
theorem FAutEQ1:
for F being Field
for E1,E2 being FieldExtension of F st E1 == E2
for f being Automorphism of E1 holds f is Automorphism of E2
proof
let F be Field, E1,E2 being FieldExtension of F;
assume AS: E1 == E2;
let f be Automorphism of E1;
reconsider g = f as Function of E2,E2 by AS;
   now let a1,a2 be Element of E2;
   reconsider b1 = a1, b2 = a2 as Element of E1 by AS;
   thus g.(a1+a2) = f.(b1+b2) by AS
                 .= f.b1 + f.b2 by VECTSP_1:def 20
                 .= g.a1 + g.a2 by AS;
   end; then
B: g is additive;
   now let a1,a2 be Element of E2;
   reconsider b1 = a1, b2 = a2 as Element of E1 by AS;
   thus g.(a1*a2) = f.(b1*b2) by AS
                 .= f.b1 * f.b2 by GROUP_6:def 6
                 .= g.a1 * g.a2 by AS;
   end; then
C: g is multiplicative;
   g.(1_E2) = f.(1_E1) by AS .= 1_E2 by AS,GROUP_1:def 13; then
g is unity-preserving;
hence thesis by AS,B,C;
end;
