Join Succeeds Operands
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Theorem
Let $\struct {S, \preceq}$ be an ordered set.
Let $a, b \in S$ admit a join $a \vee b \in S$.
Then:
- $a \preceq a \vee b$
- $b \preceq a \vee b$
That is, $a \vee b$ succeeds its operands $a$ and $b$.
Proof
By definition of join:
- $a \vee b = \sup \set {a, b}$
where $\sup$ denotes supremum.
Since a supremum is a fortiori an upper bound:
- $a \preceq \sup \set {a, b}$
- $b \preceq \sup \set {a, b}$
as desired.
$\blacksquare$