intros p q H' H'0 H'1.
apply floatEq; auto.
unfold FtoR in H'0.
apply Rpow_eq_inv with (r := IZR radix); auto 6 with real zarith.
apply Rlt_dichotomy_converse; right; red in |- *.
unfold Rabs in |- *; case (Rcase_abs radix).
intros H'2; Contradict H'2; apply Rle_not_lt; apply Ropp_le_cancel;
auto with real.
intros H'2; replace 1%R with (IZR 1); auto with real zarith.
apply Rmult_eq_reg_l with (r := IZR (Fnum p)); auto with real.
pattern (Fnum p) at 2 in |- *; rewrite H'1; auto.
Qed.

Theorem FtoREqInv2 :
forall p q : float, p = q :>R -> Fexp p = Fexp q -> p = q.