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Let us show that .
Let us prove using mathematical induction on the order of the elements of .
n=0: is a unique subset of of order 0, and by defenition of , .
n =1: for each the singleton , and thus contains all singletons.
Assume that contains all subsets of cardinality , and prove that it is also contains all subsets od cardinality . Indeed, let arbitrary subsets of cardinality . Then by assumption, and consequently, .
Therefore, by Mathematical Induction .