Distributivity of Connectives :

In our last analogy with arithmetic will involve a very well-technique need for playing around with algebraic properties. But assume there if we wanted to work out: 10 * (3 + 5). Than we could do it like this: 10 * (3 + 5) = 10 * 8 = 80. And may be we could do it like this: (10 * 3) + (10 * 5) = 30 + 50 = 80. In fact generally, we know that, for any numbers, a, b and c: a * (b + c) = (a * b) + (a * c). But like this type of theory, we say that multiplication is  distributive over addition. 

Now if you guessed it, that we can distribute some of more the connectives too. In generally or in particular way, ^ is distributive ? over and vice versa: ? is just distributive over ^. We can present these as equivalences as givens: 

What's more, we saw earlier like → is distributive over ^  , and the same as is true for →over ?  .  So therefore: