The first step is proved as a preliminary lemma, in which a partial order where a pair of elements x and y are incomparable is changed to make them comparable. This is done by first adding the pair xRy to the relation, which may result in a non-transitive relation, and then restoring transitivity by adding all pairs qRp such that qRx \text{ and } yRp. This is done on a single pair of incomparable elements x and y, and produces a relation that is still reflexive, antisymmetric and transitive and that strictly contains the original one.
