## Proving whether $\bf{K}$(Happy $\lor$Sad) $\implies$ $\neg \bf{K}($Happy) is satisfiable, valid or unsatisfiable

I have a question that I am stuck on reasoning about knowledge. For the following statement, I need to show whether it is valid, satisfiable or unsatisfiable:

$$\bf{K}$$(Happy $$\lor$$Sad) $$\implies$$ $$\neg \bf{K}($$Happy)

I have done the following:

Let e w be an interpretation where e is the set of possible worlds and w is the real world. Then we need to show that for any interpretation, e w satisfies $$\neg \bf{K}$$(Happy $$\lor$$Sad) $$\lor$$ $$\neg \bf{K}$$(Happy).:

Now assume that such an interpretation e w satisfies $$\bf{K}$$(Happy $$\lor$$Sad). This would mean for all w’ $$\in$$ e, e w’ satisfies $$\bf{K}$$(Happy $$\lor$$Sad). However in a world w” = {$$\neg Happy, \neg Sad$$}, then the interpretation e w” does not satisfy $$\bf{K}$$(Happy $$\lor$$ Sad).

Therefore, we have a contradiction and for any interpretation e w, e w satisfies $$\neg \bf{K}$$(Happy $$\lor$$Sad).

Similarly, assume an interpretation e w satisfies $$\bf{K}($$Happy). However, if we have a world(s) where $$\neg$$Happy is true, then e w” does not satisfy $$\bf{K}($$Happy). Therefore for any e w, e w satisfies $$\neg \bf{K}($$Happy) and thus the statement is valid.

I’m not sure whether my proof is correct and if not, could give me a hint on how to go about with this question.