Chn_RHS_eq: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b" assumes "three_composable_identities_in_bicategory_of_spans V H \<a> \<i> src trg f g h" shows "Chn (span_vertical_category.vcomp (concrete_category.COMP (Collect (horizontal_homs.obj V src)) (\<lambda>a b. {F. \<exists>f. horizontal_homs.in_hhom V src trg f a b \<and> (\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg f g) \<and> F = category.iso_class V f}) (\<lambda>_ _ _. maps_category.Comp V H)) (transformation_by_components.map (subcategory.comp (product_category.comp V V) (\<lambda>\<mu>\<nu>. partial_magma.arr V (fst \<mu>\<nu>) \<and> partial_magma.arr V (snd \<mu>\<nu>) \<and> src (fst \<mu>\<nu>) = trg (snd \<mu>\<nu>))) (span_vertical_category.vcomp (concrete_category.COMP (Collect (horizontal_homs.obj V src)) (\<lambda>a b. {F. \<exists>f. horizontal_homs.in_hhom V src trg f a b \<and> (\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg f g) \<and> F = category.iso_class V f}) (\<lambda>_ _ _. maps_category.Comp V H))) (comp (\<lambda>\<mu>\<nu>. span_bicategory.hcomp (concrete_category.COMP (Collect (horizontal_homs.obj V src)) (\<lambda>a b. {F. \<exists>f. horizontal_homs.in_hhom V src trg f a b \<and> (\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg f g) \<and> F = category.iso_class V f}) (\<lambda>_ _ _. maps_category.Comp V H)) (maps_category.PRJ\<^sub>0 V H \<a> \<i> src trg) (maps_category.PRJ\<^sub>1 V H \<a> \<i> src trg) (fst \<mu>\<nu>) (snd \<mu>\<nu>)) (weak_arrow_of_homs.FF V src trg (span_vertical_category.vcomp (concrete_category.COMP (Collect (horizontal_homs.obj V src)) (\<lambda>a b. {F. \<exists>f. horizontal_homs.in_hhom V src trg f a b \<and> (\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg f g) \<and> F = category.iso_class V f}) (\<lambda>_ _ _. maps_category.Comp V H))) (span_vertical_category.src (concrete_category.COMP (Collect (horizontal_homs.obj V src)) (\<lambda>a b. {F. \<exists>f. horizontal_homs.in_hhom V src trg f a b \<and> (\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg f g) \<and> F = category.iso_class V f}) (\<lambda>_ _ _. maps_category.Comp V H))) (span_vertical_category.trg (concrete_category.COMP (Collect (horizontal_homs.obj V src)) (\<lambda>a b. {F. \<exists>f. horizontal_homs.in_hhom V src trg f a b \<and> (\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg f g) \<and> F = category.iso_class V f}) (\<lambda>_ _ _. maps_category.Comp V H))) (bicategory_of_spans.SPN V H \<a> \<i> src trg))) (\<lambda>rs. bicategory_of_spans.CMP V H \<a> \<i> src trg (fst rs) (snd rs)) (f, H g h)) (span_vertical_category.vcomp (concrete_category.COMP (Collect (horizontal_homs.obj V src)) (\<lambda>a b. {F. \<exists>f. horizontal_homs.in_hhom V src trg f a b \<and> (\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg f g) \<and> F = category.iso_class V f}) (\<lambda>_ _ _. maps_category.Comp V H)) (span_bicategory.hcomp (concrete_category.COMP (Collect (horizontal_homs.obj V src)) (\<lambda>a b. {F. \<exists>f. horizontal_homs.in_hhom V src trg f a b \<and> (\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg f g) \<and> F = category.iso_class V f}) (\<lambda>_ _ _. maps_category.Comp V H)) (maps_category.PRJ\<^sub>0 V H \<a> \<i> src trg) (maps_category.PRJ\<^sub>1 V H \<a> \<i> src trg) (bicategory_of_spans.SPN V H \<a> \<i> src trg f) (transformation_by_components.map (subcategory.comp (product_category.comp V V) (\<lambda>\<mu>\<nu>. partial_magma.arr V (fst \<mu>\<nu>) \<and> partial_magma.arr V (snd \<mu>\<nu>) \<and> src (fst \<mu>\<nu>) = trg (snd \<mu>\<nu>))) (span_vertical_category.vcomp (concrete_category.COMP (Collect (horizontal_homs.obj V src)) (\<lambda>a b. {F. \<exists>f. horizontal_homs.in_hhom V src trg f a b \<and> (\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg f g) \<and> F = category.iso_class V f}) (\<lambda>_ _ _. maps_category.Comp V H))) (comp (\<lambda>\<mu>\<nu>. span_bicategory.hcomp (concrete_category.COMP (Collect (horizontal_homs.obj V src)) (\<lambda>a b. {F. \<exists>f. horizontal_homs.in_hhom V src trg f a b \<and> (\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg f g) \<and> F = category.iso_class V f}) (\<lambda>_ _ _. maps_category.Comp V H)) (maps_category.PRJ\<^sub>0 V H \<a> \<i> src trg) (maps_category.PRJ\<^sub>1 V H \<a> \<i> src trg) (fst \<mu>\<nu>) (snd \<mu>\<nu>)) (weak_arrow_of_homs.FF V src trg (span_vertical_category.vcomp (concrete_category.COMP (Collect (horizontal_homs.obj V src)) (\<lambda>a b. {F. \<exists>f. horizontal_homs.in_hhom V src trg f a b \<and> (\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg f g) \<and> F = category.iso_class V f}) (\<lambda>_ _ _. maps_category.Comp V H))) (span_vertical_category.src (concrete_category.COMP (Collect (horizontal_homs.obj V src)) (\<lambda>a b. {F. \<exists>f. horizontal_homs.in_hhom V src trg f a b \<and> (\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg