let C be Cartesian_category; :: thesis: for a, c, b, d being Object of C st Hom a,c <> {} & Hom b,d <> {} holds
Hom (a [x] b),(c [x] d) <> {}
let a, c, b, d be Object of C; :: thesis: ( Hom a,c <> {} & Hom b,d <> {} implies Hom (a [x] b),(c [x] d) <> {} )
assume that
A1: Hom a,c <> {} and
A2: Hom b,d <> {} ; :: thesis: Hom (a [x] b),(c [x] d) <> {}
Hom (a [x] b),b <> {} by Th21;
then A3: Hom (a [x] b),d <> {} by A2, CAT_1:52;
Hom (a [x] b),a <> {} by Th21;
then Hom (a [x] b),c <> {} by A1, CAT_1:52;
hence Hom (a [x] b),(c [x] d) <> {} by A3, Th25; :: thesis: verum