This derivation of the filter is based on the explanation by Pradu.
Our goal is to express the mean and covariance of the state vector $\mathbf{x}_t\sim \mathcal{N}(\mathbf{\hat{x}}_t,\mathbf{P}_t)$ that needs to be estimated, conditioned on the measurement vector $\mathbf{y}_t=\mathbf{m}$, i.e mean $\mathbf{x}_{t|\mathbf{y}_t=\mathbf{m}}$ and covariance $\mathbf{P}_{t|\mathbf{y}_t=\mathbf{m}}$.
The state transition equation is
$$ \mathbf{x}_{t+1} = \mathbf{A}_t\mathbf{x}_t + \mathbf{B}_t\mathbf{u}_t + \mathbf{q}_t, $$where $\mathbf{A}_t$ is a state transition matrix, $\mathbf{u}_t \sim \mathcal{N}(\mathbf{\hat{u}}_t,\mathbf{U}_t)$ is an input vector, and $\mathbf{q}_t \sim \mathcal{N}(\mathbf{0},\mathbf{Q}_t)$ is a noise. The measurement equation is
$$ \mathbf{y}_{t} = \mathbf{C}_t\mathbf{x}_t + \mathbf{D}_t \mathbf{u}_t + \mathbf{r}_t, $$where $\mathbf{y}_{t}$ is the measurement, $\mathbf{C}_t$ is the output matrix, $\mathbf{D}_t$ is the feed-forward matrix, and $\mathbf{r}_t \sim \mathcal{N}(\mathbf{0},\mathbf{R}_t)$. It is assumed that $\mathbf{q}_t$, $\mathbf{u}_t$, $\mathbf{x}_t$, and $\mathbf{r}_t$ are uncorrelated.
The mean and covariance of the a priori state estimate (The a priori state estimate is the state estimate before any measurements are taken into account.) follows directly from the mean and covariance of a linear combination of random variables (explained below),
$$ \mathbf{\hat{x}}_{t} = \mathbf{A}_{t-1} \mathbf{\hat{x}}_{t-1} + \mathbf{B}_{t-1} \mathbf{\hat{u}}_{t-1} \\ \mathbf{P}_t = \mathbf{A}_{t-1} \mathbf{P}_{t-1} { \mathbf{A}_{t-1} }^T + \mathbf{B}_{t-1} \mathbf{U}_{t-1} {\mathbf{B}_{t-1}}^T + \mathbf{Q}_{t-1} $$Let $\mathbf{y} = \mathbf{A}\mathbf{x}_1 + \mathbf{B}\mathbf{x}_2$ where $\mathbf{x}_1$ and $\mathbf{x}_2$ are multivariate random variables. The expected value of $\mathbf{y}$ is
$$ \mathbb{E}\left[{\mathbf{y}}\right] = \mathbb{E}\left[{ \mathbf{A}\mathbf{x}_1 + \mathbf{B}\mathbf{x}_2 }\right] \\ = \mathbf{A}\mathbb{E}\left[{\mathbf{x}_1}\right] + \mathbf{B}\mathbb{E}\left[{\mathbf{x}_2}\right]. $$The covariance matrix for $\mathbf{y}$ is
$$ \sigma(\mathbf{y},\mathbf{y}) = \mathbb{E}\left[{ ( \mathbf{y} - \mathbb{E}\left[{\mathbf{y}}\right] ) {( \mathbf{y} - \mathbb{E}\left[{\mathbf{y}}\right] )}^T }\right] \\ = \mathbb{E}\left[{ ( \mathbf{A}(\mathbf{x}_1 -\mathbb{E}\left[{ \mathbf{x}_1}\right]) + \mathbf{B}(\mathbf{x}_2 -\mathbb{E}\left[{ \mathbf{x}_2}\right]) ) {( \mathbf{A}(\mathbf{x}_1 -\mathbb{E}\left[{ \mathbf{x}_1}\right]) + \mathbf{B}(\mathbf{x}_2 -\mathbb{E}\left[{ \mathbf{x}_2}\right]) )}^T }\right] \\ = \mathbf{A} \sigma(\mathbf{x}_1,\mathbf{x}_1){\mathbf{A}}^T + \mathbf{B} \sigma(\mathbf{x}_2,\mathbf{x}_2){\mathbf{B}}^T +\mathbf{A} \sigma(\mathbf{x}_1,\mathbf{x}_2){\mathbf{B}}^T + \mathbf{B} \sigma(\mathbf{x}_2,\mathbf{x}_1){\mathbf{A}}^T. $$If $\mathbf{x}_1$ and $\mathbf{x}_2$ are independent, then
