
theorem Th13:
  for n being Element of NAT, x being Element of F_Complex st x is
  Integer holds (power F_Complex).(x,n) is Integer
proof
  let n be Element of NAT, x be Element of F_Complex such that
A1: x is Integer;
  defpred P[Nat] means (power F_Complex).(x,$1) is Integer;
A2: now
    reconsider i1=x as Integer by A1;
    let k be Nat;
     reconsider kk=k as Element of NAT by ORDINAL1:def 12;
    assume P[k];
    then reconsider i2=(power F_Complex).(x,k) as Integer;
    (power F_Complex).(x,kk)*x = i1*i2;
    hence P[k+1] by GROUP_1:def 7;
  end;
A3: P[0] by COMPLFLD:8,GROUP_1:def 7;
  for k being Nat holds P[k] from NAT_1:sch 2(A3,A2);
  hence thesis;
end;
