SPN_in_hom: 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 \<mu> :: "'b" assumes "bicategory_of_spans V H \<a> \<i> src trg"   and "partial_magma.arr V \<mu>" shows "partial_magma.in_hom (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))) (bicategory_of_spans.SPN V H \<a> \<i> src trg \<mu>) \<lparr>Chn = partial_magma.dom (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 (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.dom V \<mu>))) (trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.dom V \<mu>))) (category.iso_class V (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.dom V \<mu>)))), Dom = \<lparr>Leg0 = Maps.MkArr (src (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.dom V \<mu>))) (trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.dom V \<mu>))) (category.iso_class V (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.dom V \<mu>))), Leg1 = Maps.MkArr (src (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (partial_magma.dom V \<mu>))) (trg (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (partial_magma.dom V \<mu>))) (category.iso_class V (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (partial_magma.dom V \<mu>)))\<rparr>, Cod = \<lparr>Leg0 = Maps.MkArr (src (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.dom V \<mu>))) (trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.dom V \<mu>))) (category.iso_class V (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.dom V \<mu>))), Leg1 = Maps.MkArr (src (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (partial_magma.dom V \<mu>))) (trg (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (partial_magma.dom V \<mu>))) (category.iso_class V (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (partial_magma.dom V \<mu>)))\<rparr>\<rparr> \<lparr>Chn = partial_magma.dom (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 (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.cod V \<mu>))) (trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.cod V \<mu>))) (category.iso_class V (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.cod V \<mu>)))), Dom = \<lparr>Leg0 = Maps.MkArr (src (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.cod V \<mu>))) (trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.cod V \<mu>))) (category.iso_class V (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.cod V \<mu>))), Leg1 = Maps.MkArr (src (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (partial_magma.cod V \<mu>))) (trg (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (partial_magma.cod V \<mu>))) (category.iso_class V (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (partial_magma.cod V \<mu>)))\<rparr>, Cod = \<lparr>Leg0 = Maps.MkArr (src (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.cod V \<mu>))) (trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.cod V \<mu>))) (category.iso_class V (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.cod V \<mu>))), Leg1 = Maps.MkArr (src (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (partial_magma.cod V \<mu>))) (trg (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (partial_magma.cod V \<mu>))) (category.iso_class V (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (partial_magma.cod V \<mu>)))\<rparr>\<rparr>"