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)).

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.