Moore–Penrose Pseudoinverse of Squared Matrix Product

Question

We want to show the following equation: \begin{align} \left(C'R'RC\right)^+=\frac{1}{T-1}\left(A^{-1}-\frac{A^{-1}\iota_N\iota_N'A^{-1}}{\iota_N'A^{-1}\iota_N}\right), \end{align} where $C=\left(I_N-\frac{1}{N}\iota_N\iota_N'\right)$, $A=\frac{1}{T-1}R'\left(I_T-\frac{1}{T}\iota_T\iota_T'\right)R$ and $I_N$ and $I_T$ are the identity matrices with dimension $N$ and $T$ respectively. $R\in\mathbb{R}^{T\times N}$. We know that the equation holds numerically and that $C$ is idempotent.