Definition point_in_holed_interval (a b h : R) : R :=
if Req_dec_T h (Rdiv (Rplus a b) (INR 2)) then (Rdiv (Rplus a h) (INR 2))
else (Rdiv (Rplus a b) (INR 2)).

(* Specifications: *)
Lemma middle_in_interval : forall a b : R, Rlt a b -> (a < (a + b) / INR 2 < b)%R.
Lemma point_in_holed_interval_works (a b h : R) :
Rlt a b -> let p := point_in_holed_interval a b h in
Rlt a p /\ Rlt p b /\ p <> h.