Section NormedZmoduleTheory.
Variable V : normedZmodType R.
Implicit Types (u v w : V).
Hint Resolve normr_real : core.
Arguments real_ler_normlP {x y}.
Definition real_lter_norml := (real_ler_norml, real_ltr_norml).
Arguments real_ltr_normlP {x y}.
Definition real_lter_normr := (real_ler_normr, real_ltr_normr).

Lemma leif_nmul x1 x2 y1 y2 C1 C2 :
y1 <= 0 -> y2 <= 0 -> x1 <= y1 ?= iff C1 -> x2 <= y2 ?= iff C2 ->
y1 * y2 <= x1 * x2 ?= iff (x1 * x2 == 0) || C1 && C2.