
theorem Th57:
  for n,ni being non zero Element of NAT, q being Integer st ni <
  n & ni divides n for qc being Element of F_Complex st qc = q for j,k,l being
Integer st j = eval(cyclotomic_poly(n),qc) & k = eval(unital_poly(F_Complex, n)
  ,qc) & l = eval(unital_poly(F_Complex, ni),qc) holds j divides (k div l)
proof
  let n,ni be non zero Element of NAT, q being Integer such that
A1: ni < n & ni divides n;
  set ttt = (unital_poly(F_Complex,ni)*'cyclotomic_poly(n));
  consider f being FinSequence of (the carrier of Polynom-Ring F_Complex), p
  being Polynomial of F_Complex such that
A2: p = Product(f) and
A3: 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 ni or i = n implies f.i = <%1_F_Complex%>)
  & (i divides n & not i divides ni & i <> n implies f.i = cyclotomic_poly(i))
  and
A4: unital_poly(F_Complex,n) = ttt *' p by A1,Th54;
A5: now
    let i being non zero Element of NAT such that
A6: i in dom f;
    per cases;
    suppose
      not i divides n or i divides ni or i = n;
      hence f.i = <%1_F_Complex%> or f.i = cyclotomic_poly(i) by A3,A6;
    end;
    suppose
      i divides n & not i divides ni & i <> n;
      hence f.i = <%1_F_Complex%> or f.i = cyclotomic_poly(i) by A3,A6;
    end;
  end;
  let qc be Element of F_Complex;
  assume qc = q;
  then eval(p,qc) is integer by A2,A5,Th55;
  then consider m being Integer such that
A7: m = eval(p,qc);
  let j,k,l be Integer such that
A8: j = eval(cyclotomic_poly(n),qc) & k = eval(unital_poly(F_Complex,n),
  qc) & l = eval(unital_poly(F_Complex,ni),qc);
  reconsider jc = j, lc = l, mc = m as Element of COMPLEX by XCMPLX_0:def 2;
  reconsider jcf = jc, lcf = lc, mcf = mc as Element of F_Complex by
COMPLFLD:def 1;
  eval(unital_poly(F_Complex,n),qc) = eval(ttt,qc) * eval(p,qc) by A4,
POLYNOM4:24;
  then
A9: k = lcf*jcf*mcf by A8,A7,POLYNOM4:24
    .= l*j*m;
  now
    per cases;
    suppose
A10:  l <> 0;
      k = l*(j*m) by A9;
      then l divides k by INT_1:def 3;
      then k/l is integer by A10,WSIERP_1:17;
      then [\ k/l /] = k/l;
      then k div l = (j*m)*l/l by A9,INT_1:def 9;
      then k div l = j*m by A10,XCMPLX_1:89;
      hence thesis by INT_1:def 3;
    end;
    suppose
      l = 0;
      then k div l = 0;
      hence thesis by INT_2:12;
    end;
  end;
  hence thesis;
end;
