
theorem Th49:
  for f being Polynomial of F_Complex st deg(f) >= 1 & f is
Hurwitz for x being Element of F_Complex holds (Re(x) < 0 implies |.eval(f,x).|
< |.eval(f*',x).|) & (Re(x) > 0 implies |.eval(f,x).| > |.eval(f*',x).|) & (Re(
  x) = 0 implies |.eval(f,x).| = |.eval(f*',x).|)
proof
A1: now
    let y,z be Element of F_Complex;
    assume
A2: rpoly(1,z) is Hurwitz;
    z is_a_root_of rpoly(1,z) by Th30;
    then
A3: Re(z) < 0 by A2;
A4: y = Re(y) + Im(y) * <i> by COMPLEX1:13;
A5: y - z = eval(rpoly(1,z),y) by Th29;
A6: z = Re(z) + Im(z) * <i> by COMPLEX1:13;
    reconsider y9 = y, z9 = z as Element of COMPLEX by COMPLFLD:def 1;
A7: -y = -y9 by COMPLFLD:2;
    eval(rpoly(1,z)*',y) = (-y) - (z*') by Th47;
    then
A8: eval(rpoly(1,z)*',y) = (-y9) - (z9*') by A7,COMPLFLD:3;
A9: |.(y9 + (z9*')).| = |.-(y9 + (z9*')).| by COMPLEX1:52;
    assume Re(y) > 0;
    then |.y9 - z9.| > |.y9 + (z9*').| by A3,A4,A6,Lm14;
    hence |.eval(rpoly(1,z),y).| > |.eval((rpoly(1,z))*',y).| by A5,A8,A9,
COMPLFLD:3;
  end;
A10: now
    let a,y,z be Element of F_Complex;
    assume that
A11: rpoly(1,z) is Hurwitz and
A12: a <> 0.F_Complex;
    assume
A13: Re(y) > 0;
A14: |.a.| * |.eval((rpoly(1,z))*',y).| = |.a*'.| * |.eval((rpoly(1,z))*',
    y).| by COMPLEX1:53
      .= |.a*' * eval((rpoly(1,z))*',y).| by COMPLEX1:65
      .= |.eval((a*')*(rpoly(1,z))*',y).| by POLYNOM5:30
      .= |.eval((a*rpoly(1,z))*',y).| by Th43;
A15: |.eval(a*rpoly(1,z),y).| = |.a * eval(rpoly(1,z),y).| by POLYNOM5:30
      .= |.a.| * |.eval(rpoly(1,z),y).| by COMPLEX1:65;
    |.a.| > 0 by A12,COMPLEX1:47,COMPLFLD:7;
    hence
    |.eval(a*rpoly(1,z),y).|>|.eval((a*rpoly(1,z))*',y).| by A1,A11,A13,A15,A14
,XREAL_1:68;
  end;
A16: now
    let y,z be Element of F_Complex;
    assume
A17: rpoly(1,z) is Hurwitz;
    z is_a_root_of rpoly(1,z) by Th30;
    then
A18: Re(z) < 0 by A17;
A19: y = Re(y) + Im(y) * <i> by COMPLEX1:13;
A20: y - z = eval(rpoly(1,z),y) by Th29;
A21: z = Re(z) + Im(z) * <i> by COMPLEX1:13;
    reconsider y9 = y, z9 = z as Element of COMPLEX by COMPLFLD:def 1;
A22: -y = -y9 by COMPLFLD:2;
    eval(rpoly(1,z)*',y) = (-y) - (z*') by Th47;
    then
A23: eval(rpoly(1,z)*',y) = (-y9) - (z9*') by A22,COMPLFLD:3;
A24: |.(y9 + (z9*')).| = |.-(y9 + (z9*')).| by COMPLEX1:52;
    assume Re(y) < 0;
    then |.y9 - z9.| < |.y9 + (z9*').| by A18,A19,A21,Lm14;
    hence |.eval(rpoly(1,z),y).| < |.eval((rpoly(1,z))*',y).| by A20,A23,A24,
COMPLFLD:3;
  end;
A25: now
    let a,y,z be Element of F_Complex;
