, , , be four subgroups of a group with and .

Let .

I have shown the following:

(1)

(2)

(3)

(4)

If , then for and ;

define by .

I'm facing problem to show that is well-defined.

I started with ;

implies that . ----------( )

I'm stuck here.

I know I need to get ; ------------( )

implies that ;

finally .

How to continue from ( ) to ( )?