
theorem Th73:
  for p be Polynomial of F_Complex st len p > 2 ex z0 be Element
of F_Complex st for z be Element of F_Complex holds |.eval(p,z).| >= |.eval(p,
  z0).|
proof
  defpred P[set] means not contradiction;
  let p be Polynomial of F_Complex;
  set np = NormPolynomial(p);
  deffunc F(Element of F_Complex) = In(|.eval(np,$1).|,REAL);
  reconsider D = { F(z) where z is Element of F_Complex : P[z] } as Subset of
  REAL from DOMAIN_1:sch 8;
  set q = lower_bound D;
A1: D is bounded_below
  proof
    take 0;
    let b be ExtReal;
    assume b in D;
    then consider z be Element of F_Complex such that
A2:    b = In(|.eval(np,z).|,REAL);
     b = |.eval(np,z).| by A2;
    hence thesis by COMPLEX1:46;
  end;
  defpred P[Nat,object] means ex g1 be Element of F_Complex st g1 = $2
  & |.eval(np,g1).| < q+1/($1+1);
  In(|.eval(np,0.F_Complex).|,REAL) = |.eval(np,0.F_Complex).|;
   then
A3: |.eval(np,0.F_Complex).| in D;
A4: for n be Nat ex g be Complex st P[n,g]
  proof
    let n be Nat;
    consider r be Real such that
A5: r in D and
A6: r < q+1/(n+1) by A3,A1,SEQ_4:def 2;
    consider g1 be Element of F_Complex such that
A7: r = In(|.eval(np,g1).|,REAL) by A5;
    reconsider g=g1 as Element of COMPLEX by COMPLFLD:def 1;
    take g,g1;
    thus g1 = g;
    thus thesis by A6,A7;
  end;
  consider G be Complex_Sequence such that
A8: for n be Nat holds P[n,G.n] from CFCONT_1:sch 1(A4);
  deffunc G(Nat) = In(|.np.($1-'1).|,REAL);
  consider F be FinSequence of REAL such that
A9: len F = len np and
A10: for n be Nat st n in dom F holds F.n = G(n) from FINSEQ_2:sch 1;
  assume
A11: len p > 2;
  then
A12: len p = len p-'1+1 by XREAL_1:235,XXREAL_0:2;
  then p.(len p-'1) <> 0.F_Complex by ALGSEQ_1:10;
  then
A13: |.p.(len p-'1).| > 0 by COMPLFLD:59;
  G is bounded
  proof
    take r = Sum F + 1;
    let n be Nat;
    consider Gn be Element of F_Complex such that
A14: Gn = G.n and
A15: |.eval(np,Gn).| < q+1/(n+1) by A8;
    n+1>=0+1 by XREAL_1:6;
    then
A16: 1/(n+1) <= 1 by XREAL_1:211;
A17: len np = len p by A11,Th57;
    then
A18: np.(len np-'1) = 1_F_Complex by A11,Th56;
    |.G.n.| <= Sum F
    proof
A19:    eval(np,0.F_Complex) = np.0 by Th31;
      In(|.np.0 .|,REAL) = |.np.0 .|;
      then |.np.0 .| in D by A19;
      then |.np.0 .| >= q by A1,SEQ_4:def 2;
      then
A20:  |.np.0 .|+1 >= q+1/(n+1) by A16,XREAL_1:7;

A21: for n be Element of NAT st n in dom F holds F.n = |.np.(n-'1).|
     proof let n be Element of NAT;
      assume n in dom F;
       then F.n = G(n) by A10;
      hence F.n = |.np.(n-'1).|;
    end;

      assume |.G.n.| > Sum F;
      then |.eval(np,Gn).| > |.np.0 .|+1 by A11,A9,A14,A17,A18,A21,
COMPLFLD:60,Th72;
      hence contradiction by A15,A20,XXREAL_0:2;
    end;
    then |.G.n.|+0 < r by XREAL_1:8;
    hence thesis;
  end;
  then consider G1 be Complex_Sequence such that
A22: G1 is subsequence of G and
A23: G1 is convergent by COMSEQ_3:50;
  defpred P[Nat,object] means ex G1n be Element of F_Complex st G1n =
  G1.$1 & $2 = eval(np,G1n);
  lim G1 in COMPLEX by XCMPLX_0:def 2;
  then reconsider z0=lim G1 as Element of F_Complex by COMPLFLD:def 1;
A24: for n be Nat ex g be Complex st P[n,g]
  proof
    let n be Nat;
    reconsider nn=n as Element of NAT by ORDINAL1:def 12;
    reconsider G1n = G1.nn as Element of F_Complex by COMPLFLD:def 1;
    reconsider g = eval(np,G1n) as Element of COMPLEX by COMPLFLD:def 1;
    take g,G1n;
    thus G1n = G1.n;
    thus thesis;
  end;
  consider H be Complex_Sequence such that
A25: for n be Nat holds P[n,H.n] from CFCONT_1:sch 1(A24);
  reconsider enp0 = eval(np,z0) as Element of COMPLEX by COMPLFLD:def 1;
  consider g be Complex such that
A26: for p be Real st 0 < p
  ex n be Nat st for m be Nat st n <= m holds |.G1.m-g.| < p
    by A23;
A27:  g in COMPLEX by XCMPLX_0:def 2;
  then reconsider g1 = g as Element of F_Complex by COMPLFLD:def 1;
  reconsider eg = eval(np,g1) as Element of COMPLEX by COMPLFLD:def 1;
  now