$$ \sigma(\mathbf{y},\mathbf{y}) = \mathbf{A} \sigma(\mathbf{x}_1,\mathbf{x}_1){\mathbf{A}}^T + \mathbf{B} \sigma(\mathbf{x}_2,\mathbf{x}_2){\mathbf{B}}^T. $$The covariance matrix for vector-valued random variables is defined as: $$ \sigma(\mathbf{x},\mathbf{y}) = \mathbb{E}\left[{ ( \mathbf{x} - \mathbb{E}\left[{\mathbf{x}}\right] ) {( \mathbf{y} - \mathbb{E}\left[{\mathbf{y}}\right] )}^T }\right] \\ = \mathbb{E}\left[{\mathbf{x}{\mathbf{y}}^T}\right] - \mathbb{E}\left[{\mathbf{x}}\right]{\mathbb{E}\left[{\mathbf{y}}\right]}^T. $$
Let the joint distribution between $\mathbf{x}_t$ and $\mathbf{y}_t$ be
$$ \left[ \begin{array}{c} \mathbf{x}_t \\ \mathbf{y}_t \end{array} \right] \sim \mathcal{N} \left( \left[ \begin{array}{c} \mathbf{\hat{x}}_t \\ \mathbf{\hat{y}}_t \end{array} \right], \left[ \begin{array}{cc} \mathbf{\Sigma}_{xx,t} & \mathbf{\Sigma}_{xy,t} \\ \mathbf{\Sigma}_{yx,t} & \mathbf{\Sigma}_{yy,t} \end{array} \right] \right) $$The expected measurement from state $\mathbf{x}_t$, input $\mathbf{u}_t$ is
$$ \mathbf{\hat{y}}_{t} = \mathbf{C}_t\mathbf{\hat{x}}_t + \mathbf{D}_t \mathbf{\hat{u}}_t. $$The covariance sub-matrices are $$ \mathbf{\Sigma}_{xx,t} = \mathbf{P}_t \\ \mathbf{\Sigma}_{xy,t} = \mathbb{E}\left[{\mathbf{x}_t {\mathbf{y}_t}^T}\right] - \mathbb{E}\left[{\mathbf{x}_t}\right] {\mathbb{E}\left[{\mathbf{y}_t}\right]}^T \\ = \mathbb{E}\left[{\mathbf{x}_t {\left( \mathbf{C}_t\mathbf{x}_t + \mathbf{D}_t \mathbf{u}_t + \mathbf{r}_t \right)}^T}\right] - \mathbb{E}\left[{\mathbf{x}_t}\right] {\mathbb{E}\left[{\mathbf{C}_t\mathbf{x}_t + \mathbf{D}_t \mathbf{u}_t + \mathbf{r}_t}\right]}^T \\ = \mathbf{P}_t { \mathbf{C}_t }^T \\ \mathbf{\Sigma}_{yx,t} = {\mathbf{\Sigma}_{xy,t}}^T = \mathbf{C}_t \mathbf{P}_t \\ \mathbf{\Sigma}_{yy,t} = \mathbf{C}_t\mathbf{P}_t{\mathbf{C}_t}^T + \mathbf{D}_t \mathbf{U}_t {\mathbf{D}_t}^T + \mathbf{R}_t $$
where $\mathbb{E}\left[{\mathbf{x}}\right]$ denotes expectation of $\mathbf{x}$.
We can show that indeed,
$$ \mathbb{E}\left[{\mathbf{x}_t {\left( \mathbf{C}_t\mathbf{x}_t + \mathbf{D}_t \mathbf{u}_t + \mathbf{r}_t \right)}^T}\right] - \mathbb{E}\left[{\mathbf{x}_t}\right] {\mathbb{E}\left[{\mathbf{C}_t\mathbf{x}_t + \mathbf{D}_t \mathbf{u}_t + \mathbf{r}_t}\right]}^T = \mathbf{P}_t { \mathbf{C}_t }^T $$The expectations of $\mathbf{x}_t$, $\mathbf{u}_t$ and $\mathbf{r}_t$ are $\mathbb{E}\left[{\mathbf{x}_t}\right] = \mathbf{\hat{x}}_t$, $\mathbb{E}\left[{\mathbf{u}_t}\right] = \mathbf{\hat{u}}_t$ and $\mathbb{E}\left[{\mathbf{r}_t}\right] = 0$.
