Is my duality principle proof correct?

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woundedtiger4
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Have i done it correctly?
Handwritten proof in red ink on lined paper of R minus the intersection of A_alpha equals the union over alpha of (R minus A_alpha), beginning Suppose x is in R minus the intersection, and ending Hence x is in the union, QED.
 
haruspex said:
A better wording (after "i.e.") would be that there exists i such that x belongs to Ai, etc.

Thanks