Independence Complement Theorem

If sets  and  are independent, then so are  and , where  is the complement of  (i.e., the set of all possible outcomes not contained in ). Let  denote "or" and  denote "and." Then

			(1)
			(2)

where  is an abbreviation for . But  and  are independent, so

(3)

Also, since  and  are complements, they contain no common elements, which means that

(4)

for any . Plugging (4) and (3) into (2) then gives

(5)

Rearranging,

(6)

Q.E.D.