Proof with trivial.
intros; unfold Rsqr_Rmin in |- *; case (Req_EM_var (Ax2_R intruder) 0); intro...
elim H; apply div_tau_0_Ax2_R...
rewrite Rsqr_minus; rewrite (tmin_0 intruder evader H); rewrite Rsqr_div...
unfold Rdiv in |- *; rewrite Rsqr_R; rewrite <- Rsqr_neg; rewrite Rsqr_mult;
replace (Rsqr 2) with 4%R...
rewrite (Rinv_mult_distr 2 (Ax2_R intruder))...
rewrite (Rinv_mult_distr 4 (Ax2_R intruder))...
rewrite Rinv_mult_distr...
unfold Rminus in |- *; do 2 rewrite Rmult_plus_distr_l...
rewrite <-
(Rmult_comm (Rsqr (Bx_R intruder evader) * (/ 4 * / Rsqr (Ax2_R intruder))))
; unfold Rsqr in |- *;
rewrite (Rinv_mult_distr (Ax2_R intruder) (Ax2_R intruder))...
repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym...
rewrite Rmult_1_r...
repeat rewrite Rplus_assoc; apply Rplus_eq_compat_l...
rewrite
(Rmult_plus_distr_r (4 * (Ax2_R intruder * C_R intruder evader))
(- (Bx_R intruder evader * Bx_R intruder evader))
(/ 4 * / Ax2_R intruder))...
replace
(Ax2_R intruder *
- (2 * (t * (- Bx_R intruder evader * (/ 2 * / Ax2_R intruder)))))%R with
(Bx_R intruder evader * t)%R...
replace
(4 * (Ax2_R intruder * C_R intruder evader) * (/ 4 * / Ax2_R intruder))%R
with (C_R intruder evader)...
ring...
rewrite (Rmult_comm (/ 4))...
repeat rewrite Rmult_assoc...
rewrite (Rmult_comm 4)...
repeat rewrite Rmult_assoc...
rewrite <- Rinv_l_sym...
rewrite Rmult_1_r; rewrite (Rmult_comm (C_R intruder evader));
repeat rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym...
symmetry  in |- *; apply Rmult_1_l...
discrR...
rewrite (Rmult_comm (Ax2_R intruder));
replace
(- (2 * (t * (- Bx_R intruder evader * (/ 2 * / Ax2_R intruder)))))%R with
(-2 * t * - Bx_R intruder evader * / 2 * / Ax2_R intruder)%R...
repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym...
rewrite Rmult_1_r; repeat rewrite <- Rmult_assoc;
replace (-2 * t * - Bx_R intruder evader)%R with
(t * Bx_R intruder evader * 2)%R...
repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym...
ring...
discrR...
ring...
ring...
discrR...
unfold Rsqr in |- *; apply prod_neq_R0...
discrR...
discrR...
apply prod_neq_R0...
Qed.

(**********)
Lemma dx_dxdt :
forall (intruder evader : State) (t : R),
(xt intruder - xt evader + t * dxdt intruder)%R = dx intruder evader t.