
theorem fresh3P:
for p being Prime
for n being non zero Nat
for F be Field st card F = p|^n
for a being Element of F holds a in PrimeField F iff a|^p = a
proof
let p be Prime, n be non zero Nat, F be Field;
assume AS: card F = p|^n; then
H: Char F = p by T5;
let a be Element of F;
A: now assume a in PrimeField F;
   hence a = (Frob F).a by AS,Ffix2 .= a|^p by H,defFr;
   end;
now assume a|^p = a;
  then a = (Frob F).a by H,defFr;
  hence a in PrimeField F by AS,Ffix2;
  end;
hence thesis by A;
end;
