
theorem Th72:
  for p be Polynomial of F_Complex st len p > 2 & |.p.(len p-'1).|
=1 for F be FinSequence of REAL st len F = len p & for n be Element of NAT st n
in dom F holds F.n = |.p.(n-'1).| for z be Element of F_Complex st |.z.| > Sum
  F holds |.eval(p,z).| > |.p.0 .|+1
proof
  let p be Polynomial of F_Complex;
  assume that
A1: len p > 2 and
A2: |.p.(len p-'1).|=1;
  let F be FinSequence of REAL;
  assume that
A3: len F = len p and
A4: for n be Element of NAT st n in dom F holds F.n = |.p.(n-'1).|;
  set lF1 = len F-'1;
A5: lF1+1 = len F by A1,A3,XREAL_1:235,XXREAL_0:2;
  then
A6: F = F|(lF1+1) by FINSEQ_1:58
    .= F|lF1 ^ <*F/.(lF1+1)*> by A5,FINSEQ_5:82;
A7: len p > 1 by A1,XXREAL_0:2;
  then
A8: 1 in dom F by A3,FINSEQ_3:25;
A9: now
    let n be Element of NAT;
A10: dom(F|lF1) c= dom F by FINSEQ_5:18;
    assume
A11: n in dom (F|lF1);
    then (F|lF1).n = (F|lF1)/.n by PARTFUN1:def 6
      .= F/.n by A11,FINSEQ_4:70
      .= F.n by A11,A10,PARTFUN1:def 6
      .= |.p.(n-'1).| by A4,A11,A10;
    hence (F|lF1).n >= 0 by COMPLEX1:46;
  end;
A12: len (F|lF1) = lF1 by A5,FINSEQ_1:59,NAT_1:11;
  |.p.0 .| >= 0 by COMPLEX1:46;
  then
A13: |.p.0 .|+1 >= 0+1 by XREAL_1:6;
  let z be Element of F_Complex;
  consider G be FinSequence of the carrier of F_Complex such that
A14: eval(p,z) = Sum G and
A15: len G = len p and
A16: for n be Element of NAT st n in dom G holds G.n = p.(n-'1)*(power
  F_Complex).(z,n-'1) by POLYNOM4:def 2;
  set lF2 = len F-'2;
  assume
A17: |.z.| > Sum F;
A18: len F in dom F by A7,A3,FINSEQ_3:25;
  then F/.(lF1+1) = F.(lF1+1) by A5,PARTFUN1:def 6
    .= 1 by A2,A3,A4,A5,A18;
  then
A19: Sum F = Sum(F|lF1) + 1 by A6,RVSUM_1:74;
A20: len F >= 1+1+0 by A1,A3;
  then lF1 >= 1 by A5,XREAL_1:6;
  then
A21: 1 in dom (F|lF1) by A12,FINSEQ_3:25;
  then (F|lF1).1 = (F|lF1)/.1 by PARTFUN1:def 6
    .= F/.1 by A21,FINSEQ_4:70
    .= F.1 by A8,PARTFUN1:def 6
    .= |.p.(1-'1).| by A4,A8
    .= |.p.0 .| by XREAL_1:232;
  then Sum(F|lF1) >= |.p.0 .| by A21,A9,Th4;
  then
A22: Sum F >= |.p.0 .|+1 by A19,XREAL_1:6;
  then
A23: z <> 0.F_Complex by A17,A13,COMPLFLD:59;
  G = G|(lF1+1) by A3,A15,A5,FINSEQ_1:58
    .= G|lF1 ^ <*G/.(lF1+1)*> by A3,A15,A5,FINSEQ_5:82;
  then
A24: Sum G = Sum(G|lF1) + G/.(lF1+1) by FVSUM_1:71;
A25: dom F = dom G by A3,A15,FINSEQ_3:29;
  then G/.(lF1+1) = G.(lF1+1) by A5,A18,PARTFUN1:def 6
    .= p.lF1*(power F_Complex).(z,lF1) by A16,A5,A18,A25;
