Localization: Textdomain of Child teme

I am creating a series of WP child themes which are dependent of a parent theme which I will be using as a framework.

I need to have these themes (both parent and children) localized.

from http://codex.wordpress.org/Function_Reference/load_theme_textdomain I understood I need to add the following to my theme:

add_action('after_setup_theme', 'my_theme_setup');
function my_theme_setup(){
    load_theme_textdomain('mytextdomain', get_template_directory() . '/lang');
}

this goes in the functions.php, I suppose

however, what about children themes? functions.php of a children theme overrides functions.php of a parent theme

should textdomain (“mytextdomain”) of the child theme be the same as parent theme or be different (and also call load_theme_textdomain function with a different function name (see above code “my_theme_Setup()”)?

what is the correct way of localizing both a parent and child theme?

thank you for clarifying this 🙂