let C be Category; :: thesis: for E being Subcategory of C
for a, b being Object of E
for a9, b9 being Object of C
for f being Morphism of a,b st a = a9 & b = b9 & Hom a,b <> {} holds
f is Morphism of a9,b9
let E be Subcategory of C; :: thesis: for a, b being Object of E
for a9, b9 being Object of C
for f being Morphism of a,b st a = a9 & b = b9 & Hom a,b <> {} holds
f is Morphism of a9,b9
let a, b be Object of E; :: thesis: for a9, b9 being Object of C
for f being Morphism of a,b st a = a9 & b = b9 & Hom a,b <> {} holds
f is Morphism of a9,b9
let a9, b9 be Object of C; :: thesis: for f being Morphism of a,b st a = a9 & b = b9 & Hom a,b <> {} holds
f is Morphism of a9,b9
let f be Morphism of a,b; :: thesis: ( a = a9 & b = b9 & Hom a,b <> {} implies f is Morphism of a9,b9 )
assume ( a = a9 & b = b9 & Hom a,b <> {} ) ; :: thesis: f is Morphism of a9,b9
then ( f in Hom a,b & Hom a,b c= Hom a9,b9 ) by Def4, CAT_1:def 7;
hence f is Morphism of a9,b9 by CAT_1:21; :: thesis: verum