  then |.G/.(lF1+1).| = 1*|.(power F_Complex).(z,lF1).| by A2,A3,COMPLFLD:71;
  then
A26: |.eval(p,z).| >= |.(power F_Complex).(z,lF1).|- |.Sum(G|lF1).| by A14,A24,
COMPLFLD:64;
A27: len F-1 >= 0 by A7,A3;
A28: len (F|lF1) = lF1 by A5,FINSEQ_1:59,NAT_1:11
    .= len (G|lF1) by A3,A15,A5,FINSEQ_1:59,NAT_1:11;
  then
A29: F|lF1|(len (F|lF1)) = F|lF1 & G|lF1|(len (F|lF1)) = G|lF1 by FINSEQ_1:58;
  defpred P[Nat] means
|.Sum(G|lF1|$1).| <= (Sum (F|lF1|$1))*|.(
  power F_Complex).(z,lF2).|;
  len F-2 >=0 by A20,XREAL_1:19;
  then
A30: lF2+1 = len F-2+1 by XREAL_0:def 2
    .= lF1 by A27,XREAL_0:def 2;
  then (power F_Complex).(z,lF1) = (power F_Complex).(z,lF2)*z by GROUP_1:def 7
;
  then
A31: |.(power F_Complex).(z,lF1).|- |.(power F_Complex).(z,lF2).|*Sum (F|lF1
) = |.(power F_Complex).(z,lF2).|*|.z.|- |.(power F_Complex).(z,lF2).|*Sum (F|
  lF1) by COMPLFLD:71
    .= |.(power F_Complex).(z,lF2).|*(|.z.|-Sum (F|lF1));
A32: |.z.| > |.p.0 .|+1 by A17,A22,XXREAL_0:2;
  then
A33: |.z.| > 1 by A13,XXREAL_0:2;
A34: dom (F|lF1) = dom (G|lF1) by A28,FINSEQ_3:29;
    reconsider GG = G|lF1 as FinSequence of the carrier of F_Complex;
    reconsider FF = F|lF1 as FinSequence of REAL;
A35: for n be Nat st P[n] holds P[n+1]
  proof
    let n be Nat;
     reconsider nn=n as Element of NAT by ORDINAL1:def 12;
    assume
A36: |.Sum(G|lF1|n).| <= (Sum (F|lF1|n))*|.(power F_Complex).(z,lF2).|;
    then
A37: |.Sum(GG|n).|+|.GG/.(n+1).| <= (Sum (FF|n))*|.(power
    F_Complex).(z,lF2).|+|.GG/.(n+1).| by XREAL_1:6;
    per cases;
    suppose
A38:  n+1 <= len (G|lF1);
      then n+1 <= lF2+1 by A3,A15,A5,A30,FINSEQ_1:59,NAT_1:11;
      then n <= lF2 by XREAL_1:6;
      then |.z.| to_power n <= |.z.| to_power lF2 by A33,PRE_FF:8;
      then |.z.| to_power n <= |.power(z,lF2).| by A23,Th7;
      then
A39:  |.p.nn.| >= 0 & |.power(z,n).|<=|.power(z,lF2).|
           by A23,Th7,COMPLEX1:46;
      GG|(n+1) = GG|n ^ <*GG/.(n+1)*> by A38,FINSEQ_5:82;
      then Sum(GG|(n+1)) = Sum(GG|n) + GG/.(n+1) by FVSUM_1:71;
      then |.Sum(GG|(n+1)).| <= |.Sum(GG|n).|+|.GG/.(n+1).| by COMPLFLD:62;
      then
A40:  |.Sum(GG|(n+1)).| <= (Sum (FF|n))*|.(power F_Complex).(z,lF2)
      .|+|.GG/.(n+1).| by A37,XXREAL_0:2;
A41:  dom GG c= dom G by FINSEQ_5:18;
      n+1 >= 1 by NAT_1:11;
      then
A42:  n+1 in dom GG by A38,FINSEQ_3:25;
      then
A43:  (F|lF1)/.(n+1) = F/.(n+1) by A34,FINSEQ_4:70
        .= F.(n+1) by A25,A42,A41,PARTFUN1:def 6
