
theorem
  for x be Real st x > 0 for n be Nat holds (power F_Complex)
  .([**x,0**],n) = [**x to_power n,0**]
proof
  let x be Real;
  defpred P[Nat] means
    (power F_Complex).([**x,0**],$1) = [**x to_power $1,0**];
  assume
A1: x > 0;
A2: now
    let n be Nat;
    assume P[n];
    then
    (power F_Complex).([**x,0**],n+1) = [**x to_power n,0**]*[**x,0**] by
GROUP_1:def 7
      .= [**(x to_power n)*(x to_power 1),0**] by POWER:25
      .= [**x to_power (n+1),0**] by A1,POWER:27;
    hence P[n+1];
  end;
  (power F_Complex).([**x,0**],0) = 1r+0*<i> by COMPLFLD:8,GROUP_1:def 7
    .= [**x to_power 0,0**] by POWER:24;
  then
A3: P[0];
  thus for n be Nat holds P[n] from NAT_1:sch 2(A3,A2);
end;
