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

theorem Lm47:
  for x,z1,z2 be Element of F_Complex st
  z1 in FQ(x) & z2 in FQ(x) holds z1 * z2 in FQ(x)
  proof
  let x,z1,z2 be Element of F_Complex;
  assume that
A1: z1 in FQ(x) and
A2: z2 in FQ(x);
    consider f1 be object such that
A3: f1 in dom hom_Ext_eval(x,F_Rat) and
A4: z1 = hom_Ext_eval(x,F_Rat).f1  by A1,FUNCT_1:def 3;
   consider f2 be object such that
A5: f2 in dom hom_Ext_eval(x,F_Rat) and
A6: z2 = hom_Ext_eval(x,F_Rat).f2 by A2,FUNCT_1:def 3;
A7: dom hom_Ext_eval(x,F_Rat) =
    the carrier of Polynom-Ring F_Rat by FUNCT_2:def 1;
reconsider g1 = f1, g2 = f2 as Polynomial of F_Rat by A3,A5,POLYNOM3:def 10;
A8:  z1 * z2 = Ext_eval(g1,x) * hom_Ext_eval(x,F_Rat).f2 by Def9,A6,A4
    .= Ext_eval(g1,x) * Ext_eval(g2,x) by Def9
    .= Ext_eval(g1*'g2,x) by Th3,Th24
    .= hom_Ext_eval(x,F_Rat).(g1*'g2) by Def9;
    set g = g1*'g2;
    g in dom hom_Ext_eval(x,F_Rat) by A7,POLYNOM3:def 10;
    hence thesis by A8,FUNCT_1:def 3;
  end;
