theorem Th90:
for F being Field, E being Subfield of F holds PrimeField F is Subfield of E
proof
  let F be Field, E be Subfield of F;
  the carrier of PrimeField F c= the carrier of E
  proof
    let x be object;
    assume x in the carrier of PrimeField F;
    then x in carrier/\ F by Def10;
    then ex y being Element of F st
    x = y & for K being Subfield of F holds y in K;
    then x in E;
    hence thesis;
  end;
  hence thesis by EC_PF_1:6;
end;
