# Verify: Prove that $(A times B) cap (C times D) = (A cap C) times (B cap D)$

Mathematics Asked on December 24, 2020

Prove that $$(A times B) cap (C times D) = (A cap C) times (B cap D)$$

$$x in (A times B) cap (C times D) Leftrightarrow xin (A times B) land xin (Ctimes D)$$ (1)

Every $$x$$ is therefore a tuple of the form $$(x_1,x_2)$$.

$$xin A times B implies x_1 in A land x_2in B$$

$$xin C times D implies x_1 in C land x_2 in D$$

Therefore: $$(x_1 in A land x_2 in B) land (x_1 in C land x_2 in D) \ implies (x_1 in A land x_2 in C) land (x_2 in B land x_2 in D) \ implies (x_1 in Acap C) land (x_2 in B cap D) \ implies x = (x_1,x_2) =(Acap C)times (Bcap D)$$

Is this a sound proof?

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