
theorem
for p being Prime
for F being p-characteristic Field
holds F is perfect iff Frob F is Automorphism of F
proof
let p be Prime, F be p-characteristic Field;
H: Char F = p by RING_3:def 6;
A: now assume F is perfect; then
   B: F == F|^p by FIELD_15:91;
   C: rng(Frob F) c= the carrier of F by RELAT_1:def 19;
   now let o be object;
     assume o in the carrier of F; then
     reconsider b = o as Element of F;
     b in the carrier of F|^p &
     the carrier of F|^p = the set of all a|^p where a is Element of F
        by B,FIELD_15:def 1; then
     consider a being Element of F such that
     C: a|^p = b;
     D: dom(Frob F) = the carrier of F by FUNCT_2:def 1;
     (Frob F).a = b by H,C,defFr;
     hence o in rng(Frob F) by D,FUNCT_1:def 3;
     end;
   then Frob F is onto by C,TARSKI:2;
   hence Frob F is Automorphism of F;
   end;
now assume A: Frob F is Automorphism of F;
   B: the carrier of F|^p c= the carrier of F by EC_PF_1:def 1;
   now let o be object;
     assume o in the carrier of F; then
     reconsider b = o as Element of F;
     rng(Frob F) = the carrier of F by A,FUNCT_2:def 3; then
     consider a being object such that
     C: a in dom(Frob F) & (Frob F).a = b by FUNCT_1:def 3;
     reconsider a as Element of F by C;
     D: b = a|^p by H,C,defFr;
     the carrier of F|^p = the set of all a|^p where a is Element of F
        by FIELD_15:def 1;
     hence o in the carrier of F|^p by D;
     end;
   then E: the carrier of F c= the carrier of F|^p;
   then the carrier of F = the carrier of F|^p by B,XBOOLE_0:def 10;
   then the addF of F|^p = (the addF of F)||(the carrier of F) &
        the multF of F|^p = (the multF of F)||(the carrier of F) &
        1.(F|^p) = 1.F & 0.(F|^p) = 0.F by FIELD_15:def 1;
   then F == F|^p by E,EC_PF_1:def 1;
   hence F is perfect by FIELD_15:91;
   end;
hence thesis by A;
end;
