intros r min max p H' H'0 H'1 H'2.
case (ClosestMinOrMax r p); auto; intros H'3.
unfold FtoRradix in |- *; apply MinEq with (1 := H'3); auto.
absurd (Rabs (max - r) <= Rabs (min - r))%R.
apply Rgt_not_le.
rewrite (Faux.Rabsolu_left1 (min - r)).
rewrite Rabs_right.
replace (- (min - r))%R with (r - min)%R; [ idtac | ring ].
red in |- *; apply Rplus_lt_reg_l with (r := FtoRradix min).
repeat rewrite Rplus_minus; auto.
apply Rplus_lt_reg_l with (r := r).
replace (r + r)%R with (2%nat * r)%R; [ idtac | simpl in |- *; ring ].
replace (r + (min + (max - r)))%R with (min + max)%R; [ idtac | ring ]; auto.
apply Rle_ge; apply Rplus_le_reg_l with (r := r).
repeat rewrite Rplus_minus; auto.
rewrite Rplus_0_r; auto.
apply isMax_inv1 with (1 := H'0); auto.
apply Rplus_le_reg_l with (r := r).
repeat rewrite Rplus_minus; auto.
rewrite Rplus_0_r; auto.
apply isMin_inv1 with (1 := H'); auto.
cut (Closest r max).
intros H'4; case H'4.
intros H'5 H'6; apply H'6; auto.
case H'; auto.
apply ClosestCompatible with (1 := H'2); auto.
apply MaxEq with (1 := H'3); auto.
case H'0; auto.
Qed.

Theorem ClosestMaxEq :
forall (r : R) (min max p : float),
isMin b radix r min ->
isMax b radix r max ->
(min + max < 2%nat * r)%R -> Closest r p -> p = max :>R.