    assume that
A26: rpoly(1,z) is Hurwitz and
A27: a <> 0.F_Complex;
    assume
A28: Re(y) < 0;
A29: |.a.| * |.eval((rpoly(1,z))*',y).| = |.a*'.| * |.eval((rpoly(1,z))*',
    y).| by COMPLEX1:53
      .= |.a*' * eval((rpoly(1,z))*',y).| by COMPLEX1:65
      .= |.eval((a*')*(rpoly(1,z))*',y).| by POLYNOM5:30
      .= |.eval((a*rpoly(1,z))*',y).| by Th43;
A30: |.eval(a*rpoly(1,z),y).| = |.a * eval(rpoly(1,z),y).| by POLYNOM5:30
      .= |.a.| * |.eval(rpoly(1,z),y).| by COMPLEX1:65;
    |.a.| > 0 by A27,COMPLEX1:47,COMPLFLD:7;
    hence |.eval(a*rpoly(1,z),y).|<|.eval((a*rpoly(1,z))*',y).| by A16,A26,A28
,A30,A29,XREAL_1:68;
  end;
  defpred P[Nat] means for f being Polynomial of F_Complex st deg(f) >= 1 & f
  is Hurwitz & deg(f) = $1 for x being Element of F_Complex holds (Re(x) < 0
  implies |.eval(f,x).| < |.eval(f*',x).|) & (Re(x) > 0 implies |.eval(f,x).| >
  |.eval(f*',x).|) & (Re(x) = 0 implies |.eval(f,x).| = |.eval(f*',x).|);
  let f be Polynomial of F_Complex;
  assume that
A31: deg(f) >= 1 and
A32: f is Hurwitz;
A33: now
    let y,z be Element of F_Complex;
    assume
A34: rpoly(1,z) is Hurwitz;
    z is_a_root_of rpoly(1,z) by Th30;
    then
A35: Re(z) < 0 by A34;
A36: y = Re(y) + Im(y) * <i> by COMPLEX1:13;
A37: y - z = eval(rpoly(1,z),y) by Th29;
A38: z = Re(z) + Im(z) * <i> by COMPLEX1:13;
    reconsider y9 = y, z9 = z as Element of COMPLEX by COMPLFLD:def 1;
A39: -y = -y9 by COMPLFLD:2;
    eval(rpoly(1,z)*',y) = (-y) - (z*') by Th47;
    then
A40: eval(rpoly(1,z)*',y) = (-y9) - (z9*') by A39,COMPLFLD:3;
A41: |.(y9 + (z9*')).| = |.-(y9 + (z9*')).| by COMPLEX1:52;
    assume Re(y) = 0;
    then |.y9 - z9.| = |.y9 + (z9*').| by A35,A36,A38,Lm14;
    hence |.eval(rpoly(1,z),y).| = |.eval((rpoly(1,z))*',y).| by A37,A40,A41,
COMPLFLD:3;
  end;
A42: now
    let a,y,z be Element of F_Complex;
    assume that
A43: rpoly(1,z) is Hurwitz and
    a <> 0.F_Complex;
A44: |.eval(a*rpoly(1,z),y).| = |.a * eval(rpoly(1,z),y).| by POLYNOM5:30
      .= |.a.| * |.eval(rpoly(1,z),y).| by COMPLEX1:65;
A45: |.a.| * |.eval((rpoly(1,z))*',y).| = |.a*'.| * |.eval((rpoly(1,z))*',
    y).| by COMPLEX1:53
      .= |.a*' * eval((rpoly(1,z))*',y).| by COMPLEX1:65
      .= |.eval((a*')*(rpoly(1,z))*',y).| by POLYNOM5:30
      .= |.eval((a*rpoly(1,z))*',y).| by Th43;
    assume Re(y) = 0;
    hence |.eval(a*rpoly(1,z),y).| = |.eval((a*rpoly(1,z))*',y).| by A33,A43
,A44,A45;
  end;
A46: now
    let k be Nat;
    assume
A47: P[k];
    now
      let f be Polynomial of F_Complex;
