In his book 'Likelihood based inference in cointegrated Var', in order to get the expression for the Granger's representation theorem,, Johansen claims that:
(1)
$$\beta \bot(\alpha' \bot \beta \bot )^{-1} \alpha' \bot + \alpha (\beta' \alpha)^{-1} \beta' = I $$
Where:
$\alpha$ is NxR $\text{rank}(\alpha) =R$
$\beta$ is NxR $\text{rank}(\beta) =R$
$\beta \bot $ is NxN-R $\text{rank}(\beta \bot) =N-R$
$\alpha \bot $ is NxN-R $\text{rank}(\alpha \bot) =N-R$
$\alpha' \alpha \bot =0$
$\beta' \beta \bot =0$
I am not able to prove (1). Can you help me, please?