1. Define a general loss random variable

Suppose the loss random variable $L$ has
$$\mathbb{E} [L] = \mu \quad, \quad \text{var}(L) = \sigma^2$$
then as a function of $L$, we can define the Value at Risk at confidence level $\alpha$, $\text{VaR}_{\alpha}$ as,
$$ \text{VaR}_\alpha (L) = \mu + \sigma z_\alpha = F^{-1} (\alpha)\tag{1}$$

where $z_{\alpha}$ is the $\alpha$-quantile of the standardized loss random variable $Z$.

$$ Z = \dfrac{L \; – \mu}{\sigma}\tag{2}$$

2. Daily loss rate random vector $\mathbf{l}$ of a portfolio, comprising losses $l_i$ for $n$ constituent assets $i$. (Based on the definition of the general loss random variable $L$)

Key assumption, let
$$ \mathbf{l} \sim \ (\mathbf{e}, \mathbf{V})$$

be the daily loss rate random vector of joint normally distributed daily losses. Where,

\begin{align}
\mathbf{e} = \mathbb{E} \bigl[ \mathbf{l} \bigr] \quad \text{with} \quad
\mathbf{e} = \left[\begin{array}{c}\mu_1\\ \vdots\\ \mu_n\end{array}\right] \quad \text{and} \quad \mathbf{l} = \left[\begin{array}{c}l_1\\ \vdots\\ l_n\end{array}\right]
\end{align}
and
$$\mathbf{V} = \mathbb{E} \bigl[ \underbrace{(\mathbf{l} \; – \mathbf{e})(\mathbf{l} \; – \mathbf{e})^T }_{\text{n $\times$ n outer product}} \bigr]$$
of which the $(i,j)^{\text{th}}$ entry is
$$({l_i} \; – {\mu_i})({l_i} \; – {\mu_i}) \qquad (1 \leqslant i, j \leqslant n)$$

3. Value at Risk of a portfolio of n-weighted assets.

From the assumptions made, we can say any linear combination of the components of $l$ is a normal random variable. That is the portfolio loss,
$$l^{(p)} = \sum_{i=1}^n w_i l_i \sim N(\mu, \sigma^2)$$
where we know from linear algebra that the sum of all $w_i l_i$ coompnents is the inner product
$$\mu = \mathbf{w}^T\mathbf{e} \quad \text{and similarly} \quad \sigma^2 = \mathbf{w}^T\mathbf{w}$$
In view of this we can write
$$l^{(p)} = \mu + \sigma Z \quad \text{where} \quad Z \sim N(0,1)$$
and portfolio loss r.v.
$$\text{L} = \text{V}_t l^{(p)} = \text{V}_t (\mu + \sigma Z)$$
such that VaR(L)
$$\text{VaR}_{\alpha}(\text{L)} = \mu + \sigma z_\alpha \tag{3}$$
which is the same as $(1)$ with $z_{\alpha}$ being the $\alpha$-quantile of the standardized loss random variable $Z$. This means
$$\Phi(z_{\alpha}) = \dfrac{1}{2\pi} \int_{-\infty}^{z_{\alpha}} e^{-{x^2/2}} dx \;=\; \alpha$$

statistical tables document values of the integral at discrete values of $z_{\alpha}$.