Lower Sections of Well-Ordered Classes are Order Isomorphic at most Uniquely

Theorem
Let $\struct {A, \preccurlyeq_A}$ and $\struct {B, \preccurlyeq_B}$ be well-ordered classes.

Let $L_A$ be a lower section of $\struct {A, \preccurlyeq_A}$.

Then:
 * $(1): \quad L_A$ is order isomorphic to at most one lower section $L_B$ of $\struct {B, \preccurlyeq_B}$
 * $(2): \quad$ If such an $L_B$ exists, there exists exactly one order isomorphism from $L_A$ to $L_B$.