Constructive Dilemma for Join Semilattices
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Theorem
Let $\struct {S, \vee, \preceq}$ be a join semilattice.
Let $a, b, c, d \in S$.
Let $a \preceq b$.
Let $c \preceq d$.
Then $\paren {a \vee c} \preceq \paren {b \vee d}$.
Proof
By Join Semilattice is Ordered Structure, $\preceq$ is compatible with $\vee$.
By the definition of ordering, $\preceq$ is transitive.
Thus the theorem holds by Operating on Transitive Relationships Compatible with Operation.
$\blacksquare$