The first higher Stasheff-Tamari orders are quotients of the higher Bruhat orders.

We prove two related conjectures concerning the higher Bruhat orders and the first higher Stasheff-Tamari orders. Namely, we prove the conjecture of Danilov, Karzanov, and Koshevoy that every triangulation of a cyclic polytope arises as a vertex figure of a cubillage of a cyclic zonotope. This gives an order-preserving surjection from the higher Bruhat orders to the first higher Stasheff-Tamari orders. We then go further by showing that this map is full, which proves the conjecture of Dimakis and M\"uller-Hoissen that the higher Tamari orders are the same posets as the first higher Stasheff-Tamari orders. We show that order-preserving maps which are surjective and full correspond to quotients of posets. This notion of quotient posets is more general than those that have been considered previously. The equivalence of the first higher Stasheff-Tamari orders and the higher Tamari orders entails that the first higher Stasheff-Tamari orders play a role in the theories of KP solitons and polygon equations from mathematical physics.