Definition CReal_from_cauchy (xn : nat -> CReal) (xcau : Un_cauchy_mod xn) : CReal :=
{|
  seq := CReal_from_cauchy_seq xn xcau;
  scale := CReal_from_cauchy_scale xn xcau;
  cauchy := CReal_from_cauchy_cauchy xn xcau;
  bound := CReal_from_cauchy_bound xn xcau
|}.

Lemma CRealComplete : forall xn : nat -> CReal, (forall p : positive, {n : nat | forall i j : nat, (n <= i)%nat -> (n <= j)%nat -> (CReal_abs (xn i + - xn j)) <= (inject_Q (1 # p))}) -> {l : CReal & forall p : positive, {n : nat | forall i : nat, (n <= i)%nat -> (CReal_abs (xn i + - l)) <= (inject_Q (1 # p))}}.