
theorem Th29:
  for I being non degenerated domRing-like commutative Ring for u
being Element of Quot.I st u <> q0.I holds (quotmult(I)).(u,(quotmultinv(I)).(u
  )) = q1.I & (quotmult(I)).((quotmultinv(I)).(u),u) = q1.I
proof
  let I be non degenerated domRing-like commutative Ring;
  let u be Element of Quot.I;
  assume
A1: u <> q0.I;
A2: (quotmult(I)).((quotmultinv(I)).(u),u) = (quotmult(I)).(qmultinv(u),u)
  by Def15
    .= qmult(qmultinv(u),u) by Def13
    .= q1.I by A1,Th18;
  (quotmult(I)).(u,(quotmultinv(I)).(u)) = (quotmult(I)).(u,qmultinv(u))
  by Def15
    .= qmult(u,qmultinv(u)) by Def13
    .= q1.I by A1,Th18;
  hence thesis by A2;
end;
