reserve a,b,c,d,a9,b9,c9,d9,y,x1,u,v for Real,
  s,t,h,z,z1,z2,z3,s1,s2,s3 for Complex;

theorem Th34:
  for z be Complex for n,k be Element of NAT st n <> 0
holds z = ((n-root |.z.|)*cos((Arg z+2*PI*k)/n) + (n-root |.z.|)*sin((Arg z+2*
  PI*k)/n)*<i>)|^ n
proof
  let z be Complex;
  let n,k be Element of NAT;
  assume
A1: n <> 0;
  then
A2: n >= 0+1 by NAT_1:13;
  per cases;
  suppose
    z <> 0;
    then
A3: |.z.| > 0 by COMPLEX1:47;
    thus ((n-root |.z.|)*cos((Arg z+2*PI*k)/n)+ (n-root |.z.|)*sin((Arg z+2*PI
*k)/n)*<i>)|^ n = ((n-root |.z.|)* (cos((Arg z+2*PI*k)/n)+sin((Arg z+2*PI*k)/n)
    *<i>))|^ n
      .= ((n-root |.z.|)|^ n) *((cos((Arg z+2*PI*k)/n)+sin((Arg z+2*PI*k)/n)
    *<i>)|^ n) by NEWTON:7
      .= ((n-root |.z.|)|^ n)*(cos Arg z+(sin Arg z)*<i>) by A1,Th33
      .= |.z.|*(cos Arg z+(sin Arg z)*<i>) by A1,A3,COMPTRIG:4
      .= |.z.|*cos Arg z-0*sin Arg z+(|.z.|*sin Arg z+0*cos Arg z)*<i>
      .= z by COMPTRIG:62;
  end;
  suppose
A4: z = 0;
    hence
    ((n-root |.z.|)*cos((Arg z+2*PI*k)/n)+ (n-root |.z.|)*sin((Arg z+2*PI
*k)/n)*<i>)|^ n =((0*cos((Arg z+2*PI*k)/n)+ (n-root 0)*sin((Arg z+2*PI*k)/n)*
    <i>)|^ n) by A2,COMPLEX1:44,POWER:5
      .= (0*sin((Arg z+2*PI*k)/n)*<i>)|^ n by A2,POWER:5
      .= z by A1,A4,Lm2;
  end;
end;
