Moore-Penrose Inverse of sum of two matrices using spectral decomposition

Question

Suppose $X$ is $n \times n$ symmetric with $rank(X)=p$.

If Y is $n \times n$ matrix with $rank(Y) = n-p$, such that $XY=(0)$ then it follows that $X^{+}X + Y^{+}Y = I_{n}$

How to show this? I believe we need to use spectral decomposition of X to show the result.