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
  for z,s being Element of COMPLEX, n being Element of NAT st s<>0 & z<>
  0 & n>=1 & s|^ n=z|^ n holds |.s.| = |.z.|
proof
  let z,s be Element of COMPLEX, n be Element of NAT;
  assume that
A1: s<>0 and
A2: z<>0 and
A3: n>=1 and
A4: s|^ n=z|^ n;
  z|^ n = (|.s.|*cos Arg s + |.s.|*sin Arg s *<i>)|^ n by A4,COMPTRIG:62
    .=(|.s.|*(cos Arg s+sin Arg s *<i>))|^ n
    .=(|.s.||^ n)*((cos Arg s+sin Arg s *<i>)|^ n) by NEWTON:7
    .=(|.s.| to_power n)*(cos (n*Arg s)+sin (n*Arg s)*<i>) by Th31
    .= ((|.s.| to_power n)*cos (n*Arg s) )+((|.s.| to_power n)*sin (n*Arg s)
  )*<i>;
  then
A5: ((|.s.| to_power n)*cos (n*Arg s)) +((|.s.| to_power n)*sin (n*Arg s ))*
  <i> = (|.z.|*cos Arg z + |.z.|*sin Arg z *<i>)|^ n by COMPTRIG:62
    .=(|.z.|*(cos Arg z+sin Arg z *<i>))|^ n
    .=(|.z.||^ n)*((cos Arg z+sin Arg z *<i>)|^ n) by NEWTON:7
    .=(|.z.| to_power n)*(cos (n*Arg z)+sin (n*Arg z)*<i>) by Th31
    .= ((|.z.| to_power n)*cos (n*Arg z))+((|.z.| to_power n) *sin (n*Arg z)
  )*<i>;
  then (|.s.| to_power n)*cos (n*Arg s)=(|.z.| to_power n)*cos (n*Arg z) by
COMPLEX1:77;
  then
  (|.s.| to_power n) ^2*(cos (n*Arg s))^2+ ((|.s.| to_power n)*sin (n*Arg
s))^2 =((|.z.| to_power n)*cos (n*Arg z))^2+ ((|.z.| to_power n)*sin (n*Arg z))
  ^2 by A5,SQUARE_1:9;
  then (|.s.| to_power n) ^2*((cos (n*Arg s))^2+(sin (n*Arg s))^2) =((|.z.|
  to_power n))^2*((cos (n*Arg z))^2+(sin (n*Arg z))^2);
  then (|.s.| to_power n) ^2*((cos (n*Arg s))^2+(sin (n*Arg s))^2) =((|.z.|
  to_power n))^2*1 by SIN_COS:29;
  then
A6: (|.s.| to_power n) ^2*1=((|.z.| to_power n))^2 by SIN_COS:29;
A7: |.s.|>0 by A1,COMPLEX1:47;
  then |.s.| to_power n > 0 by POWER:34;
  then
A8: (|.s.| to_power n) =sqrt ((|.z.| to_power n))^2 by A6,SQUARE_1:22;
A9: |.z.|>0 by A2,COMPLEX1:47;
  then |.z.| to_power n > 0 by POWER:34;
  then |.s.| |^ n=|.z.| |^ n by A8,SQUARE_1:22;
  then |.s.| = n-root (|.z.| |^ n) by A3,A7,POWER:4;
  hence thesis by A3,A9,POWER:4;
end;
