reserve o,m for set;
reserve C for Cartesian_category;
reserve a,b,c,d,e,s for Object of C;

theorem
  Hom(a,c) <> {} & Hom(b,d) <> {} implies Hom(a[x]b,c[x]d) <> {}
proof
  assume that
A1: Hom(a,c) <> {} and
A2: Hom(b,d) <> {};
  Hom(a[x]b,b) <> {} by Th19;
  then
A3: Hom(a[x]b,d) <> {} by A2,CAT_1:24;
  Hom(a[x]b,a) <> {} by Th19;
  then Hom(a[x]b,c) <> {} by A1,CAT_1:24;
  hence thesis by A3,Th23;
end;
