
theorem DP3:
for F being Field
for a being non zero Element of F
for n,m being Nat holds anpoly(a,m)`^n = anpoly(a|^n,n*m)
proof
let F be Field, a be non zero Element of F, n,m be Nat;
defpred P[Nat] means
  for m being Nat holds anpoly(a,m)`^($1) = anpoly(a|^($1),($1)*m);
now let m be Nat;
  thus anpoly(a,m)`^0
     = 1_.(F) by POLYNOM5:15
    .= (1_F)|F by RING_4:14
    .= anpoly(1_F,0) by FIELD_1:7
    .= anpoly(a|^0,0*m) by BINOM:8;
  end; then
IA: P[0];
IS: now let k be Nat;
    assume IV: P[k];
    now let m be Nat;
      thus anpoly(a,m)`^(k+1)
         = (anpoly(a,m)`^k) *' anpoly(a,m) by POLYNOM5:19
        .= anpoly(a|^k,k*m) *' anpoly(a,m) by IV
        .= anpoly((a|^k)*a,k*m+m) by FIELD_1:10
        .= anpoly((a|^k)*(a|^1),k*m+m) by BINOM:8
        .= anpoly(a|^(k+1),(k+1)*m) by BINOM:10;
      end;
    hence P[k+1];
    end;
for k being Nat holds P[k] from NAT_1:sch 2(IA,IS);
hence thesis;
end;
