intros H' H'0; split.
apply oppBounded; auto.
intros f H'1.
replace (Rabs (Fopp p - - r)) with (Rabs (p - r)).
replace (Rabs (f - - r)) with (Rabs (Fopp f - r)).
apply H'0; auto.
apply oppBounded; auto.
unfold FtoRradix in |- *; rewrite Fopp_correct.
pattern r at 1 in |- *; replace r with (- - r)%R; [ idtac | ring ].
replace (- FtoR radix f - - - r)%R with (- (FtoR radix f - - r))%R;
[ idtac | ring ].
apply Rabs_Ropp; auto.
unfold FtoRradix in |- *; rewrite Fopp_correct.
replace (- FtoR radix p - - r)%R with (- (FtoR radix p - r))%R;
[ idtac | ring ].
apply sym_eq; apply Rabs_Ropp.
Qed.

(* Specifications: *)
Theorem EvenClosestSymmetric : SymmetricP EvenClosest.
red in |- *; intros r p H'; case H'; clear H'.