Conditions for Lexicographic Order on Pair of Ordered Sets to be Lattice/Lemma 1

Theorem
Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.

Let $\preccurlyeq_l$ denote the lexicographic order on $S_1 \times S_2$''':
 * $\tuple {x_1, x_2} \preccurlyeq_l \tuple {y_1, y_2} \iff \tuple {x_1 \prec_1 y_1} \lor \tuple {x_1 = y_1 \land x_2 \preccurlyeq_2 y_2}$

Let $\tuple {x_1, x_2} \preccurlyeq_l \tuple {y_1, y_2}$.

Then:
 * $x_1 \preccurlyeq_1 y_1$

Proof
Recall the definition of lexicographic order:

Then: