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For finite lists, Cartesian lifting of a binary operation to lists preserves associativity. Both associations enumerate triples in the same order: the entry from xs changes slowest, then the entry from ys, then the entry from zs changes fastest. The results are respectivelyandAssociativity makes corresponding entries equal; their order and multiplicity already agree. Empty lists make both sides empty. Therefore

Each entry contributes a choice between omission and inclusion. Use list concatenation as $\oplus$, so that Cartesian lifting of a binary operation to lists joins every choice from an earlier position to every choice from later positions:

crossAppend :: [[a]] -> [[a]] -> [[a]]
crossAppend xss yss = [xs ++ ys | xs <- xss, ys <- yss]

f :: a -> [[a]]
f x = [[],[x]]

subs :: [a] -> [[a]]
subs xs = foldr crossAppend [[]] (map f xs)

The seed is [[]], because the empty list has one subsequence, namely [], and concatenating the empty list changes nothing. If rest lists the subsequences of the tail, crossAppend [[],[x]] rest lists all omissions of x first and all inclusions second. For [a,b,c], the last position changes fastest, producing the specified order [], [c], [b], [b,c], [a], [a,c], [a,b], [a,b,c]. Repeated input values can give equal subsequences at distinct choices of positions; those repetitions are retained. Thus