
theorem multi00:
for F being Field,
    p being Polynomial of F
for E being FieldExtension of F,
    q being Polynomial of E
for n being Element of NAT st q = p holds q`^n = p`^n
proof
let F be Field, p be Polynomial of F;
let E be FieldExtension of F, q be Polynomial of E;
let n be Element of NAT;
assume AS: q = p;
defpred P[Nat] means q`^($1) = p`^($1);
   q`^0 = 1_.(E) by POLYNOM5:15 .= 1_.(F) by  FIELD_4:14
       .= p`^0 by POLYNOM5:15; then
A: P[0];
B: now let k be Nat;
   assume IV: P[k];
   q`^(k+1) = (q`^k) *' q by POLYNOM5:19
           .= (p`^k) *' p by AS,IV,FIELD_4:17
           .= p`^(k+1) by POLYNOM5:19;
   hence P[k+1];
   end;
for k being Nat holds P[k] from NAT_1:sch 2(A,B);
hence thesis;
end;
