let C be Cartesian_category; :: thesis: for a, b being Object of C holds <:(pr1 a,b),(pr2 a,b):> = id (a [x] b)
let a, b be Object of C; :: thesis: <:(pr1 a,b),(pr2 a,b):> = id (a [x] b)
A1: Hom (a [x] b),b <> {} by Th21;
then A2: (pr2 a,b) * (id (a [x] b)) = pr2 a,b by CAT_1:58;
A3: Hom (a [x] b),a <> {} by Th21;
then (pr1 a,b) * (id (a [x] b)) = pr1 a,b by CAT_1:58;
hence <:(pr1 a,b),(pr2 a,b):> = id (a [x] b) by A3, A1, A2, Def11; :: thesis: verum