        .= |.p.(n+1-'1).| by A4,A25,A42,A41
        .= |.p.nn.| by NAT_D:34;
      GG/.(n+1) = G/.(n+1) by A42,FINSEQ_4:70
        .= G.(n+1) by A42,A41,PARTFUN1:def 6
        .= p.(n+1-'1)*(power F_Complex).(z,n+1-'1) by A16,A42,A41
        .= p.nn*(power F_Complex).(z,n+1-'1) by NAT_D:34
        .= p.nn*(power F_Complex).(z,n) by NAT_D:34;
      then |.GG/.(n+1).| = (F|lF1)/.(n+1)*|.(power F_Complex).(z,n).|
      by A43,COMPLFLD:71;
      then |.GG/.(n+1).| <= (F|lF1)/.(n+1)*|.(power F_Complex).(z,lF2).|
      by A43,A39,XREAL_1:64;
      then
A44:  (Sum(F|lF1|n))*|.(power F_Complex).(z,lF2).|+|.GG/.(n+1).| <=
      (Sum((F|lF1)|n))*|.(power F_Complex).(z,lF2).|+ ((F|lF1)/.(n+1))*|.(power
      F_Complex).(z,lF2).| by XREAL_1:6;
      (F|lF1)|(n+1) = (F|lF1)|n ^ <*(F|lF1)/.(n+1)*> by A28,A38,FINSEQ_5:82;
      then Sum(F|lF1|(n+1)) = Sum ((F|lF1)|n) + (F|lF1)/.(n+1) by RVSUM_1:74;
      hence thesis by A40,A44,XXREAL_0:2;
    end;
    suppose
A45:  n+1 > len GG;
      then n >= len GG by NAT_1:13;
      then
A46:  GG|n = GG & (F|lF1)|n = (F|lF1) by A28,FINSEQ_1:58;
      GG|(n+1) = GG by A45,FINSEQ_1:58;
      hence thesis by A28,A36,A45,A46,FINSEQ_1:58;
    end;
  end;
  G|lF1|0 = <*>the carrier of F_Complex;
  then
A47: P[0] by COMPLFLD:57,RLVECT_1:43,RVSUM_1:72;
  for n be Nat holds P[n] from NAT_1:sch 2(A47,A35);
  then |.Sum(G|lF1).| <= (Sum (F|lF1))*|.(power F_Complex).(z,lF2).| by A29;
  then
  |.(power F_Complex).(z,lF1).|- |.Sum(G|lF1).| >= |.(power F_Complex).(z
  ,lF1).|-(Sum (F|lF1))*|.(power F_Complex).(z,lF2).| by XREAL_1:13;
  then
A48: |.eval(p,z).| >= |.(power F_Complex).(z,lF1).|- |.(power F_Complex).(z,
  lF2).|*Sum (F|lF1) by A26,XXREAL_0:2;
  len F >= 2+1 by A1,A3,NAT_1:13;
  then len F-2 >= 1 by XREAL_1:19;
  then lF2 >= 1 by XREAL_0:def 2;
  then |.z.| to_power lF2 >= |.z.| to_power 1 by A33,PRE_FF:8;
  then |.power(z,lF2).| >= |.z.| to_power 1 by A23,Th7;
  then
  |.(power F_Complex).(z,lF2).| >= |.power(z,1).| by A23,Th7;
  then
A49: |.(power F_Complex).(z,lF2).| >= |.z.| by GROUP_1:50;
  |.(power F_Complex).(z,lF2).| >= 0 & |.z.|-Sum (F|lF1) > 1 by A17,A19,
COMPLEX1:46,XREAL_1:20;
  then
  |.(power F_Complex).(z,lF2).|*(|.z.|-Sum (F|lF1)) >= |.(power F_Complex
  ).(z,lF2).|*1 by XREAL_1:64;
  then |.eval(p,z).| >= |.(power F_Complex).(z,lF2).| by A48,A31,XXREAL_0:2;
  then |.eval(p,z).| >= |.z.| by A49,XXREAL_0:2;
  hence thesis by A32,XXREAL_0:2;
end;
