
theorem Th25:
  for L being Abelian add-associative right_zeroed
right_complementable distributive domRing-like non empty doubleLoopStr, p, q,
  r being Polynomial of L st p <> 0_. L & p*'q = p*'r holds q = r
proof
  let L be Abelian add-associative right_zeroed right_complementable
distributive domRing-like non empty doubleLoopStr, p, q, r be Polynomial of L;
  assume
A1: p <> 0_. L;
  assume p*'q = p*'r;
  then p*'q-p*'r = 0_. L by POLYNOM3:29;
  then p*'(q-r) = 0_. L by Th23;
  then q-r = 0_. L by A1,Th18;
  hence thesis by Th24;
end;
