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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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