
theorem Th6:
  for n being Ordinal, L being Abelian right_zeroed add-associative
  right_complementable well-unital distributive associative commutative non
trivial doubleLoopStr, p,q being Polynomial of n,L holds -(p *' q) = (-p) *' q
  & -(p *' q) = p *' (-q)
proof
  let n be Ordinal, L be Abelian right_zeroed add-associative
  right_complementable well-unital distributive associative commutative non
  trivial doubleLoopStr, p,q be Polynomial of n,L;
  reconsider p9 = p, q9 = q as Element of Polynom-Ring(n,L) by POLYNOM1:def 11;
  reconsider p9,q9 as Element of Polynom-Ring(n,L);
A1: p9 * q9 = p *' q by POLYNOM1:def 11;
  -p = -p9 by Lm7;
  then
A2: (-p9) * q9 = (-p) *' q by POLYNOM1:def 11;
  -q = -q9 by Lm7;
  then
A3: p9 * (-q9) = p *' (-q) by POLYNOM1:def 11;
  -(p9 * q9) = (-p9) * q9 & -(p9 * q9) = p9 * (-q9) by VECTSP_1:9;
  hence thesis by A2,A3,A1,Lm7;
end;
