
theorem ThA:
for F being Field
for E being FieldExtension of F
for a being Element of E
for n being Nat holds a|^n in the carrier of FAdj(F,{a})
proof
let F be Field, E be FieldExtension of F, a be Element of E, n be Nat;
H: {a} is Subset of FAdj(F,{a}) by FIELD_6:35;
defpred P[Nat] means a|^($1) in the carrier of FAdj(F,{a});
    a|^0 = 1_E by BINOM:8 .= 1.FAdj(F,{a}) by FIELD_6:def 6; then
IA: P[0];
IS: now let k be Nat;
    assume IV: P[k];
    I: FAdj(F,{a}) is Subring of E by FIELD_5:12;
    a in {a} by TARSKI:def 1; then
    reconsider b1 = a, b2 = a |^ k as Element of FAdj(F,{a}) by IV,H;
    a |^ (k+1) = a |^k * a |^ 1 by BINOM:10
              .= a |^k * a by BINOM:8
              .= b2 * b1 by I,FIELD_6:16;
    hence P[k+1];
    end;
for n being Nat holds P[n] from NAT_1:sch 2(IA,IS);
hence thesis;
end;