If we multiply in the first term with $\mathbf{x}_t$, we get
$$ \mathbb{E}\left[{\mathbf{x}_t {\left( \mathbf{C}_t\mathbf{x}_t + \mathbf{D}_t \mathbf{u}_t + \mathbf{r}_t \right)}^T}\right] = \mathbb{E}\left[\mathbf{x}_t {\mathbf{x}_t}^T { \mathbf{C}_t}^T + \mathbf{x}_t {\mathbf{u}_t}^T {\mathbf{D}_t}^T + \mathbf{x}_t {\mathbf{r}_t}^T\right] = \\ \mathbb{E}\left[\mathbf{x}_t {\mathbf{x}_t}^T\right]{ \mathbf{C}_t}^T + \mathbb{E}\left[\mathbf{x}_t {\mathbf{u}_t}^T\right]{\mathbf{D}_t}^T + \mathbb{E}\left[\mathbf{x}_t {\mathbf{r}_t}^T\right] $$If we assume that $\mathbf{x}_t$, $\mathbf{u}_t$ and $\mathbf{x}_t$, $\mathbf{r}_t$ are independent random variables,
$$ \mathbb{E}\left[\mathbf{x}_t {\mathbf{x}_t}^T\right] = \sigma(\mathbf{x}_t,\mathbf{x}_t) + \mathbb{E}\left[\mathbf{x}_t\right] \mathbb{E}\left[\mathbf{x}_t\right]^T = \mathbf{P}_t + \mathbf{\hat{x}}_t {\mathbf{\hat{x}}_t}^T \\ \mathbb{E}\left[\mathbf{x}_t {\mathbf{u}_t}^T\right] = \mathbb{E}\left[\mathbf{x}_t\right] \mathbb{E}\left[\mathbf{u}_t\right]^T = \mathbf{\hat{x}}_t {\mathbf{\hat{u}}_t}^T \\ \mathbb{E}\left[\mathbf{x}_t {\mathbf{r}_t}^T\right] = \mathbb{E}\left[\mathbf{x}_t\right] \mathbb{E}\left[\mathbf{r}_t\right]^T = \mathbf{\hat{x}}_t 0 = 0 $$hence the first term is
$$ \mathbf{P}_t {\mathbf{C}_t}^T + \mathbf{\hat{x}}_t {\mathbf{\hat{x}}_t}^T {\mathbf{C}_t}^T + \mathbf{\hat{x}}_t {\mathbf{\hat{u}}_t}^T {\mathbf{D}_t}^T .$$The second part of the second term is
$$ {\mathbb{E}\left[{\mathbf{C}_t\mathbf{x}_t + \mathbf{D}_t \mathbf{u}_t + \mathbf{r}_t}\right]}^T = \\ \mathbb{E}\left[\mathbf{x}_t\right]^T {\mathbf{C}_t}^T + \mathbb{E}\left[\mathbf{u}_t\right]^T {\mathbf{D}_t}^T + \mathbb{E}\left[\mathbf{r}_t\right]^T = \\ {\mathbf{\hat{x}}_t}^T {\mathbf{C}_t}^T + {\mathbf{\hat{u}}_t}^T {\mathbf{D}_t}^T ,$$therefore the second term evaluates to
$$ \mathbf{\hat{x}}_t {\mathbf{\hat{x}}_t}^T {\mathbf{C}_t}^T + \mathbf{\hat{x}}_t {\mathbf{\hat{u}}_t}^T {\mathbf{D}_t}^T . $$If we subtract this term from the first, we get $\mathbf{P}_t {\mathbf{C}_t}^T$ as expected.
