The statement to be proved is that C(n, n-k)= C(n, k). I assume that "C(n, i)" is defined in your text but you don't give the definition here!
I think "C(n, i)" is the "binomial coefficient". In many texts, that is defined as . In that case, the proof would be just a straightforward computation. But it appears that, here, the proof of C(n, k) is "the number of subsets, of a set containing n elements, that contain k elements". To prove that C(n, n-k)= C(n, k), using that definition, observe that if subset B, of set A (containing n elements), contains k elements, then the complement of B contains the other n- k elements of A. That is, to every subset of A with k elements, there is associated one with n-k elements.
That's the point of the two functions. is the function that to each subset B assigns its complement and is the inverse function.