apply Rsqr_incr_0.
rewrite Rsqr_plus.
cut
(Rsqr (i t) =
(Rsqr l + Rsqr (e t) +
2 *
((xi intr t - xe) * (xe - xi intr 0) + (yi intr t - ye) * (ye - yi intr 0)))%R).
intro.
rewrite H.
apply Rplus_le_compat_l.
repeat rewrite Rmult_assoc.
apply Rmult_le_compat_l.
left; prove_sup0.
cut (l = sqrt (Rsqr (xe - xi intr 0) + Rsqr (ye - yi intr 0))).
intro.
rewrite H0.
unfold e in |- *.
unfold Rsqr_evader_distance in |- *.
rewrite <-
(Rmult_comm (sqrt (Rsqr (xi intr t - xe) + Rsqr (yi intr t - ye))))
.
apply sqrt_cauchy.
apply Rsqr_inj.
apply l_is_pos.
apply sqrt_positivity; apply Rplus_le_le_0_compat; apply Rle_0_sqr.
rewrite Rsqr_sqrt.
apply L.
apply Rplus_le_le_0_compat; apply Rle_0_sqr.
rewrite L.
unfold e, i in |- *.
repeat rewrite Rsqr_sqrt.
unfold Rsqr_evader_distance, Rsqr_intruder_distance in |- *.
replace (xi intr t - xi intr 0)%R with (xi intr t - xe + (xe - xi intr 0))%R.
replace (yi intr t - yi intr 0)%R with (yi intr t - ye + (ye - yi intr 0))%R.
repeat rewrite Rsqr_plus.
ring.
ring.
ring.
apply Rsqr_evader_distance_pos.
apply Rsqr_intruder_distance_pos.
unfold i in |- *.
apply sqrt_positivity.
apply Rsqr_intruder_distance_pos.
apply Rplus_le_le_0_compat.
apply l_is_pos.
unfold e in |- *.
apply sqrt_positivity.
apply Rsqr_evader_distance_pos.
Qed.

Lemma theta_inv :
forall t : R,
(0 <= t)%R ->
(thetat intr 0 - rho_vi * t <= thetat intr t)%R /\
(thetat intr t <= thetat intr 0 + rho_vi * t)%R.