
theorem lemAuthelp:
for p being Prime
for n being non zero Nat
for F being Field st card F = p|^n
for k being Nat st 0 < k & k <= n-1 holds (Frob F)`^k <> id F
proof
let p be Prime, n be non zero Nat, F be Field;
assume AS1: card F = p|^n;
let k be Nat;
assume AS2: 0 < k & k <= n-1; then
reconsider k as non zero Nat;
H: k + 0 < (n - 1) + 1 by AS2,XREAL_1:8;
I: Char F = p by AS1,T5; then
reconsider F as p-characteristic finite Field by AS1,RING_3:def 6;
set g = (Frob F)`^k;
now assume AS3: g = id F;
  now let o be object;
    assume o in the carrier of F; then
    reconsider a = o as Element of F;
    a|^(p|^k) = g.a by I,T4 .= a by AS3;
    then a is_a_root_of X^(p|^k,F) by thXX;
    hence o in Roots X^(p|^k,F) by POLYNOM5:def 10;
    end; then
  the carrier of F c= Roots X^(p|^k,F); then
  A: card(the carrier of F) <= card Roots X^(p|^k,F) by NAT_1:43;
  deg X^(p|^k,F) = p|^k by Lm12; then
  card Roots X^(p|^k,F) <= p|^k by RING_5:22; then
  C: p|^n <= p|^k by A,AS1,XXREAL_0:2;
  p|^k <= p|^n by H,TT7; then
  p|^k = p|^n by C,XXREAL_0:1;
  hence contradiction by H,lemp;
  end;
hence thesis;
end;