Given a measurement $\mathbf{m}$, the conditional distribution for $\mathbf{x}$ given $\mathbf{y}$ is a normal distribution with the following properties
$$ \mathbf{\hat{x}}_{t|\mathbf{y}_t=\mathbf{m}} = \mathbb{E}\left[{\mathbf{x}_t|\mathbf{y}_t=\mathbf{m}}\right] = \mathbf{\hat{x}_t} + \mathbf{\Sigma}_{xy,t} \mathbf{\Sigma}_{yy,t}^{-1}( \mathbf{m} - {\hat{\mathbf{y}}_t} ) \\ \mathbf{P}_{t|\mathbf{y}_t=\mathbf{m}} = \text{Var}(\mathbf{x}_t|\mathbf{y}_t=\mathbf{m}) = \mathbf{\Sigma}_{xx,t} - \mathbf{\Sigma}_{xy,t} \mathbf{\Sigma}_{yy,t}^{-1} \mathbf{\Sigma}_{yx,t} \\ $$Substituting, we get the final equations of the Kalman filter $$ \mathbf{\hat{x}}_{t|\mathbf{y}_t=\mathbf{m}} = \mathbf{\hat{x}_t} + \mathbf{P}_t { \mathbf{C}_t }^T \left( \mathbf{C}_t\mathbf{P}_t{\mathbf{C}_t}^T + \mathbf{D}_t \mathbf{U}_t {\mathbf{D}_t}^T + \mathbf{R}_t\right)^{-1}( \mathbf{m} - \mathbf{C}_t\mathbf{\hat{x}}_t - \mathbf{D}_t \mathbf{\hat{u}}_t ) \\ \mathbf{P}_{t|\mathbf{y}_t=\mathbf{m}} = \mathbf{P}_t - \mathbf{P}_t { \mathbf{C}_t }^T \left( \mathbf{C}_t\mathbf{P}_t{\mathbf{C}_t}^T + \mathbf{D}_t \mathbf{U}_t {\mathbf{D}_t}^T + \mathbf{R}_t\right)^{-1} \mathbf{C}_t \mathbf{P}_t \\ $$
The only thing remained is to show the conditional expectation and covariance of multivariate normal random variables, as it was used in the above derivation.
Let $\mathbf{x}$, $\mathbf{y}$ be jointly normal with means $\mathbb{E}\left[{\mathbf{x}}\right]=\mathbf{\mu}_{\mathbf{x}}$, $\mathbb{E}\left[{\mathbf{y}}\right] = \mathbf{\mu}_{\mathbf{y}}$ and covariance $$ \left[ \begin{array}{cc} \mathbf{\Sigma}_{xx} & \mathbf{\Sigma}_{xy} \\ \mathbf{\Sigma}_{yx} & \mathbf{\Sigma}_{yy} \end{array} \right]. $$
Let us introduce a new variable $\mathbf{z}$, which is a linear combination of variables $\mathbf{x}$ and $\mathbf{y}$. It is Gaussian, since it is a linear combination of Gaussian random variables. $$ \mathbf{z} = \mathbf{x} - \mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1} \mathbf{y}. $$
$\mathbf{z}$ and $\mathbf{y}$ are independent because $\mathbf{z}$ and $\mathbf{y}$ are jointly normal and
$$ \sigma( \mathbf{z}, \mathbf{y} ) = \sigma( \mathbf{x} - \mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1} \mathbf{y}, \mathbf{y} ) \\ \sigma( \mathbf{z}, \mathbf{y} ) = \mathbb{E}\left[{ \left(\mathbf{x} - \mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1} \mathbf{y} \right) {\mathbf{y}}^T }\right] - \mathbb{E}\left[{\mathbf{x} - \mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1} \mathbf{y} }\right] {\mathbb{E}\left[{\mathbf{y}}\right]}^T \\ \sigma( \mathbf{z}, \mathbf{y} ) = \mathbb{E}\left[{ \mathbf{x}{\mathbf{y}}^T }\right] - \mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1} \mathbb{E}\left[{ \mathbf{y} {\mathbf{y}}^T}\right]- \mathbb{E}\left[{\mathbf{x}}\right]{\mathbb{E}\left[{\mathbf{y}}\right]}^T -\mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1} \mathbb{E}\left[{\mathbf{y}}\right] {\mathbb{E}\left[{\mathbf{y}}\right]}^T \\ \sigma( \mathbf{z}, \mathbf{y} ) = \mathbf{\Sigma}_{xy} - \mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1} \mathbf{\Sigma}_{yy} = \mathbf{0}\\ $$Let $\mathbf{t}\sim \mathcal{N}(\mathbf{\mu}_{\mathbf{t}},\mathbf{T})$. Note that because $\mathbf{z}$ and $\mathbf{y}$ are independent, $\mathbb{E}\left[{\mathbf{z}|\mathbf{y}=\mathbf{t}}\right] = \mathbb{E}\left[{\mathbf{z}}\right] = \mathbf{\mu}_{\mathbf{x}} - \mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1}\mathbf{\mu}_{\mathbf{y}}$. The conditional expectation for $\mathbf{x}$ given $\mathbf{y}$ is
