def13
    .= qadd(qmult(u,w),qmult(v,w)) by Th15
    .= qadd((quotmult(I)).(u,w),qmult(v,w)) by Def13
    .= qadd((quotmult(I)).(u,w),(quotmult(I)).(v,w)) by Def13
    .= (quotadd(I)).((quotmult(I)).(u,w),(quotmult(I)).(v,w)) by Def12;
  hence thesis;
end;
