reserve z,z1,z2,z3,z4 for Element of F_Complex;

theorem Th74:
  for e being Element of F_Complex, n be Nat holds
  (power F_Complex).(e,n) = e|^n
proof
  let e be Element of F_Complex;
  defpred P[Nat] means (power F_Complex).(e,$1) = e|^$1;
A1: now
    let n be Nat;
    reconsider n9 = n as Element of NAT by ORDINAL1:def 12;
    reconsider p = (power F_Complex).(e,n9) as Element of F_Complex;
    assume
A2: P[n];
    (power F_Complex).(e,n+1) = p*e by GROUP_1:def 7
      .= e|^(n+1) by A2,NEWTON:6;
    hence P[n+1];
  end;
  (power F_Complex).(e,0) = 1_F_Complex by GROUP_1:def 7
  .= 1 by Def1;
  then
A3: P[0] by NEWTON:4;
  thus for n being Nat holds P[n] from NAT_1:sch 2(A3,A1);
end;
