Question about a paper on Calculating All Possible Regressions I am currently reading the paper "Computational Efficiency in All Possible Regressions" by Liu and it mentions the following.

A quick explanation of what I understand: We have a set of $k$ variables. $\Omega$ denotes the set of these variables, $\Omega_2$ denotes the set of the variables already in the model and $\Omega_1$ denotes the set of variables to be added in the model.
[Edit - Equations]
Equation (2.1)
$$\Theta_{i_1,i_2,...,i_m} = SSE\left(\{x_i\}_{i\in\Omega_2}\right) - SSE\left(\{x_i\}_{i\in\Omega}\right) = SSR\left(\{x_i\}_{i\in\Omega_1} \mid \{x_i\}_{i\in\Omega_2}\right)$$
Question 1
From the definition in (2.1), I understand that $\Theta_{i_1,i_2,...,i_m}$ is the difference between the Sum of Squares due to Error (SSE) of the current model (containing $\Omega_2$) and the full model (containing $\Omega$, i.e. all the available variables). The author then says that this is equal to the "extra Sum of Squares due to Regression for adding $\{x_i\},i\in\Omega_1$, given the variables $\{x_i\}, i\in\Omega_2$ are already in the model."  How do we arrive in this equality?
Question 2
When we are to add one variable, i.e. $\Omega_1 = \{j\}$, then $\Theta_j = b_j^2/c_{jj}$. Again, how can one prove this? I noticed that this quantity is similar to the t-statistic $t = \frac{\hat{\beta_j}}{\sqrt{\hat{V}\left(\hat{\beta_j}\right)}} = \frac{\hat{\beta_j}}{S\sqrt{c_{jj}}}$ which is used when testing the null hypothesis $H_0: \beta_j = 0$
 A: I believe I have the answer for Question 2.
We have that
$$t_{n-k-1} = \frac{\hat{\beta_j}}{\sqrt{\hat{V}\left(\hat{\beta_j}\right)}} = \frac{\hat{\beta_j}}{S\sqrt{c_{jj}}}$$
where
$$S^2 =\frac{SSE}{n-k-1}$$
Note that $t_{n-k-1}$ is the statistic for the $t$-test for the hypothesis below.
I will denote the full model of $k$ variables as $M$ and the current model with the $\Omega_2$ variables as $M_2$. Of course, $M_2 \subseteq M$.
Then,
$$\Theta_{i_1,i_2,...,i_m} = SSE\left(\{x_i\}_{i\in\Omega_2}\right) - SSE\left(\{x_i\}_{i\in\Omega}\right) = SSE_2 - SSE$$
We assume that the current model $M_2$ excludes only one variable, say $j$, from the full model $M$ (i.e.$\beta_j  = 0$), and that the rest of the variables with coefficients $\beta_i, i\neq j$ are unrestricted.
We also notice that
$$ \Theta_j = SSE_2 - SSE \implies \frac{ \Theta_j}{S^2} = \frac{SSE_2 - SSE}{S^2} = \frac{SSE_2 - SSE}{\frac{SSE}{n-k-1}} \sim F_{1,n-p} $$
Which means that $\frac{ \Theta_j}{S^2}$ is the $F$-statistic for testing the hypothesis
$$
H_0: \beta_j = 0 , \beta_i \text{ unrestricted } \forall i \neq j \left(M_2\right)
$$
$$
H_1: \beta_i \text{ unrestricted } \forall i  \left(M\right)
$$
Also,
$$
F_{1,n-k-1} \equiv t^2_{n-k-1} = \frac{ \left(\hat{\beta_j} - \beta_{j_\left(0\right)} \right)^2}{S^2c_{jj}} = \frac{ \hat{\beta_j}^2}{S^2c_{jj}}
$$
Finally,
$$
\frac{\Theta_j}{S^2} =  \frac{ \hat{\beta_j}^2}{S^2c_{jj}} \implies 
\Theta_j =  \frac{ \hat{\beta_j}^2}{c_{jj}}
$$
Edit:
I believe what I wrote in the last two lines is not entirely correct. It is true that if
$$ X \sim t_{n-k-1} \implies X^2 \sim F_{1,n-k-1}$$
However, it is NOT generally true that if
$$ X \sim F_{1,n-k-1} \text{ and } Y \sim F_{1,n-k-1} \text{ then } X \equiv Y$$
which is what I have used for the result. Does anyone have a hint on this?
