 reserve o for object;
 reserve F for non almost_trivial Field;
 reserve x,a for Element of F;
reserve n for non zero Nat;

theorem
  for F being non almost_trivial Field
  ex K being non polynomial_disjoint Field st K,F are_isomorphic
  proof
    let F be non almost_trivial Field;
    set x = the non trivial Element of F;
    reconsider o = <%0.F,1.F%> as object;
    per cases;
      suppose
A1:     not o in [#]F; then
        reconsider S = ExField(x,o) as Field
         by Th7,Th8,Th9,Th10,Th11,Th12;
        [#]S /\ [#]Polynom-Ring S <> {} by Th13; then
        reconsider S as non polynomial_disjoint Field by Def3;
        take S;
        isoR(x,o) is additive multiplicative unity-preserving by A1,Th15;
        hence thesis by MOD_4:def 12,QUOFIELD:def 23;
      end;
      suppose ex a being Element of F st a = <%0.F,1.F%>; then
        consider a being Element of F such that
A2:     a = <%0.F,1.F%>;
        a in [#]Polynom-Ring(F) by A2,POLYNOM3:def 10; then
        a in [#]F /\ [#]Polynom-Ring(F)
        by XBOOLE_0:def 4; then
        reconsider S = F as non polynomial_disjoint Field by Def3;
        take S;
        thus thesis;
      end;
    end;
