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

theorem
  for f be Element of Polynom-Ring F_Rat, n be non zero Nat
    st f is irreducible holds n*f is irreducible
   proof
     let f be Element of Polynom-Ring F_Rat, n be non zero Nat;
     assume
A1:  f is irreducible;
reconsider f0 = f as Element of the carrier of Polynom-Ring F_Rat;
reconsider n0 = In(n,F_Rat) as Element of F_Rat;
reconsider n1 = n as Element of NAT by ORDINAL1:def 12;
A2:   n0 <> 0.F_Rat by GAUSSINT:def 14;
A3:   {n1*f}-Ideal c= {f}-Ideal by Lm2, RING_2:19;
A4:   n0*f0 = n*f by Lm1;
reconsider nf = n*f as Element of the carrier of Polynom-Ring F_Rat;
      (n0")*nf = ((n0")*n0)*f0 by A4,RING_4:11
      .= (1.F_Rat)*f0 by A2,VECTSP_1:def 10 .= f; then
A5:   f = @((n0"|F_Rat)*'~nf) by FIELD_8:2
      .= (@(n0"|F_Rat)) * (@(~nf)) by POLYNOM3:def 10
      .= (@(n0"|F_Rat)) * nf; then
      {f}-Ideal = {nf}-Ideal by A3,RING_2:19,GCD_1:def 1; then
A7:   {nf}-Ideal is maximal by A1,RING_2:26;
      reconsider nf0 = n*f as non zero Element
        of Polynom-Ring F_Rat by A5,A1;
      nf0 is irreducible by A7,RING_2:26;
      hence thesis;
    end;
