theorem
  for K be strict finite Field,
  SK1 be strict Subfield of K st
  card K = card SK1 holds SK1 = K
  proof
    let K be strict finite Field, SK1 be strict Subfield of K;
    assume
A1: card K = card SK1;
A2: the carrier of SK1 = the carrier of K
    proof
      assume A3: the carrier of SK1 <> the carrier of K;
 A4: the carrier of SK1 c= the carrier of K by Def1; then
      the carrier of SK1 c< the carrier of K by A3,XBOOLE_0:def 8;
      hence contradiction by A1,A4,CARD_2:48;
    end;
    K is Subfield of K by Th1;
    hence thesis by A2,Th8;
  end;
