
theorem Th43:
  for n being Element of NAT, z being Element of F_Complex st z is
  Real ex x being Real st x = z & (power F_Complex).(z,n) = x |^ n
proof
  let n be Element of NAT;
  let z be Element of F_Complex;
  assume z is Real;
  then reconsider x=z as Real;
  per cases;
  suppose
A1: x = 0;
    then
A2: z = 0.F_Complex by COMPLFLD:def 1;
    thus thesis
    proof
      per cases;
      suppose
A3:     n = 0;
        then (power F_Complex).(z,n) = 1 by COMPLFLD:8,GROUP_1:def 7
          .= x |^ n by A3,NEWTON:4;
        hence thesis;
      end;
      suppose
A4:     n > 0;
        then
A5:     n >= 0+1 by NAT_1:13;
        (power F_Complex).(z,n) = 0.F_Complex by A2,A4,VECTSP_1:36
          .= x|^n by A1,A5,COMPLFLD:7,NEWTON:11;
        hence thesis;
      end;
    end;
  end;
  suppose
A6: x <> 0;
    defpred P[Nat] means (power F_Complex).(z,$1) = x |^ $1;
A7: for n being Nat st P[n] holds P[n+1]
    proof
      let n be Nat such that
A8:   P[n];
      reconsider nn=n as Element of NAT by ORDINAL1:def 12;
      (power F_Complex).(z,n+1) = (power F_Complex).(z,nn)*z by GROUP_1:def 7
        .= (x #Z n) * x by A8,PREPOWER:36
        .= (x#Z n) * (x #Z 1) by PREPOWER:35
        .= (x #Z (n+1)) by A6,PREPOWER:44
        .= (x |^ (n+1)) by PREPOWER:36;
      hence thesis;
    end;
    (power F_Complex).(z,0) = 1r by COMPLFLD:8,GROUP_1:def 7
      .= x #Z 0 by PREPOWER:34
      .= x |^ 0 by PREPOWER:36;
    then
A9: P[0];
    for n being Nat holds P[n] from NAT_1:sch 2(A9,A7);
    then (power F_Complex).(z,n) = x |^ n;
    hence thesis;
  end;
end;
