
theorem Th19:
  for L be Abelian add-associative right_zeroed
  right_complementable well-unital commutative distributive non empty
     doubleLoopStr for p be Polynomial of L
   for n be Nat holds p`^(n+1) = (p`^n)*'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;
  let n be Nat;
  reconsider nn=n as Element of NAT by ORDINAL1:def 12;
  reconsider p1=p as Element of Polynom-Ring L by POLYNOM3:def 10;
  thus p`^(n+1) = (power Polynom-Ring L).(p1,nn)*p1 by GROUP_1:def 7
    .= (p`^n)*'p by POLYNOM3:def 10;
end;
