- apply if_inv in H.
- set (if_comp H (@step1_niso _ s1 _ _ s2 r)) as yy.
- set (if_comp (if_inv (@step1_niso _ s1 _ _ s2 r0)) yy) as yy'.
- apply (if_comp (@roundtrip_lemma _ s1 _ _ s2 r)).
+ set (step1_niso s1 (smme_mee s2) s2 r) as q.
+ apply if_inv in q.
+ eapply if_comp.
+ eapply if_comp; [ idtac | apply q ].
+
+ eapply if_comp.
+ apply (if_associativity
+ (RestrictToImage r)
+ (R' s1 s2 r >>>> ff_functor_section_functor me_homfunctor me_full me_faithful)
+ (HomFunctor (senr_c s2) (senr_c_i s2))).
+ apply if_inv.
+ eapply if_comp.
+ apply (if_associativity
+ (RestrictToImage r)
+ (R' s1 s2 r)
+ (FullImage_InclusionFunctor _)).
+ apply (if_respects
+ (RestrictToImage r)
+ (RestrictToImage r)
+ (R' s1 s2 r >>>> FullImage_InclusionFunctor _)
+ (((R' s1 s2 r >>>> ff_functor_section_functor me_homfunctor me_full me_faithful) >>>>
+ HomFunctor (senr_c s2) (senr_c_i s2)))).
+ apply (if_id _).
+ apply if_inv.
+ eapply if_comp.
+ apply (if_associativity
+ (R' s1 s2 r)
+ (ff_functor_section_functor me_homfunctor me_full me_faithful)
+ (HomFunctor (senr_c s2) (senr_c_i s2))).
+ apply roundtrip_lemma.
+
+ apply if_inv.
+ set (step1_niso s1 (smme_mee s2) s2 r0) as q'.
+ apply if_inv in q'.
+ eapply if_comp.
+ eapply if_comp; [ idtac | apply q' ].
+ eapply if_comp.
+ apply (if_associativity
+ (RestrictToImage r0)
+ (R' s1 s2 r0 >>>> ff_functor_section_functor me_homfunctor me_full me_faithful)
+ (HomFunctor (senr_c s2) (senr_c_i s2))).