reserve x for set;
reserve i,j for Integer;
reserve n,n1,n2,n3 for Nat;
reserve K,K1,K2,K3 for Field;
reserve SK1,SK2 for Subfield of K;
reserve ek,ek1,ek2 for Element of K;

theorem Th7:
  SK1 is Subfield of SK2 iff for x st x in SK1 holds x in SK2
   proof
     thus SK1 is Subfield of SK2 implies
     for x st x in SK1 holds x in SK2 by Th3;
     assume A1: for x st x in SK1 holds x in SK2;
     the carrier of SK1 c= the carrier of SK2
     proof
       let x be object;
       assume x in the carrier of SK1;
       then reconsider x as Element of SK1;
       x in SK1;
       then x in SK2 by A1;
       hence thesis;
     end;
     hence thesis by Th6;
   end;
