Consider the linear model $Y = X \beta + \epsilon$, where $\beta \in \Bbb R^{p}$. Suppose that $r := \mbox{rank}(X) < p $ and let $C \in \Bbb R^{m\times p}$ be a matrix satisfying the condition
$\mbox{rank}(C) = p-r$,
$\mathcal{R}(X^\mathrm{T})\cap \mathcal{R}(C^\mathrm{T})=\{0\}$.
then
$$(X^\mathrm{T}X+C^\mathrm{T}C)^{-1} \text{ is a generalized inverse of } C^\mathrm{T}C \tag{a}$$
$$C(X^\mathrm{T}X+C^\mathrm{T}C)^{-1}C^\mathrm{T}=I \tag{b}$$
I think one can be obtained by another, since if $C(X^\mathrm{T}X+C^\mathrm{T}C)^{-1}C^\mathrm{T}=I$, then $C^\mathrm{T}C(X^\mathrm{T}X+C^\mathrm{T}C)^{-1}C^\mathrm{T}=C^\mathrm{T}I$, which implies $$C^\mathrm{T}C(X^\mathrm{T}X+C^\mathrm{T}C)^{-1}C^\mathrm{T}C=C^\mathrm{T}IC=C^\mathrm{T}C,$$ Hence we prove the (a). now I am not sure how to prove (b), thank you for help me to figure it out.