
theorem
  for x be Element of F_Complex ex x1 be Element of COMPLEX st x = x1 &
  FPower(x,2) = x1(#)(id(COMPLEX)(#)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)(#)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)(#)id(COMPLEX)).y1 by VALUED_1:6
      .= x1*(id(COMPLEX).y1*id(COMPLEX).y1) by VALUED_1:5
      .= x1*(y1*id(COMPLEX).y1)
      .= x*(y*y)
      .= x*power(y,2) by GROUP_1:51;
  end;
  hence thesis by Def12;
end;
