
theorem FA4:
for F being finite Field
for f being Automorphism of F
for a being Element of PrimeField F holds f.a = a
proof
let F be finite Field, f be Automorphism of F, a be Element of PrimeField F;
set P = PrimeField F;
M: the carrier of P c= the carrier of F by EC_PF_1:def 1; then
reconsider g = f|(the carrier of P) as
                Function of the carrier of P,the carrier of F by FUNCT_2:32;
H: now let o be object;
   assume H1: o in the carrier of P; then
   reconsider a = o as Element of F by M;
   a in P by H1; then
   f.a in P by FA4a;
   hence g.o in the carrier of P by H1,FUNCT_1:49;
   end;
   dom g = the carrier of P by FUNCT_2:def 1; then
reconsider g as Function of the carrier of P,the carrier of P
   by H,FUNCT_2:3;
A: F is P-extending by FIELD_4:7;
B: g is one-to-one by FUNCT_1:52;
   card(the carrier of P) = card(the carrier of P); then
   g is onto by B,FINSEQ_4:63; then
F: g = id P by A,PrimAUT;
now let a be Element of P;
  thus f.a = g.a by FUNCT_1:49 .= a by F;
  end;
hence thesis;
end;
