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

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