unfold MinDistance_lb, MinDistance in |- *; unfold Rminus in |- *;
rewrite (Rplus_comm (m_rho_ub MinT)); rewrite (Rplus_comm (m T));
apply Rplus_le_compat_l...
cut (MinT <= MinT)%R...
intro H; apply Rle_trans with (m MinT)...
apply (m_rho_ub_0 (mkTimeT MinT H MinT_MaxT))...
apply (m_le (mkTimeT MinT H MinT_MaxT) T)...
simpl in |- *; apply (cond_1 T)...
right...
Qed.

(**********)
Lemma MinDistance_lb_pos : (0 < MinDistance_lb)%R.
Proof with trivial.