+lemma unwind2_term_fsubst_pic_sn (f) (t) (u) (p): p ฯต ๐ โ
+ (โผ[f]t)[โ(โp)โโผ[โถ[f]p]u] โ โผ[f](t[โpโu]).
+#f #t #u #p #Hp #ql * *
+[ #rl * #r #Hr #H1 #H2 destruct
+ >unwind2_path_append_pic_sn
+ /4 width=3 by in_comp_unwind2_path_term, or_introl, ex2_intro/
+| * #q #Hq #H1 #H0
+ @(ex2_intro โฆ H1) @or_intror @conj // *
+ [ <list_append_empty_sn #H2 destruct
+ elim (unwind2_path_root f q) #r #_ #Hr /2 width=2 by/
+ | #l #r #H2 destruct
+ /3 width=2 by unwind2_path_append_pic_sn/
+ ]
+]
+qed-.
+
+lemma unwind2_term_fsubst_pic_dx (f) (t) (u) (p): p ฯต ๐ โ p ฯต โตt โ t ฯต ๐ โ
+ โผ[f](t[โpโu]) โ (โผ[f]t)[โ(โp)โโผ[โถ[f]p]u].
+#f #t #u #p #Hp #H1p #H2p #ql * #q * *
+[ #r #Hu #H1 #H2 destruct
+ /5 width=3 by unwind2_path_append_pic_sn, ex2_intro, or_introl/
+| #Hq #H0 #H1 destruct
+ @or_intror @conj [ /2 width=1 by in_comp_unwind2_path_term/ ] *
+ [ <list_append_empty_sn #Hr @(H0 โฆ (๐)) -H0
+ <list_append_empty_sn @H2p -H2p
+ /2 width=2 by unwind2_path_des_structure, prototerm_in_comp_root/
+ | #l #r #Hr
+ elim (unwind2_path_inv_append_ppc_dx โฆ Hr) -Hr // #s1 #s2 #Hs1 #_ #H1 destruct
+ lapply (H2p โฆ Hs1) -H2p -Hs1 /2 width=2 by ex_intro/
+ ]
+]
+qed-.
+
+lemma unwind2_term_fsubst_pic (f) (t) (u) (p): p ฯต ๐ โ p ฯต โตt โ t ฯต ๐ โ
+ (โผ[f]t)[โ(โp)โโผ[โถ[f]p]u] โ โผ[f](t[โpโu]).
+/4 width=3 by unwind2_term_fsubst_pic_sn, conj, unwind2_term_fsubst_pic_dx/ qed.
+
+(* Constructions with fsubst and ppc ****************************************)
+
+lemma unwind2_term_fsubst_ppc_sn (f) (t) (u) (p): u ฯต ๐ โ