
theorem Th7:
  for z be Element of F_Complex st z <> 0.F_Complex
   for n be Nat holds |.power(z,n).| = |.z.| to_power n
proof
  let z be Element of F_Complex;
  defpred P[Nat] means |.power(z,$1).| = |.z.|
  to_power $1;
  assume z <> 0.F_Complex;
  then
A1: |.z.| <> 0 by COMPLFLD:58;
A2: |.z.| >= 0 by COMPLEX1:46;
A3: for n be Nat st P[n] holds P[n+1]
  proof
    let n be Nat;
    assume
A4: |.power(z,n).| = |.z.| to_power n;
    thus |.power(z,n+1).| = |.(power F_Complex).(z,n)*z.| by
GROUP_1:def 7
      .= (|.z.| to_power n)*|.z.| by A4,COMPLFLD:71
      .= (|.z.| to_power n)*(|.z.| to_power 1) by POWER:25
      .= |.z.| to_power (n+1) by A1,A2,POWER:27;
  end;
  |.(power F_Complex).(z,0).| = 1 by COMPLEX1:48,COMPLFLD:8,GROUP_1:def 7
    .= |.z.| to_power 0 by POWER:24;
  then
A5: P[0];
  thus for n be Nat holds P[n] from NAT_1:sch 2(A5,A3);
end;
