
theorem Th23:
  for L being Abelian add-associative right_zeroed
  right_complementable distributive non empty doubleLoopStr, p, q, r being (
  Polynomial of L) holds p*'q-p*'r = p*'(q-r)
proof
  let L be Abelian add-associative right_zeroed right_complementable
  distributive non empty doubleLoopStr, p, q, r be (Polynomial of L);
  set PRL = Polynom-Ring L;
  reconsider pc = p, qc = q, rc = r as Element of PRL by POLYNOM3:def 10;
A1: qc-rc = q-r by Th22;
  p*'q = pc*qc & p*'r = pc*rc by POLYNOM3:def 10;
  hence p*'q-p*'r = pc*qc - pc*rc by Th22
    .= pc*(qc-rc) by VECTSP_1:11
    .= p*'(q-r) by A1,POLYNOM3:def 10;
end;
