
theorem Th11:
  for L being add-associative right_zeroed right_complementable
Abelian distributive non empty doubleLoopStr for p1,p2 being Polynomial of L
  holds -(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;
A1: -(p19+p29) = -p19+-p29 by RLVECT_1:31;
A2: -p2=-p29 by Lm4;
  p1 + p2 = p19 + p29 by POLYNOM3:def 10;
  then
A3: -(p1+p2)=-(p19+p29) by Lm4;
  -p1=-p19 by Lm4;
  hence thesis by A3,A2,A1,POLYNOM3:def 10;
end;
