
theorem Th56:
  for n being non zero Element of NAT, j, k, q being Integer, qc
  being Element of F_Complex st qc = q & j = eval(cyclotomic_poly(n),qc) & k =
  eval(unital_poly(F_Complex, n),qc) holds j divides k
proof
  let n be non zero Element of NAT, j,k,q being Integer, qc being Element of
  F_Complex such that
A1: qc = q and
A2: j = eval(cyclotomic_poly(n),qc) and
A3: k = eval(unital_poly(F_Complex, n),qc);
  set jfc = eval(cyclotomic_poly(n),qc);
  per cases by NAT_1:53;
  suppose
    n = 1;
    hence thesis by A2,A3,Th48;
  end;
  suppose
A4: n > 1;
    set eup1fc = eval(unital_poly(F_Complex,1),qc);
    reconsider eup1 = eup1fc as Integer by A1,Th47;
    consider f being FinSequence of (the carrier of Polynom-Ring F_Complex), p
    being Polynomial of F_Complex such that
A5: p = Product(f) and
A6: dom f = Seg n & for i being non zero Element of NAT st i in Seg n
holds ( not i divides n or i divides 1 or i = n implies f.i = <%1_F_Complex%>)
    & (i divides n & not i divides 1 & i <> n implies f.i = cyclotomic_poly(i))
    and
A7: unital_poly(F_Complex,n) = unital_poly(F_Complex,1)*'(
    cyclotomic_poly n)*'p by A4,Th54,NAT_D:6;
    set epfc = eval(p,qc);
    now
      let i be non zero Element of NAT;
      assume
A8:   i in dom f;
      per cases;
      suppose
        not i divides n or i divides 1 or i = n;
        hence f.i = <%1_F_Complex%> or f.i = cyclotomic_poly(i) by A6,A8;
      end;
      suppose
        i divides n & not i divides 1 & i <> n;
        hence f.i = <%1_F_Complex%> or f.i = cyclotomic_poly(i) by A6,A8;
      end;
    end;
    then reconsider ep = epfc as Integer by A1,A5,Th55;
    k = eval((unital_poly(F_Complex,1)*'cyclotomic_poly(n)),qc) * epfc by A3,A7
,POLYNOM4:24;
    then k = eup1fc * jfc * epfc by POLYNOM4:24;
    then k = eup1 * ep * j by A2;
    hence thesis by INT_1:def 3;
  end;
end;
