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))}}.