Way Below Relation is Transitive

Theorem

Let $\left({S, \preceq}\right)$ be an ordered set.

Let $x, y, z \in S$ such that

$x \ll y \ll z$

Then

$x \ll z$

Proof

By Way Below implies Preceding:

$x \preceq y$

By definition of reflexivity:

$z \preceq z$

Thus by Preceding and Way Below implies Way Below:

$x \ll z$

$\blacksquare$

Sources

• 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices

• Mizar article WAYBEL_3:5