      assume that
A48:  deg(f) >= 1 and
A49:  f is Hurwitz and
A50:  deg(f) = k+1;
      let x be Element of F_Complex;
      per cases by A48,A50,XXREAL_0:1;
      suppose
        k+1 = 1;
        then consider z1,z2 being Element of F_Complex such that
A51:    z1 <> 0.F_Complex and
A52:    f = z1 * rpoly(1,z2) by A50,Th28;
        rpoly(1,z2) is Hurwitz by A49,A51,A52,Th40;
        hence (Re(x) < 0 implies |.eval(f,x).| < |.eval(f*',x).|) & (Re(x) > 0
implies |.eval(f,x).| > |.eval(f*',x).|) & (Re(x) = 0 implies |.eval(f,x).| =
        |.eval(f*',x).|) by A25,A10,A42,A51,A52;
      end;
      suppose
A53:    k+1 > 1;
A54:    k + 1 + 1 > k + 1 + 0 by XREAL_1:6;
        then
A55:    f <> 0_.(F_Complex) by A50,POLYNOM4:3;
        len f > 1 by A48,A50,A54,XXREAL_0:2;
        then f is with_roots by POLYNOM5:74;
        then consider z being Element of F_Complex such that
A56:    z is_a_root_of f by POLYNOM5:def 8;
        consider f1 being Polynomial of F_Complex such that
A57:    f = rpoly(1,z) *' f1 by A56,Th33;
A58:    f1 <> 0_.(F_Complex) by A57,A55,POLYNOM3:34;
        rpoly(1,z) <> 0_.(F_Complex) by A57,A55,POLYNOM3:34;
        then
A59:    deg f = deg(rpoly(1,z)) + deg(f1) by A57,A58,Th23
          .= 1 + deg(f1) by Th27;
A60:    rpoly(1,z) is Hurwitz by A49,A57,Th41;
A61:    f1 is Hurwitz by A49,A57,Th41;
A62:    k >= 1 by A53,NAT_1:13;
A63:    now
          reconsider r19 = eval(rpoly(1,z)*',x), e19 = eval(f1*',x) as Element
          of COMPLEX by COMPLFLD:def 1;
          reconsider r9 = eval(rpoly(1,z),x), e9 = eval(f1,x) as Element of
          COMPLEX by COMPLFLD:def 1;
A64:      eval(rpoly(1,z)*',x) * eval(f1*',x) = eval((rpoly(1,z)*')*'((
          f1)*'),x) by POLYNOM4:24;
          assume
A65:      Re(x) > 0;
          then
A66:      |.e9.| > |.e19.| by A47,A50,A59,A61,A62;
          0 <= |.r19.| by COMPLEX1:46;
          then
A67:      |.r19.| * |.e9.| >= |.r19.| * |.e19.| by A66,XREAL_1:64;
          0 <= |.e19.| by COMPLEX1:46;
          then |.r9.| * |.e9.| > |.r19.| * |.e9.| by A1,A60,A65,A66,XREAL_1:68;
          then |.r9.| * |.e9.| > |.r19.| * |.e19.| by A67,XXREAL_0:2;
          then |.r9 * e9.| > |.r19.| * |.e19.| by COMPLEX1:65;
          then
A68:      |.eval(rpoly(1,z),x) * eval(f1,x).| > |.eval((rpoly(1,z))*',x)
          * eval(f1*',x).| by COMPLEX1:65;
          eval(rpoly(1,z),x) * eval(f1,x) = eval(rpoly(1,z)*'f1,x) by
POLYNOM4:24;
          hence |.eval(f,x).| > |.eval(f*',x).| by A57,A68,A64,Th46;
        end;
