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- 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.
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