
theorem
  for x be Element of F_Complex ex x1 be Element of COMPLEX st x = x1 &
  FPower(x,1) = x1(#)id(COMPLEX)
proof
  let x be Element of F_Complex;
  reconsider x1=x as Element of COMPLEX by COMPLFLD:def 1;
  take x1;
  thus x = x1;
  the carrier of F_Complex = COMPLEX by COMPLFLD:def 1;
  then reconsider f=x1(#)id(COMPLEX) as Function of F_Complex,F_Complex;
  now
    let y be Element of F_Complex;
    reconsider y1=y as Element of COMPLEX by COMPLFLD:def 1;
    thus f.y = x1*id(COMPLEX).y1 by VALUED_1:6
      .= x*y
      .= x*power(y,1) by GROUP_1:50;
  end;
  hence thesis by Def12;
end;
