
theorem
  for n being non zero Element of NAT st 1 < n for q being Element of
NAT st 1 < q for qc being Element of F_Complex st qc = q for i being Integer st
  i = eval(cyclotomic_poly(n),qc) holds |.i.| > q - 1
proof
  set MGFC = MultGroup F_Complex;
  set cMGFC = the carrier of MultGroup F_Complex;
  let n be non zero Element of NAT such that
A1: 1 < n;
  consider S being non empty finite Subset of F_Complex such that
A2: S = {y where y is Element of cMGFC : ord y = n} and
A3: cyclotomic_poly(n) = poly_with_roots((S,1)-bag) by Def5;
  rng canFS(S) = S by FUNCT_2:def 3;
  then reconsider fs = canFS(S) as FinSequence of F_Complex by FINSEQ_1:def 4;
  let q be Element of NAT such that
A4: 1 < q;
  let qc be Element of F_Complex such that
A5: qc = q;
  deffunc F(set) = |.qc - fs/.$1.|;
  consider p1 being FinSequence such that
A6: len p1 = len fs & for i being Nat st i in dom p1 holds p1.i = F(i)
  from FINSEQ_1:sch 2;
A7: for i being Element of NAT, c being Element of F_Complex st i in dom p1
  & c = (canFS(S)).i holds p1.i = |.qc - c.|
  proof
    let i be Element of NAT, c being Element of F_Complex such that
A8: i in dom p1 and
A9: c = (canFS(S)).i;
    i in dom fs by A6,A8,FINSEQ_3:29;
    then fs/.i = (canFS(S)).i by PARTFUN1:def 6;
    hence thesis by A6,A8,A9;
  end;
  for x being object st x in rng p1 holds x in REAL
  proof
    let x be object;
    assume x in rng p1;
    then consider i being Nat such that
A10: i in dom p1 and
A11: p1.i = x by FINSEQ_2:10;
    i in dom fs by A6,A10,FINSEQ_3:29;
    then fs/.i = (canFS(S)).i by PARTFUN1:def 6;
    then x = F(i) by A7,A10,A11;
    hence thesis by XREAL_0:def 1;
  end;
  then rng p1 c= REAL;
  then reconsider ps=p1 as non empty FinSequence of REAL by A6,FINSEQ_1:def 4;
  len fs = card S by FINSEQ_1:93;
  then
A12: |.eval(cyclotomic_poly(n),qc).| = Product(ps) by A3,A6,A7,Th3;
A13: rng fs = S by FUNCT_2:def 3;
A14: for i being Element of NAT st i in dom ps holds ps.i > q - 1
  proof
    let i be Element of NAT such that
A15: i in dom ps;
    i in dom fs by A6,A15,FINSEQ_3:29;
    then fs/.i = (canFS(S)).i by PARTFUN1:def 6;
    then
A16: ps.i = |.[**q,0**] - fs/.i.| by A5,A7,A15;
A17: i in dom fs by A6,A15,FINSEQ_3:29;
    then fs.i in rng fs by FUNCT_1:3;
    then fs/.i in rng fs by A17,PARTFUN1:def 6;
    then
A18: ex y being Element of MGFC st fs/.i = y & ord y = n by A2,A13;
A19: now
      assume
A20:  fs/.i = [**1,0**];
      1_MultGroup F_Complex = [**1, 0**] by Th17,COMPLFLD:8;
      hence contradiction by A1,A18,A20,GROUP_1:42;
    end;
    fs/.i in n-roots_of_1 by A18,Th34;
    then |.fs/.i.| = 1 by Th23;
    hence thesis by A4,A16,A19,Th6;
  end;
  reconsider qi=q as Integer;
  1+1 <= qi by A4,INT_1:7;
  then
A21: 1+1+-1 <= qi +-1 by XREAL_1:7;
  let i be Integer;
   reconsider x = q-1 as Real;
  assume i = eval(cyclotomic_poly(n),qc);
  then |.i.| > x by A21,A12,A14,Th7;
  hence |.i.| > q - 1;
end;
