+#p #IH @exteq_repl
+/3 width=5 by compose_repl_fwd_dx, compose_repl_fwd_sn, exteq_canc_dx/
+qed.
+
+lemma piter_compose (A) (B) (f) (g) (h) (p):
+ h ∘ f ≐ g ∘ h → h ∘ (f^{A}p) ≐ (g^{B}p) ∘ h.
+#A #B #f #g #h #p elim p -p
+[ #H @exteq_repl
+ /2 width=5 by compose_repl_fwd_sn, compose_repl_fwd_dx/
+| #p #IH #H @exteq_repl
+ [4: @compose_repl_fwd_dx [| @piter_succ ]
+ |5: @compose_repl_fwd_sn [| @piter_succ ]
+ |1,2: skip
+ ]
+ @exteq_trans [2: @compose_assoc |1: skip ]
+ @exteq_trans [2: @(compose_repl_fwd_sn … H) | 1:skip ]
+ @exteq_canc_sn [2: @compose_assoc |1: skip ]
+ /3 width=1 by compose_repl_fwd_dx/
+]