Suggestion:
Make the JFrame instance a field of the MainWindow class, and provide an accessor method for it.
public final class MainWindow{
    private final JFrame main_f;
    public MainWindow(){
        main_f = new JFrame("xx");
        main_f.getContentPane().setLayout(new BorderLayout());
        main_f.setDefaultCloseOperation(JFrame.DISPOSE_ON_CLOSE);
        .
        .
        .
    }
    public final JFrame getMainFrame(){
        return main_f;
    }
    .
    .
    .
}
And then in the Disposing class, you should have a MainWindow instance, where you'll simply do the following to dispose of its JFrame instance:
mainWindowInstance.getMainFrame().dispose();
Recommendation:
Edit:
This is to address the errors that you're seeing:
- variable main_f might not have been initialized
- cannot find symbol "mainWindowInstance"
With regard to the first error, this is because in the example I provided, I used the final modifier. This field must be initialized upon object creation. Therefore, you must have more than one constructor. To resolve this, either remove the final modifier, or initialize the main_f field in every constructor of MainWindow.
With regard to the second error, mainWindowInstance is something that I left for you to create. Here's a "for instance" -
public class Disposing{
    private MainWindow mainWindowInstance;
    public Disposing(){
        mainWindowInstance = new MainWindow();
        .
        .
        .
    }
    public void diposeMainFrame(){
        mainWindowInstance.getMainFrame().dispose();
    }
}