
theorem F16:
for F being Field
for a being non zero Element of F
for n being Nat holds (a")|^n = (a|^n)"
proof
let F be Field, a be non zero Element of F, n be Nat;
defpred P[Nat] means (a")|^($1) = (a|^($1))";
(a")|^0 = (1_F)" by BINOM:8 .= (a|^0)" by BINOM:8; then
IA: P[0];
IS: now let k be Nat;
    assume IV: P[k];
    A: a <> 0.F & a|^k <> 0.F;
    (a")|^(k+1) = ((a")|^k) * ((a")|^1) by BINOM:10
               .= ((a")|^k) * (a") by BINOM:8
               .= (a * a|^k)" by IV,A,VECTSP_2:11
               .= (a|^k * a|^1)" by BINOM:8
               .= (a|^(k+1))" 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;
