Way Above Closure is Upper
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Theorem
Let $\struct {S, \preceq}$ be an ordered set.
Let $x \in S$.
Then $x^\gg$ is upper
where $x^\gg$ denotes the way above closure of $x$.
Proof
Let $y \in x^\gg$, $z \in S$ such that
- $y \preceq z$
By definition of way above closure:
- $x \ll y$
By Preceding and Way Below implies Way Below:
- $x \ll z$
Thus by definition of way above closure:
- $z \in x^\gg$
$\blacksquare$
Sources
- Mizar article WAYBEL_3:funcreg 3