there are three more theorems that deserve our attention so the first theorem is the uniting theorem so it states X and Y or X and Y Prime equals to X and the Dual is X or Y and X or Y Prime = to X so the proof is pretty straightforward so take the left hand side take X as is the common term you'll be left with Y or Y Prime so so this is X and Y or Y Prime is 1 so this is X uh you can do the same thing for the right hand side one so just multiply it out have X and X or X and Y prime or X and Y are Y and Y Prime as you know Y and Y Prime is zero so you have x and x is X or XY prime or XY now out of these two terms I'm going to take X as the common factor one or Y prime or X Y so this is X or Y so this is X and one or XY so that's X or XY so take X as the common term again so this is X and one which is X uh the next theorem is kind of important it's known as the elimination theorem so as per elimination theorem you have X or X Prime y = to X or Y and the Dual is X and X prime or y equals XY and then you have the absorption theorem so you have X or x and y equal x and finally the Dual is x and x or y equal x