 Why would one expect this latter to work? Well, the chain produced is naturally a subset well-ordering, for it is both linear and also such that the intersection of arbitrary elements of members of the chain is itself a member of the chain, and thus there is a natural subset-least element for each subset of members of the chain. But the wider explanation is hinted at towards the end of Zermelo's proof. Suppose a set M is (speaking informally) de facto well-ordered by an ordering relation ≺. Call the set ℜ≼(a) = {x ∈ M : a ≼ x} the ‘remainder [Rest]’ determined by a and the ordering ≺. Consider now the set of ‘remainders’ given by this ordering, i.e., {ℜ≼(x) : x ∈ M}. This set is in fact well-ordered by reverse inclusion, where the successor remainder to ℜ≼(a) is just the remainder determined by a's successor a′ under ≺, and where intersections are taken at the limit elements (the intersection of a set of remainders is again a remainder). But not only is this set well-ordered by reverse inclusion, the ordering is isomorphic to the ordering ≺ on M, that is: 