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
  for K1, K2, K3 be Field st
  K1 is Subfield of K2 & K2 is Subfield of K3 holds K1 is Subfield of K3
  proof
    let K1, K2, K3 be Field;
    assume A1: K1 is Subfield of K2 & K2 is Subfield of K3;
    set C1 = the carrier of K1;
    set C2 = the carrier of K2;
    set C = the carrier of K3;
    set ADD = the addF of K3;
    set MULT = the multF of K3;
A2: C1 c= C2 by A1,Def1;
    then A3: [:C1,C1:] c= [:C2,C2:] by ZFMISC_1:96;
    C2 c= C by A1,Def1; then
A4: C1 c= C by A2;
A5: the addF of K2 = ADD || C2 by A1,Def1;
A6: the addF of K1 = (the addF of K2) || C1 by A1,Def1
    .= ADD || C1 by A3,A5,FUNCT_1:51;
A7: the multF of K2 = MULT || C2 by A1,Def1;
A8: the multF of K1 = (the multF of K2) || C1 by A1,Def1
    .= MULT || C1 by A3,A7,FUNCT_1:51;
    1.K1 = 1.K2 & 0.K1 = 0.K2 by A1,Def1;
    then 1.K1 = 1.K3 & 0.K1 = 0.K3 by A1,Def1;
    hence thesis by A4,A6,A8,Def1;
  end;
