I want to write a sentence like
We consider the equation f(x)=0. We show (the) existence of (a) solution and (the) uniqueness of (the) solution.
I want to know if the above sentence uses correctly "the" and "a". In particular, I don't know if I should put "the" in front of "existence" or "uniqueness".
I know that "the" is specific and "a" is general. But I cannot decide from this which or no article is good for the above sentence.