Condition for Woset to be Isomorphic to Ordinal/Mistake

Source Work
$\S 1$: Naive Set Theory:
 * $\S 1.7$: Well-Orderings and Ordinals:
 * Theorem $1.7.11$

Mistake

 * Also, since:

we have
 * $(2): \quad \paren {g_y X_x}: X_x \cong \paren {\map Z y}_{\map {g_y} x}$.

Now, $\map Z y$ is an ordinal, so by Theorem $1.7.6$, $\paren {\map Z y}_{\map {g_y} x}$ is an ordinal.

...

Correction
Statement $(2)$ has a misprint.

It should read:
 * $(2): \quad \paren {g_y \restriction X_x}: X_x \cong \paren {\map Z y}_{\map {g_y} x}$

thereby providing the crucial information that the mapping under consideration is the restriction of $g_y$ to $X_x$.