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 Th4:
  for K1,K2 be strict Field st
  K1 is Subfield of K2 & K2 is Subfield of K1 holds K1 = K2
  proof
    let K1,K2 be strict Field;
    assume A1: K1 is Subfield of K2
    & K2 is Subfield of K1;
A2: the carrier of K1 c= the carrier of K2 &
    the carrier of K2 c= the carrier of K1 by A1,Def1; then
A3: the carrier of K1 = the carrier of K2 by XBOOLE_0:def 10;
A4: the addF of K2 = (the addF of K2) || the carrier of K1 by A3,RELSET_1:19
    .= the addF of K1 by A1,Def1;
A5: the multF of K2 = (the multF of K2) || the carrier of K1
    by A3,RELSET_1:19
    .= the multF of K1 by A1,Def1;
    1.K1 = 1.K2 & 0.K1 = 0.K2 by A1,Def1;
    hence thesis by A4,A5,A2,XBOOLE_0:def 10;
  end;
