reserve p,q for Rational;
reserve g,m,m1,m2,n,n1,n2 for Nat;
reserve i,i1,i2,j,j1,j2 for Integer;
reserve R for Ring, F for Field;

theorem
for F being strict Field
holds F is prime iff (F,F_Rat are_isomorphic or
                      ex p being Prime st F,Z/p are_isomorphic)
proof
let F be strict Field;
thus F is prime implies F, F_Rat are_isomorphic or
 ex p being Prime st F, Z/p are_isomorphic
proof
  assume A1: F is prime;
  per cases by Th85;
  suppose Char F = 0;
    then F is 0-characteristic;
    then PrimeField F, F_Rat are_isomorphic by Th100;
    hence thesis by EC_PF_1:def 2,A1;
  end;
  suppose Char F is prime;
    then consider p being Prime such that A2: Char F = p;
    F is p-characteristic by A2;
    then A3: PrimeField F, Z/p are_isomorphic by Th107;
    PrimeField F, F are_isomorphic by EC_PF_1:def 2,A1;
    hence thesis by A3,Th43;
  end;
end;
assume A4: F, F_Rat are_isomorphic or
          ex p being Prime st F, Z/p are_isomorphic;
per cases by A4;
suppose F, F_Rat are_isomorphic;
  then consider f being Function of F,F_Rat such that
  A5: f is RingIsomorphism;
  A6: F_Rat is F-isomorphic by A5;
  then reconsider EK1 = F_Rat as F-homomorphic Field;
  reconsider f as Homomorphism of F,EK1 by A5;
  now let K be Field;
    assume A7: K is strict Subfield of F;
    then reconsider EK = F_Rat as K-homomorphic Field by A6,Th56;
    reconsider g = f|K as Homomorphism of K,EK by A7,Th57;
    A8: Image g = F_Rat by EC_PF_1:def 2;
    A9: the carrier of K c= the carrier of F by A7,EC_PF_1:def 1;
    now let x be object;
      assume x in the carrier of F;
      then reconsider a = x as Element of the carrier of F;
      f.a in Image g by A8;
      then f.a in rng g by RING_2:def 6;
      then consider y being object such that
      A10: y in dom g & g.y = f.a by FUNCT_1:def 3;
      reconsider y as Element of the carrier of K by A10;
      A11: y in the carrier of F by A9;
      (f|the carrier of K).y = f.y by FUNCT_1:49;
      then a = y by A10,A11,FUNCT_2:19;
      hence x in the carrier of K;
      end;
    then the carrier of F c= the carrier of K;
    then A12: the carrier of F = the carrier of K by A7,EC_PF_1:def 1;
    the addF of K = (the addF of F) || the carrier of K
      & the multF of K = (the multF of F) || the carrier of K
      & 1.F = 1.K & 0.F = 0.K by A7,EC_PF_1:def 1;
    hence K = F by A7,A12;
    end;
hence thesis by EC_PF_1:def 2;
end;
suppose ex p being Prime st F, Z/p are_isomorphic;
  then consider p being Prime such that A13: F, Z/p are_isomorphic;
  consider f being Function of F, Z/p such that
  A14: f is RingIsomorphism by A13;
  A15: Z/p is F-isomorphic by A14;
  then reconsider EK1 = Z/p as F-homomorphic Field;
  reconsider f as Homomorphism of F,EK1 by A14;
  now let K be Field;
    assume A16: K is strict Subfield of F;
    then reconsider EK = Z/p as K-homomorphic Field by A15,Th56;
    reconsider g = f|K as Homomorphism of K,EK by A16,Th57;
    A17: Image g = Z/p by EC_PF_1:def 2;
    A18: the carrier of K c= the carrier of F by A16,EC_PF_1:def 1;
    now let x be object;
      assume x in the carrier of F;
      then reconsider a = x as Element of the carrier of F;
      f.a in Image g by A17;
      then f.a in rng g by RING_2:def 6;
      then consider y being object such that
      A19: y in dom g & g.y = f.a by FUNCT_1:def 3;
      reconsider y as Element of the carrier of K by A19;
      A20: y in the carrier of F by A18;
      (f|the carrier of K).y = f.y by FUNCT_1:49;
      then a = y by A19,A20,FUNCT_2:19;
      hence x in the carrier of K;
      end;
    then the carrier of F c= the carrier of K;
    then A21: the carrier of F = the carrier of K by A16,EC_PF_1:def 1;
    the addF of K = (the addF of F) || the carrier of K
      & the multF of K = (the multF of F) || the carrier of K
      & 1.F = 1.K & 0.F = 0.K by A16,EC_PF_1:def 1;
    hence K = F by A16,A21;
    end;
hence thesis by EC_PF_1:def 2;
end;
end;
