
theorem Th69:
  for x be Element of F_Complex for n be Nat
   ex f be Function of COMPLEX,COMPLEX
    st f = FPower(x,n) & FPower(x,n+1) = f(#)id(COMPLEX)
proof
  let x be Element of F_Complex;
  let n be Nat;
A1: the carrier of F_Complex = COMPLEX by COMPLFLD:def 1;
  then reconsider f=FPower(x,n) as Function of COMPLEX,COMPLEX;
  reconsider g=f(#)id(COMPLEX) as Function of F_Complex,F_Complex by A1;
  take f;
  thus f = FPower(x,n);
  now
    let y be Element of F_Complex;
    reconsider y1=y as Element of COMPLEX by COMPLFLD:def 1;
    thus g.y = f.y1*id(COMPLEX).y1 by VALUED_1:5
      .= FPower(x,n).y*y
      .= x*power(y,n)*y by Def12
      .= x*((power F_Complex).(y,n)*y)
      .= x*power(y,n+1) by GROUP_1:def 7;
  end;
  hence thesis by Def12;
end;
