 This theorem should not be too surprising. If \(\preceq\) is complete and transitive over \(S\), then the options in \(S\) can be put in an order, from the most to the least preferred, where some options may fall in the same position (if they are deemed equally desirable) but where there are no cycles, loops, or gaps.  Theorem 1  just says that we can assign numbers to the options in \(S\) in a way that represents this order. (For a simple proof of Theorem 1, except for a strict rather than a weak preference relation, consult Peterson 2009: 95.)