Assume that a linear model is $y = X\beta + \varepsilon$ then the OLS is $\hat{\beta} = (X^{T}X)^{-1}X^{T}y$.
Consider the block matrix $X = (X_1,X_2)$ which is corresponding to $y = X_1\beta_1 + X_2\beta_2 + \varepsilon$.
Let
$\tilde{\beta_2}(\beta_1) = argmin_{\beta_2}||y-X_1\beta_1-X_2\beta_2||^2$
and
$\tilde{\beta_1} = argmin_{\beta_1} |||y-X_2\tilde{\beta_2}(\beta_1) - X_1\beta_1||^2$
Is it true that
\begin{equation} \left( \begin{array}{cc} \tilde{\beta_1} \\ \tilde{\beta_2}(\tilde{\beta_1}) \end{array} \right) = \hat{\beta} \end{equation}