About statements of the implicit function theorem in "Calculus - Several Variables -" by Shoshichi Kobayashi. Is this statement necessary?

Question

The following statements of the implicit function theorem is from "Calculus - Several Variables -" by Shoshichi Kobayashi (in Japanese).

Let $f(x,y,z)$ be a function defined on an open set which contains a point $(x_0,y_0,z_0)$.
Let $f_x,f_y,f_z$ be continuous functions.
Let $f(x_0,y_0,z_0)=0, f_z(x_0,y_0,z_0)\neq 0$.
There exist $a, b>0$ and a unique $C^1$ function $z=h(x,y)$ defined on $|x-x_0|<a,|y-y_0|<a$ such that

1. $z_0=h(x_0,y_0)$,
2. $|h(x,y)-z_0|<b,\,\,\,\,\,\,\,\,|x-x_0|<a, |y-y_0|<a,$
3. $f(x,y,h(x,y))\equiv 0,\,\,\,\,\,\,\,\,|x-x_0|<a, |y-y_0|<a,$
4. $f_z(x,y,h(x,y))\neq 0,\,\,\,\,\,\,\,\,|x-x_0|<a, |y-y_0|<a,$
5. $h_x(x,y)=-\frac{f_x(x,y,h(x,y))}{f_z(x,y,h(x,y))},\,\,\,\,\,\,\,\,|x-x_0|<a, |y-y_0|<a,$
6. $h_y(x,y)=-\frac{f_y(x,y,h(x,y))}{f_z(x,y,h(x,y))},\,\,\,\,\,\,\,\,|x-x_0|<a, |y-y_0|<a.$

I think we don't need the following statement:

2. $|h(x,y)-z_0|<b,\,\,\,\,\,\,\,\,|x-x_0|<a, |y-y_0|<a,$

I think the following statements are sufficient:

Let $f(x,y,z)$ be a function defined on an open set which contains a point $(x_0,y_0,z_0)$.
Let $f_x,f_y,f_z$ be continuous functions.
Let $f(x_0,y_0,z_0)=0, f_z(x_0,y_0,z_0)\neq 0$.
There exist $a>0$ and a unique $C^1$ function $z=h(x,y)$ defined on $|x-x_0|<a,|y-y_0|<a$ such that

1. $z_0=h(x_0,y_0)$,
2. $f(x,y,h(x,y))\equiv 0,\,\,\,\,\,\,\,\,|x-x_0|<a, |y-y_0|<a,$
3. $f_z(x,y,h(x,y))\neq 0,\,\,\,\,\,\,\,\,|x-x_0|<a, |y-y_0|<a,$
4. $h_x(x,y)=-\frac{f_x(x,y,h(x,y))}{f_z(x,y,h(x,y))},\,\,\,\,\,\,\,\,|x-x_0|<a, |y-y_0|<a,$
5. $h_y(x,y)=-\frac{f_y(x,y,h(x,y))}{f_z(x,y,h(x,y))},\,\,\,\,\,\,\,\,|x-x_0|<a, |y-y_0|<a.$

Why did the author write 2.?