reserve i,j for Nat;
reserve A,B for Ring;
reserve K, L for Field;

theorem Th84:
  for x,a be Element of F_Complex st x is algebraic & a <> 0.F_Complex &
  a in the carrier of FQ_Ring(x)
  ex f,g be Element of Polynom-Ring F_Rat
  st {f}-Ideal = Ann_Poly(x,F_Rat) & not(g in Ann_Poly(x,F_Rat)) &
  a = hom_Ext_eval(x,F_Rat).g &
  {f}-Ideal,{g}-Ideal are_co-prime
  proof
    let x,a be Element of F_Complex;
    assume that
A1:  x is algebraic and
A2:  a <> 0.F_Complex and
A3:  a in the carrier of FQ_Ring(x);
     consider f be Element of Polynom-Ring F_Rat such that
A4:  {f}-Ideal = Ann_Poly(x,F_Rat) by Th34,Th3;
     consider g be Element of Polynom-Ring F_Rat such that
A5:  not(g in Ann_Poly(x,F_Rat)) and
A6:  a = hom_Ext_eval(x,F_Rat).g by Th83,A2,A3;
A7:  {f}-Ideal is prime by A4,A1,Th3,Th39;
A8:  f <> 0.Polynom-Ring F_Rat by A1,A4,Th35,IDEAL_1:47;
     {f}-Ideal,{g}-Ideal are_co-prime by A4,A5,A7,A8,Th81;
     hence thesis by A4,A5,A6;
  end;
