Proof: Let A, B, C, D be sets such that A and B are nonempty and (A x C) ⊆ (B x D).  Let (x, y) ∈ A x C.  By definition of Cartesian product, x  ∈ A and y  ∈ C.  Since (A x C) ⊆ (B x D), (x,y) ∈ B x D by definition of subset.  By definition of Cartesian product, x ∈ B and y ∈ D.  Since x ∈ A and B, and y ∈ C and D, and (A x C) ⊆ (B x D), A ⊆ B and C ⊆ D.