Variance calculation in matrix notation for $var(z-Ax)$ I noted from a post here that
$$var(z - \mathbf{A}x)=var(z)+var(\mathbf{A}x)-\mathbf{A}cov(z, -x)-cov(z,-x)\mathbf{A}^T (Eq. 1)$$ (I dropped the conditional part in the original formula from the post since I believe it irrelevant to my question. If I write the above formula incorrectly, please refer to the original formula in the aforementioned post.)
Additionally, I noted:
$$var(x_1 + \mathbf{A}x_2) = var(x_1)+\mathbf{A}var(x_2)\mathbf{A}^T+\mathbf{A}cov(x_1, x_2)+cov(x_2,x_1)\mathbf{A}^T  (Eq. 2)$$
I do not particularly understand the $\mathbf{A}cov(z, -x)-cov(z,-x)\mathbf{A}^T$ part in $Eq.1$ and the $\mathbf{A}cov(x_1, x_2)+cov(x_2,x_1)\mathbf{A}^T$ part in $Eq.2$.
I wish to see a step-by-step proof of how the two $cov$ components are derived, particularly for $Eq.1$. (No proof of $Eq.2$ is needed if the proofs are similar.)

I followed User1865345's approach for Eq1:
$$
var(z-Ax) = E[ [(z-\mu_z)-A(x-\mu_x)] [(z-\mu_z)-A(x-\mu_x)]^T ]
= E[ (z-\mu_z)(z-\mu_z)^T - (z-\mu_z)(x-\mu_x)^TA^T - A(x-\mu_x)(z-\mu_z)^T + A(x-\mu_x)(x-\mu_x)^TA^T  ]
= var(z) - cov(z, x)A^T - Acov(x, z) + var(Ax)
$$
Unfortunately, the above result is not identical from Eq.1. As I mentioned in my question, I did omitted conditional expression in Eq1 from the original post. I wish to know if there is anything wrong with my proof above.

Discussing about $\mathrm{Eq}. ~(2),$ with $\mathbf X_i, ~\mathbb E\mathbf X_i:=\boldsymbol\mu_i, ~i\in\{1, 2\}$ and $\bf A$ being a non-stochastic matrix,
\begin{align}
\mathbb{Var}(\mathbf X_1 + \mathbf{ AX_2}) &= \mathbb E[(\mathbf X_1-\boldsymbol \mu_1) + \mathbf A(\mathbf X_2-\boldsymbol\mu_2) ][(\mathbf X_1-\boldsymbol \mu_1) + \mathbf A(\mathbf X_2-\boldsymbol\mu_2) ]^\mathsf T\\ &= \mathbb E\left[(\mathbf X_1-\boldsymbol \mu_1) (\mathbf X_1-\boldsymbol \mu_1) ^\mathsf T+\mathbf A(\mathbf X_2-\boldsymbol\mu_2)(\mathbf X_1-\boldsymbol \mu_1) ^\mathsf T+(\mathbf X_1-\boldsymbol \mu_1)(\mathbf X_2-\boldsymbol\mu_2)^\mathsf T\mathbf A^\mathsf T + \mathbf A(\mathbf X_2-\boldsymbol\mu_2)(\mathbf X_2-\boldsymbol\mu_2)^\mathsf T\mathbf A^\mathsf T\right]\\&= \mathbb {Var}(\mathbf X_1) +\mathbf A\mathbb{Cov}(\mathbf X_2, \mathbf X_1) +\mathbb{Cov}(\mathbf X_1, \mathbf X_2)\mathbf A^\mathsf  T+ \mathbf A\mathbb{Var}(\mathbf X_2) \mathbf A^\mathsf T. 
\end{align}
