+lemma cnv_inv_appl_SO_cpes (a) (h) (G) (L):
+ ∀V,T. ⦃G, L⦄ ⊢ ⓐV.T ![a, h] →
+ ∃∃n,p,W,U. a = Ⓣ → n = 1 & ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
+ ⦃G, L⦄ ⊢ V ⬌*[h,1,0] W & ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W.U.
+#a #h #G #L #V #T #H
+elim (cnv_inv_appl_SO … H) -H #n #p #W #U #Hn #HV #HT #HVW #HTU
+/3 width=7 by cpms_div, ex5_4_intro/
+qed-.
+
+lemma cnv_inv_appl_true_cpes (h) (G) (L):
+ ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T ![h] →
+ ∃∃p,W,U. ⦃G,L⦄ ⊢ V ![h] & ⦃G,L⦄ ⊢ T ![h] &
+ ⦃G,L⦄ ⊢ V ⬌*[h,1,0] W & ⦃G,L⦄ ⊢ T ➡*[1,h] ⓛ{p}W.U.
+#h #G #L #V #T #H
+elim (cnv_inv_appl_SO_cpes … H) -H #n #p #W #U #Hn
+>Hn -n [| // ] #HV #HT #HVW #HTU
+/2 width=5 by ex4_3_intro/
+qed-.
+