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 5;
then ( f in Hom (a9,b9) & Hom (a9,b9) <> {} ) ;
hence f is Morphism of a9,b9 by CAT_1:def 5; :: thesis: verum