The positive formulation of the Golden Rule is “Do unto others as you would have them do unto you”.
The negative formulation (according to Rabbi Hillel) is “If something is hateful to you, do not do it to others”.
Here is the proof that the negative formulation implies the positive formulation:
If there is an action X that you find hateful, then by definition you would not want someone else to do X to you.
Therefore if you do X to another, you are not doing unto them as you would have them do unto you.
This therefore violates the positive formulation of the Golden Rule.
Q. E. D.
Now, does the positive formulation imply the negative one? Yes, but that is more complex to prove, so it will have to wait until I have the time for another post.