What is the purpose of proving that Lady Macbeth is a fourth witch in Macbeth?I mean she does act like a witch and has similar characteristics to...

I think that an attempt to "prove" that Lady Macbeth is a
fourth witch in Macbeth would involve a misinterpretation of both
the play and the role of the supernatural in culture and literature.  First, Lady
Macbeth does not have similar character traits when compared to the witches--the witches
are simply harbingers of fate and they do not direct Macbeth to act one way or another. 
Lady Macbeth, on the other hand, does influence her husband to act, and she largely
persuades Macbeth to murder King Duncan.  Next, the number three is symbolic when
looking at elements of fate and destiny, and this would be broken by adding a fourth. 
Finally, Lady Macbeth is round character in the play as she goes from being quite greedy
and driven to being remorseful when she realizes that many people have died at Macbeth's
hand.  The witches remain constant throughout the play.