    let p be Real;
    consider fPF be Function of COMPLEX,COMPLEX such that
A28: fPF = Polynomial-Function(F_Complex,np) and
A29: fPF is_continuous_on COMPLEX by Th71;
    assume 0 < p;
    then consider p1 be Real such that
A30: 0 < p1 and
A31: for x1 be Complex st x1 in COMPLEX & |.x1-g.| < p1
    holds |.fPF/.x1 - fPF/.g.| < p by A29,CFCONT_1:39,A27;
    consider n be Nat such that
A32: for m be Nat st n <= m holds |.G1.m-g.| < p1 by A26,A30;
    take n;
    let m be Nat;
     reconsider mm= m as Element of NAT by ORDINAL1:def 12;
    assume n <= m;
    then
A33: |.G1.m-g.| < p1 by A32;
    ex G1m be Element of F_Complex st G1m = G1.m & H.m = eval(np,G1m) by A25;
    then
A34: H.m = fPF/.(G1.mm) by A28,Def13;
    eg = fPF/.g by A28,Def13;
    hence |.H.m-eg.| < p by A31,A33,A34;
  end;
  then
A35: H is convergent;
  consider PF be Function of COMPLEX,COMPLEX such that
A36: PF = Polynomial-Function(F_Complex,np) and
A37: PF is_continuous_on COMPLEX by Th71;
  now
    let a be Real;
A38:  lim G1 in COMPLEX by XCMPLX_0:def 2;
    assume 0 < a;
    then consider s be Real such that
A39: 0 < s and
A40: for x1 be Complex st x1 in COMPLEX & |.x1-lim G1.| < s
    holds |.PF/.x1 - PF/.lim G1.| < a by A37,CFCONT_1:39,A38;
    consider n be Nat such that
A41: for m be Nat st n <= m holds |.G1.m-lim G1.| < s by A23,A39,
COMSEQ_2:def 6;
    take n;
    let m be Nat;
     reconsider mm=m as Element of  NAT by ORDINAL1:def 12;
    assume n <= m;
    then
A42: |.G1.m-lim G1.| < s by A41;
    ex G1m be Element of F_Complex st G1m = G1.m & H.m = eval(np,G1m) by A25;
    then
A43: PF/.(G1.mm) = H.m by A36,Def13;
    PF/.lim G1 = eval(np,z0) by A36,Def13;
    hence |.H.m - enp0 .| < a by A40,A42,A43;
  end;
  then
A44: enp0 = lim H by A35,COMSEQ_2:def 6;
  deffunc F(Nat) = 1/($1+1);
  consider R be Real_Sequence such that
A45: for n be Nat holds R.n = F(n) from SEQ_1:sch 1;
  take z0;
  let z be Element of F_Complex;
  reconsider v = |.eval(np,z).| as Element of REAL by XREAL_0:def 1;
 set Rcons = seq_const |.eval(np,z).|;
  consider Nseq be increasing sequence of NAT such that
A46: G1 = G*Nseq by A22,VALUED_0:def 17;
  In(|.eval(np,z).|,REAL) = |.eval(np,z).|;
  then |.eval(np,z).| in D;
  then
A47: |.eval(np,z).| >= q by A1,SEQ_4:def 2;
A48: now
    let n be Nat;
A49:  n in NAT by ORDINAL1:def 12;
    consider G1n be Element of F_Complex such that
A50: G1n = G1.n and
A51: H.n = eval(np,G1n) by A25;
    consider gNn be Element of F_Complex such that
A52: gNn = G.(Nseq.n) and
A53: |.eval(np,gNn).| < q+1/((Nseq.n)+1) by A8;
    Nseq.n >= n by SEQM_3:14;
    then Nseq.n+1 >= n+1 by XREAL_1:6;
    then 1/(Nseq.n+1) <= 1/(n+1) by XREAL_1:85;
    then q+1/(Nseq.n+1) <= q+1/(n+1) by XREAL_1:6;
    then |.eval(np,gNn).| < q+1/(n+1) by A53,XXREAL_0:2;
    then q > |.eval(np,gNn).|-1/(n+1) by XREAL_1:19;
    then
A54: Rcons.n = |.eval(np,z).| & |.eval(np,z).| > |.eval(np,gNn).|-1/(n+1)
    by A47,SEQ_1:57,XXREAL_0:2;
    dom (|.H.|-R) = NAT by FUNCT_2:def 1;
    then (|.H.|-R).n = |.H.|.n-R.n by VALUED_1:13,A49
      .= |.H.|.n-1/(n+1) by A45
      .= |.eval(np,G1n).|-1/(n+1) by A51,VALUED_1:18;
    hence Rcons.n >= (|.H.|-R).n by A46,A50,A52,A54,FUNCT_2:15,A49;
  end;
A55: R is convergent by A45,SEQ_4:31;
  then |.H.|-R is convergent by A35;
  then Rcons.0 = |.eval(np,z).| & lim (|.H.|-R) <= lim Rcons by A48,SEQ_1:57
,SEQ_2:18;
  then
A56: lim (|.H.|-R) <= |.eval(np,z).| by SEQ_4:25;
  lim (|.H.|-R) = lim |.H.| - lim R by A35,A55,SEQ_2:12
    .= |.lim H.| - lim R by A35,SEQ_2:27
    .= |.lim H.| - 0 by A45,SEQ_4:31;
  then |.eval(p,z)/p.(len p-'1).| >= |.eval(np,z0).| by A11,A56,A44,Th58;
  then |.eval(p,z)/p.(len p-'1).| >= |.eval(p,z0)/p.(len p-'1).| by A11,Th58;
  then |.eval(p,z).|/|.p.(len p-'1).| >= |.eval(p,z0)/p.(len p-'1).| by A12,
ALGSEQ_1:10,COMPLFLD:73;
  then |.eval(p,z).|/|.p.(len p-'1).| >= |.eval(p,z0).|/|.p.(len p-'1).| by A12
,ALGSEQ_1:10,COMPLFLD:73;
  hence thesis by A13,XREAL_1:74;
end;
