
theorem multi0:
for F being Field,
    p being Polynomial of F
for i,j being Element of NAT holds p`^(i+j) = (p`^i) *' (p`^j)
proof
let F be Field, p be Polynomial of F; let i,j be Element of NAT;
set G = Polynom-Ring F;
reconsider p1 = p as Element of G by POLYNOM3:def 10;
A: p`^i = power(G).(p1,i) & p`^j = power(G).(p1,j) by POLYNOM5:def 3;
thus p`^(i+j) = power(G).(p1,i+j) by POLYNOM5:def 3
  .= p1|^(i+j) by BINOM:def 2
  .= (p1|^i) * (p1|^j) by BINOM:10
  .= power(G).(p1,i) * (p1|^j) by BINOM:def 2
  .= power(G).(p1,i) * power(G).(p1,j) by BINOM:def 2
  .= (p`^i) *' (p`^j) by A,POLYNOM3:def 10;
end;
