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

theorem Lm48:
  for x be Element of F_Complex, a be Element of F_Rat holds
  a in FQ(x)
  proof
  let x be Element of F_Complex;
  let a be Element of F_Rat;
  reconsider f = <% a %> as Polynomial of F_Rat;
A2: dom hom_Ext_eval(x,F_Rat) =
      the carrier of Polynom-Ring F_Rat by FUNCT_2:def 1;
A3: Ext_eval(f,x) = In(a,F_Complex) by Th3,Th25;
    reconsider f as Element of Polynom-Ring F_Rat by POLYNOM3:def 10;
    In(a,F_Complex) = hom_Ext_eval(x,F_Rat).f by A3,Def9;
    hence thesis by A2,FUNCT_1:def 3;
  end;