f g) \<and> F = category.iso_class V f}) (\<lambda>_ _ _. maps_category.Comp V H))) (span_vertical_category.trg (concrete_category.COMP (Collect (horizontal_homs.obj V src)) (\<lambda>a b. {F. \<exists>f. horizontal_homs.in_hhom V src trg f a b \<and> (\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg f g) \<and> F = category.iso_class V f}) (\<lambda>_ _ _. maps_category.Comp V H))) (bicategory_of_spans.SPN V H \<a> \<i> src trg))) (\<lambda>rs. bicategory_of_spans.CMP V H \<a> \<i> src trg (fst rs) (snd rs)) (g, h))) (span_bicategory.assoc (concrete_category.COMP (Collect (horizontal_homs.obj V src)) (\<lambda>a b. {F. \<exists>f. horizontal_homs.in_hhom V src trg f a b \<and> (\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg f g) \<and> F = category.iso_class V f}) (\<lambda>_ _ _. maps_category.Comp V H)) (maps_category.PRJ\<^sub>0 V H \<a> \<i> src trg) (maps_category.PRJ\<^sub>1 V H \<a> \<i> src trg) (bicategory_of_spans.SPN V H \<a> \<i> src trg f) (bicategory_of_spans.SPN V H \<a> \<i> src trg g) (bicategory_of_spans.SPN V H \<a> \<i> src trg h)))) = concrete_category.COMP (Collect (horizontal_homs.obj V src)) (\<lambda>a b. {F. \<exists>f. horizontal_homs.in_hhom V src trg f a b \<and> (\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg f g) \<and> F = category.iso_class V f}) (\<lambda>_ _ _. maps_category.Comp V H) (Maps.MkArr (src (arrow_of_tabulations_in_maps.chine V H \<a> \<i> src trg (composite_tabulation_in_maps.tab V H \<a> \<i> src trg f (identity_in_bicategory_of_spans.tab V H \<a> \<i> src trg f) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg f) (H g h) (identity_in_bicategory_of_spans.tab V H \<a> \<i> src trg (H g h)) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H g h)) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (H g h))) (H (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H g h)) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg f)) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (H g h))))) (H f (H g h)) (identity_in_bicategory_of_spans.tab V H \<a> \<i> src trg (H f (H g h))) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H f (H g h))) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (H f (H g h))) (H f (H g h)))) (trg (arrow_of_tabulations_in_maps.chine V H \<a> \<i> src trg (composite_tabulation_in_maps.tab V H \<a> \<i> src trg f (identity_in_bicategory_of_spans.tab V H \<a> \<i> src trg f) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg f) (H g h) (identity_in_bicategory_of_spans.tab V H \<a> \<i> src trg (H g h)) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H g h)) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (H g h))) (H (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H g h)) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg f)) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (H g h))))) (H f (H g h)) (identity_in_bicategory_of_spans.tab V H \<a> \<i> src trg (H f (H g h))) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H f (H g h))) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (H f (H g h))) (H f (H g h)))) (category.iso_class V (arrow_of_tabulations_in_maps.chine V H \<a> \<i> src trg (composite_tabulation_in_maps.tab V H \<a> \<i> src trg f (identity_in_bicategory_of_spans.tab V H \<a> \<i> src trg f) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg f) (H g h) (identity_in_bicategory_of_spans.tab V H \<a> \<i> src trg (H g h)) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H g h)) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (H g h))) (H (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H g h)) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg f)) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (H g h))))) (H f (H g h)) (identity_in_bicategory_of_spans.tab V H \<a> \<i> src trg (H f (H g h))) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H f (H g h))) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (H f (H g h))) (H f (H g h))))) (concrete_category.COMP (Collect (horizontal_homs.obj V src)) (\<lambda>a b. {F. \<exists>f. horizontal_homs.in_hhom V src trg f a b \<and> (\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg f g) \<and> F = category.iso_class V f}) (\<lambda>_ _ _. maps_category.Comp V H) (Maps.MkArr (src (arrow_of_tabulations_in_maps.chine V H \<a> \<i> src trg (composite_tabulation_in_maps.tab V H \<a> \<i> src trg f (identity_in_bicategory_of_spans.tab V H \<a> \<i> src trg f) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg f) (H g h) (composite_tabulation_in_maps.tab V H \<a> \<i> src trg g (identity_in_bicategory_of_spans.tab V H \<a> \<i> src trg g) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg g) h (identity_in_bicategory_of_spans.tab V H \<a> \<i> src trg h) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg h) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg