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 Th89:
for F being Field,
    E being strict Subfield of PrimeField F holds E = PrimeField F
proof
let F be Field, E be strict Subfield of PrimeField F;
set K = PrimeField F;
A1:  the carrier of E c= the carrier of K
  & the addF of E = (the addF of K) || the carrier of E
  & the multF of E = (the multF of K) || the carrier of E
  & 1.E = 1.K & 0.E = 0.K by EC_PF_1:def 1;
E is Subfield of F by EC_PF_1:5;
then carrier/\ F c= the carrier of E by Lm13;
then the carrier of E = the carrier of PrimeField F by EC_PF_1:def 1,Def10;
hence thesis by A1;
end;
