
theorem Th5:
  for z being Element of F_Complex for k being Element of NAT holds
  (power(F_Complex).(z,k))*' = power(F_Complex).(z*',k)
proof
  let z be Element of F_Complex, k be Element of NAT;
  defpred P[Nat] means ex j be Element of NAT st j = $1 & (power(F_Complex).(z
  ,j))*' = power(F_Complex).(z*',j);
A1: now
    let k be Nat;
     reconsider kk=k as Element of NAT by ORDINAL1:def 12;
    assume
A2: P[k];
    (power(F_Complex).(z,k+1))*' = (power(F_Complex).(z,kk) * z)*' by
GROUP_1:def 7
      .= (power(F_Complex).(z*',kk)) * (z*') by A2,COMPLFLD:54
      .= power(F_Complex).(z*',k+1) by GROUP_1:def 7;
    hence P[k+1];
  end;
  (power(F_Complex).(z,0))*' = (1_F_Complex)*' by GROUP_1:def 7
    .= 1_F_Complex by Lm2,COMPLEX1:38
    .= power(F_Complex).(z*',0) by GROUP_1:def 7;
  then
A3: P[0];
  for k be Nat holds P[k] from NAT_1:sch 2(A3,A1);
  then ex j being Element of NAT st j = k & (power(F_Complex).(z,j))*' =
  power(F_Complex).(z*',j);
  hence thesis;
end;
