
theorem
for F being Field
for a being Element of F
for n being Nat holds (a|F)`^n = (a|^n)|F
proof
let F be Field, a be Element of F, n be Nat;
defpred P[Nat] means (a|F)`^($1) = (a|^($1))|F;
  (a|F)`^0
    = 1_.(F) by POLYNOM5:15
   .= (1_F)|F by RING_4:14
   .= (a|^0)|F by BINOM:8; then
IA: P[0];
IS: now let k be Nat;
    assume IV: P[k];
    (a|F)`^(k+1)
         = ((a|^k)|F) *' (a|F) by IV,POLYNOM5:19
        .= ((a|^k) * a)|F by RING_4:18
        .= ((a|^k) * (a|^1))|F by BINOM:8
        .= (a|^(k+1))|F by BINOM:10;
    hence P[k+1];
    end;
for k being Nat holds P[k] from NAT_1:sch 2(IA,IS);
hence thesis;
end;
