I need to prove that the expected value of MSA is equal to the expression to the right. Where MSA is the mean square of a factor a. The equations are easier to understand in the link of the images.
$$ \mathrm{MSA} = \frac{\mathrm{SSA}}{\mathrm{gla}}, \qquad \mathrm{SSA} = \sum_{i} (\bar{Y}_i - \bar{y})^2 $$
$$ \mu = \frac{1}{a}\sum_{i=1}^{a} \mu_i, $$
$$ \mathbb{E}[\mathrm{MSA}]=\sigma^2+\frac{n}{a-1}\sum_{i=1}^{a}(\mu_i−\mu)^2. $$ asume the experiment is balanced.