 reserve n,k for Nat;
 reserve L for comRing;
 reserve R for domRing;
 reserve x0 for positive Real;

theorem
  for f be Element of the carrier of Polynom-Ring F_Rat, i be Nat
    holds
    denominator (f.i) in denomi-set(f) & ex z be Integer st
    z*(denominator (f.i)) = Product(denomi-seq(f))
    proof
      let f be Element of the carrier of Polynom-Ring F_Rat;
      let i be Nat;
      dom f = NAT by FUNCT_2:def 1; then
A2:   f.i in rng f by FUNCT_1:3,ORDINAL1:def 12;
      f.i in RAT by RAT_1:def 2; then
      f.i in dom TRANQN by FUNCT_2:def 1; then
A4:   TRANQN.(f.i) in TRANQN.:(rng f) by A2,FUNCT_1:def 6; then
      denominator(f.i) in denomi-set(f) by DIOPHAN2:def 4; then
      denominator(f.i) in rng (denomi-seq(f)) by FUNCT_2:def 3; then
      consider j be object such that
A6:   j in dom(denomi-seq(f)) & (denomi-seq(f)).j = denominator(f.i)
      by FUNCT_1:def 3;
      ex z being Integer st z * (denomi-seq(f)).j = Product(denomi-seq(f))
      by A6,INT_6:10;
      hence thesis by DIOPHAN2:def 4,A4,A6;
    end;
