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 Th92:
for F,K being Field holds
K = PrimeField F iff (K is strict Subfield of F &
                      for E being Subfield of F holds K is Subfield of E)
proof
let F,K be Field;
now assume A1: K is strict Subfield of F &
              for E being Subfield of F holds K is Subfield of E;
   then A2: the carrier of K c= the carrier of F
           & 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 EC_PF_1:def 1;
   A3: now let x be object;
      assume x in carrier/\ F;
      then consider y being Element of F such that
      A4: x = y & for E being Subfield of F holds y in E;
      x in K by A4,A1;
      hence x in the carrier of K;
      end;
   now let x be object;
      assume A5: x in the carrier of K;
      for E being Subfield of F holds x in E
        proof
        let E be Subfield of F;
        K is Subfield of E by A1;
        then the carrier of K c= the carrier of E by EC_PF_1:def 1;
        hence x in E by A5;
        end;
      hence x in carrier/\ F by A2,A5;
      end;
   then the carrier of K = carrier/\ F by A3,TARSKI:2;
   hence K = PrimeField F by A1,A2,Def10;
   end;
hence thesis by Th90;
end;