h)) (H (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg h) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg g)) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg h)))) (H (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg g) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (H (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg g)) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg h))))) (H (H (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg h) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg g)) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg h)))) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg f)) (H (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg g) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (H (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg g)) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg h))))))) (H f (H g h)) (composite_tabulation_in_maps.tab V H \<a> \<i> src trg f (identity_in_bicategory_of_spans.tab V H \<a> \<i> src trg f) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg f) (H g h) (identity_in_bicategory_of_spans.tab V H \<a> \<i> src trg (H g h)) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H g h)) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (H g h))) (H (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H g h)) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg f)) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (H g h))))) (H (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg f) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (H (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg f)) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (H g h))))) (H f (H g h)))) (trg (arrow_of_tabulations_in_maps.chine V H \<a> \<i> src trg (composite_tabulation_in_maps.tab V H \<a> \<i> src trg f (identity_in_bicategory_of_spans.tab V H \<a> \<i> src trg f) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg f) (H g h) (composite_tabulation_in_maps.tab V H \<a> \<i> src trg g (identity_in_bicategory_of_spans.tab V H \<a> \<i> src trg g) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg g) h (identity_in_bicategory_of_spans.tab V H \<a> \<i> src trg h) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg h) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg h)) (H (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg h) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg g)) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg h)))) (H (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg g) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (H (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg g)) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg h))))) (H (H (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg h) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg g)) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg h)))) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg f)) (H (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg g) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (H (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg g)) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg h))))))) (H f (H g h)) (composite_tabulation_in_maps.tab V H \<a> \<i> src trg f (identity_in_bicategory_of_spans.tab V H \<a> \<i> src trg f) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg f) (H g h) (identity_in_bicategory_of_spans.tab V H \<a> \<i> src trg (H g h)) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H g h)) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (H g h))) (H (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H g h)) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg f)) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (H g h))))) (H (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg f) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (H (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg f)) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (H g h))))) (H f (H g h)))) (category.iso_class V (arrow_of_tabulations_in_maps.chine V H \<a> \<i> src trg (composite_tabulation_in_maps.tab V H \<a> \<i> src trg f (identity_in_bicategory_of_spans.tab V H \<a> \<i> src trg f) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg f) (H g h) (composite_tabulation_in_maps.tab V H \<a> \<i> src trg g (identity_in_bicategory_of_spans.tab V H \<a> \<i> src trg g) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg g) h (identity_in_bicategory_of_spans.tab V H \<a> \<i> src trg h) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg h) (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg h)) (H (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg h) (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg g)) (bicategory_of_spans.tab\<^sub>1 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