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 E,F being Field
holds E includes F iff ex K being strict Subfield of E st K,F are_isomorphic
proof
  let E,F be Field;
  hereby assume A1: E includes F;
   reconsider EE = E as Ring;
   consider K being strict Subring of EE such that
   A2: K,F are_isomorphic by A1;
   ex f being Function of F,K st f is RingIsomorphism by A2,QUOFIELD:def 23;
   then reconsider KK = K as F-isomorphic Ring by Def4;
   the carrier of KK c= the carrier of E
   & the addF of KK = (the addF of E) || the carrier of KK
   & the multF of KK = (the multF of E) || the carrier of KK
   & 1.E = 1.KK & 0.E = 0.KK by C0SP1:def 3;
   then KK is strict Subfield of E by EC_PF_1:def 1;
   hence ex K being strict Subfield of E st K,F are_isomorphic by A2;
  end;
  given K being strict Subfield of E such that A3: K,F are_isomorphic;
  the carrier of K c= the carrier of E
  & the addF of K = (the addF of E) || the carrier of K
  & the multF of K = (the multF of E) || the carrier of K
  & 1.E = 1.K & 0.E = 0.K by EC_PF_1:def 1;
  then K is strict Subring of E by C0SP1:def 3;
  hence E includes F by A3;
end;