A69:    now
          reconsider r19 = eval(rpoly(1,z)*',x), e19 = eval(f1*',x) as Element
          of COMPLEX by COMPLFLD:def 1;
          reconsider r9 = eval(rpoly(1,z),x), e9 = eval(f1,x) as Element of
          COMPLEX by COMPLFLD:def 1;
A70:      0 <= |.r9.| by COMPLEX1:46;
          assume
A71:      Re(x) < 0;
          then
A72:      |.r9.| < |.r19.| by A16,A60;
          0 <= |.e9.| by COMPLEX1:46;
          then
A73:      |.r9.| * |.e9.| <= |.r19.| * |.e9.| by A72,XREAL_1:64;
          |.e9.| < |.e19.| by A47,A50,A59,A61,A62,A71;
          then |.r19.| * |.e9.| < |.r19.| * |.e19.| by A72,A70,XREAL_1:68;
          then |.r9.| * |.e9.| < |.r19.| * |.e19.| by A73,XXREAL_0:2;
          then |.r9 * e9.| < |.r19.| * |.e19.| by COMPLEX1:65;
          then
A74:      |.eval(rpoly(1,z),x) * eval(f1,x).| < |.eval((rpoly(1,z))*',x)
          * eval(f1*',x).| by COMPLEX1:65;
A75:      eval(rpoly(1,z)*',x) * eval(f1*',x) = eval((rpoly(1,z)*')*'((f1
          )*'),x) by POLYNOM4:24;
          eval(rpoly(1,z),x) * eval(f1,x) = eval(rpoly(1,z)*'f1,x) by
POLYNOM4:24;
          hence |.eval(f,x).| < |.eval(f*',x).| by A57,A74,A75,Th46;
        end;
        now
          reconsider r19 = eval(rpoly(1,z)*',x), e19 = eval(f1*',x) as Element
          of COMPLEX by COMPLFLD:def 1;
          reconsider r9 = eval(rpoly(1,z),x), e9 = eval(f1,x) as Element of
          COMPLEX by COMPLFLD:def 1;
A76:      eval(rpoly(1,z)*',x) * eval(f1*',x) = eval((rpoly(1,z)*')*'((
          f1)*'),x) by POLYNOM4:24;
          assume
A77:      Re(x) = 0;
          then |.eval(f1,x).| = |.eval(f1*',x).| by A47,A50,A59,A61,A62;
          then |.r9.| * |.e9.| = |.r19.| * |.e19.| by A33,A60,A77;
          then |.r9 * e9.| = |.r19.| * |.e19.| by COMPLEX1:65;
          then
A78:      |.eval(rpoly(1,z),x) * eval(f1,x).| = |.eval((rpoly(1,z))*',x)
          * eval(f1*',x).| by COMPLEX1:65;
          eval(rpoly(1,z),x) * eval(f1,x) = eval(rpoly(1,z)*'f1,x) by
POLYNOM4:24;
          hence |.eval(f,x).| = |.eval(f*',x).| by A57,A78,A76,Th46;
        end;
        hence (Re(x) < 0 implies |.eval(f,x).| < |.eval(f*',x).|) & (Re(x) > 0
implies |.eval(f,x).| > |.eval(f*',x).|) & (Re(x) = 0 implies |.eval(f,x).| =
        |.eval(f*',x).|) by A69,A63;
      end;
    end;
    hence P[k+1];
  end;
  let x be Element of F_Complex;
A79: P[0];
A80: for k be Nat holds P[k] from NAT_1:sch 2(A79,A46);
  f <> 0_.(F_Complex) by A31,Th20;
  then deg f is Element of NAT by Lm8;
  hence thesis by A31,A32,A80;
end;
