intros p q p' q' H' H'0 H'1.
elim H'1; intros H'2 H'3; elim H'3; intros H'4 H'5; apply H'5; clear H'3 H'1;
auto.
case H'0; auto.
apply Rle_trans with p; auto.
apply isMin_inv1 with (1 := H'0); auto.
apply Rlt_le; auto.
Qed.
(* What it is to be a maximum *)
Definition isMax (r : R) (max : float) :=
Fbounded b max /\
(r <= max)%R /\
(forall f : float, Fbounded b f -> (r <= f)%R -> (max <= f)%R).
(* It is a projector *)

Theorem isMax_inv1 : forall (p : float) (r : R), isMax r p -> (r <= p)%R.
intros p r H; case H; intros H1 H2; case H2; auto.