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I was suggesting that all the GUI code go into a new class that the JFrame and JApplet code would add to themselves:

public class OneClass extends JFrame{
OneClass() {
add(new...

An easy way is to rewrite the JFrame app to have a move all the the GUI stuff to a new JPanel that can either be added to the JFrame or to the JApplet.
Those two classes would only be containers for...