intros; generalize (ypt t H H0); intro; elim H1; intros; split;
[ unfold hy in H2; unfold Rminus in H2; rewrite yp0 in H2;
rewrite Rmult_0_r in H2; rewrite cos_0 in H2; rewrite Rmult_1_r in H2;
generalize
(Rplus_le_compat_r (l * sin (thetat intr 0 + beta))
(r_vi * cos (rho_vi * t) + - r_vi)
(yp t + - (l * sin (thetat intr 0 + beta))) H2);
replace (r_vi * cos (rho_vi * t) + - r_vi)%R with
(r_vi * (cos (rho_vi * t) - 1))%R;
[ repeat rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r;
intro; rewrite <- (Rplus_comm (r_vi * (cos (rho_vi * t) - 1)));
rewrite (Rplus_comm beta); assumption
| ring ]
| unfold hy in H3; unfold Rminus in H3; rewrite yp0 in H3;
rewrite Rmult_0_r in H3; rewrite cos_0 in H3; rewrite Rmult_1_r in H3;
generalize
(Rplus_le_compat_r (l * sin (thetat intr 0 + beta))
(yp t + - (l * sin (thetat intr 0 + beta)))
(r_vi + - (r_vi * cos (rho_vi * t))) H3);
replace (r_vi + - (r_vi * cos (rho_vi * t)))%R with
(- (r_vi * (cos (rho_vi * t) - 1)))%R;
[ repeat rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r;
intro; unfold Rminus in |- *; rewrite (Rplus_comm beta);
rewrite <- (Rplus_comm (- (r_vi * (cos (rho_vi * t) + -1))));
assumption
| unfold Rminus in |- *; rewrite Rmult_plus_distr_l; ring ] ].
Qed.

Lemma Dxp :
forall t : R,
derive_pt xp t (xp_derivable t) =
(vi intr * cos (thetat intr t - thetat intr 0))%R.