
theorem
  for L be Abelian add-associative right_zeroed right_complementable
  well-unital commutative distributive non empty doubleLoopStr for p be
  Polynomial of L holds p`^3 = p*'p*'p
proof
  let L be Abelian add-associative right_zeroed right_complementable
  well-unital commutative distributive non empty doubleLoopStr;
  let p be Polynomial of L;
  reconsider p1=p as Element of Polynom-Ring L by POLYNOM3:def 10;
  reconsider pp=p1*p1 as Polynomial of L by POLYNOM3:def 10;
  thus p`^3 = (power Polynom-Ring L).(p1,2+1)
    .= power(p1,1+1)*p1 by GROUP_1:def 7
    .= power(p1,0+1)*p1*p1 by GROUP_1:def 7
    .= (power Polynom-Ring L).(p1,0)*p1*p1*p1 by GROUP_1:def 7
    .= (1_Polynom-Ring L)*p1*p1*p1 by GROUP_1:def 7
    .= p1*p1*p1
    .= pp*'p by POLYNOM3:def 10
    .= p*'p*'p by POLYNOM3:def 10;
end;
