Infimum of Intersection of Upper Closures equals Join Operands

Theorem
Let $L = \struct {S, \vee, \preceq}$ be a join semilattice.

Let $x, y \in S$.

Then $\map \inf {x^\succeq \cap y^\succeq} = x \vee y$

Proof
By Intersection of Upper Closures is Upper Closure of Join Operands:
 * $x^\succeq \cap y^\succeq = \paren {x \vee y}^\succeq$

Thus by Infimum of Upper Closure of Element:
 * $\map \inf {x^\succeq \cap y^\succeq} = x \vee y$