let C be Category; :: thesis: for a, b being Object of C
for f being Morphism of a,b st Hom (a,b) <> {} holds
(id b) * f = f
let a, b be Object of C; :: thesis: for f being Morphism of a,b st Hom (a,b) <> {} holds
(id b) * f = f
let f be Morphism of a,b; :: thesis: ( Hom (a,b) <> {} implies (id b) * f = f )
assume A1: Hom (a,b) <> {} ; :: thesis: (id b) * f = f
then f in Hom (a,b) by Def7;
then cod f = b by Th18;
then A2: (id b) * f = f by Th46;
Hom (b,b) <> {} by Th55;
hence (id b) * f = f by A1, A2, Def13; :: thesis: verum