
theorem Th12:
  for L being add-associative right_zeroed right_complementable
distributive Abelian non empty doubleLoopStr for p1,p2 being Polynomial of L
  holds -(p1 *' p2) = (-p1) *' p2 & -(p1 *' p2) = p1 *' (-p2)
proof
  let L be add-associative right_zeroed right_complementable distributive
  Abelian non empty doubleLoopStr;
  let p1,p2 be Polynomial of L;
  reconsider p19=p1,p29=p2 as Element of Polynom-Ring(L) by POLYNOM3:def 10;
  p1*'p2= p19*p29 by POLYNOM3:def 10;
  then
A1: -(p1*'p2)=-(p19*p29) by Lm4;
  -p1 = -p19 by Lm4;
  then (-p1) *' p2 = (-p19) * p29 by POLYNOM3:def 10;
  hence -(p1 *' p2) = (-p1) *' p2 by A1,VECTSP_1:9;
  -p2 = -p29 by Lm4;
  then p1 *' (-p2) = p19 * (-p29) by POLYNOM3:def 10;
  hence thesis by A1,VECTSP_1:8;
end;
