
theorem finun:
for F1,F2 being finite Field st order F1 = order F2 holds F1,F2 are_isomorphic
proof
let F1,F2 be finite Field;
assume AS: order F1 = order F2;
consider p1 being Prime, n1 being non zero Nat such that
A: Char F1 = p1 & order F1 = p1|^n1 by finex2;
consider p2 being Prime, n2 being non zero Nat such that
B: Char F2 = p2 & order F2 = p2|^n2 by finex2;
set PF1 = PrimeField F1, PF2 = PrimeField F2;
p1 = p2 & n1 = n2 by AS,A,B,lemp; then
PF1,Z/p1 are_isomorphic & Z/p1,PF2 are_isomorphic by A,B,RING_3:114; then
PF1,PF2 are_isomorphic by RING_3:44; then
consider i being Function of PF1,PF2 such that
I: i is RingIsomorphism;
reconsider PF2 as PF1-homomorphic PF1-isomorphic Field
  by I,RING_2:def 4,RING_3:def 4;
reconsider i as Isomorphism of PF1,PF2 by I;
reconsider E1 = F1 as SplittingField of X^(p1|^n1,PF1) by A,split;
set E2 = the SplittingField of (PolyHom i).X^(p1|^n1,PF1);
consider f being Function of E1,E2 such that
D: f is i-extending isomorphism by FIELD_8:57;
E: E1,E2 are_isomorphic by D;
F: (PolyHom i).X^(p1|^n1,PF1) = X^(p2|^n2,PF2)
   proof
   reconsider q = (PolyHom i).X^(p1|^n1,PF1) as Polynomial of PF2;
   set r1 = X^(p1|^n1,PF1), r2 = X^(p2|^n2,PF2);
   now let k be Nat;
     per cases;
     suppose H: k = 1;
       hence q.k = i.(r1.1) by FIELD_1:def 2
                .= i.(-1.PF1) by Lm10
                .= -(i.(1_PF1)) by RING_2:7
                .= -(1_PF2) by GROUP_1:def 13
                .= r2.k by H,Lm10;
       end;
     suppose H: k = p1|^n1;
       thus q.k = i.(r1.k) by FIELD_1:def 2
               .= i.(1_PF1) by H,Lm10
               .= 1_PF2 by GROUP_1:def 13
               .= r2.k by H,AS,A,B,Lm10;
       end;
     suppose H: k <> 1 & k <> p1|^n1;
       thus q.k = i.(r1.k) by FIELD_1:def 2
               .= i.(0.PF1) by H,Lm11
               .= 0.PF2 by RING_2:6
                .= r2.k by H,AS,A,B,Lm11;
       end;
     end;
   hence thesis;
   end;
reconsider E3 = F2 as SplittingField of X^(p2|^n2,PF2) by B,split;
E2,E3 are_isomorphic_over PF2 by F,FIELD_8:58; then
consider g being Function of E2,E3 such that
G: g is isomorphism by FIELD_8:def 5;
E2,E3 are_isomorphic by G;
hence thesis by E,RING_3:44;
end;