$$ \mathbb{E}\left[{\mathbf{x}|\mathbf{y}=\mathbf{t}}\right] = \mathbb{E}\left[{\mathbf{z}+\mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1} \mathbf{y}|\mathbf{y}=\mathbf{t}}\right] \\ \mathbb{E}\left[{\mathbf{x}|\mathbf{y}=\mathbf{t}}\right] = \mathbb{E}\left[{\mathbf{z}|\mathbf{y}=\mathbf{t}}\right]+\mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1} \mathbb{E}\left[{\mathbf{y}|\mathbf{y}=\mathbf{t}}\right] \\ \mathbb{E}\left[{\mathbf{x}|\mathbf{y}=\mathbf{t}}\right] = \mathbf{\mu}_{\mathbf{x}} - \mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1}\mathbf{\mu}_{\mathbf{y}}+\mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1} \mathbf{\mu}_{\mathbf{t}} \\ \mathbb{E}\left[{\mathbf{x}|\mathbf{y}=\mathbf{t}}\right] = \mathbf{\mu}_{\mathbf{x}} + \mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1}( \mathbf{\mu}_{\mathbf{t}} - \mathbf{\mu}_{\mathbf{y}} ) \\ $$The conditional covariance is
$$ \text{Var}(\mathbf{x}|\mathbf{y}=\mathbf{t}) = \text{Var}(\mathbf{z}+\mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1} \mathbf{y}|\mathbf{y}=\mathbf{t}) \\ \text{Var}(\mathbf{x}|\mathbf{y}=\mathbf{t}) = \text{Var}(\mathbf{z}|\mathbf{y}) + \text{Var}(\mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1} \mathbf{y}|\mathbf{y}=\mathbf{t}) \\ \text{Var}(\mathbf{x}|\mathbf{y}=\mathbf{t}) = \text{Var}(\mathbf{z}) + \mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1} \mathbf{T} \mathbf{\Sigma}_{yy}^{-1} \mathbf{\Sigma}_{yx} \\ \text{Var}(\mathbf{x}|\mathbf{y}=\mathbf{t}) = \text{Var}(\mathbf{x} - \mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1} \mathbf{y}) + \mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1} \mathbf{T} \mathbf{\Sigma}_{yy}^{-1} \mathbf{\Sigma}_{yx} \\ \text{Var}(\mathbf{x}|\mathbf{y}=\mathbf{t}) = \mathbf{\Sigma}_{xx} + \mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1} \mathbf{\Sigma}_{yy} {(\mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1})}^T - \mathbf{\Sigma}_{xy} {(\mathbf{\Sigma}_{xy}\mathbf{\Sigma}_{yy}^{-1})}^T - \mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1} \mathbf{\Sigma}_{yx} + \mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1} \mathbf{T} \mathbf{\Sigma}_{yy}^{-1} \mathbf{\Sigma}_{yx} \\ \text{Var}(\mathbf{x}|\mathbf{y}=\mathbf{t}) = \mathbf{\Sigma}_{xx} + \mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1} \mathbf{\Sigma}_{yx} - \mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1}\mathbf{\Sigma}_{yx} - \mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1} \mathbf{\Sigma}_{yx} + \mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1} \mathbf{T} \mathbf{\Sigma}_{yy}^{-1} \mathbf{\Sigma}_{yx} \\ \text{Var}(\mathbf{x}|\mathbf{y}=\mathbf{t}) = \mathbf{\Sigma}_{xx} - \mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1} \mathbf{\Sigma}_{yx} + \mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1} \mathbf{T} \mathbf{\Sigma}_{yy}^{-1} \mathbf{\Sigma}_{yx} \\ \text{Var}(\mathbf{x}|\mathbf{y}=\mathbf{t}) = \mathbf{\Sigma}_{xx} + \mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1} \left(\mathbf{T} - \mathbf{\Sigma}_{yy} \right) \mathbf{\Sigma}_{yy}^{-1} \mathbf{\Sigma}_{yx} \\ $$In summary,
$$ \mathbb{E}\left[{\mathbf{x}|\mathbf{y}=\mathbf{t}}\right] = \mathbf{\mu}_{\mathbf{x}} + \mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1}( \mathbf{\mu}_{\mathbf{t}} - \mathbf{\mu}_{\mathbf{y}} ) \\ \text{Var}(\mathbf{x}|\mathbf{y}=\mathbf{t}) = \mathbf{\Sigma}_{xx} + \mathbf{\Sigma}_{xy} \mathbf{\Sigma}_{yy}^{-1}\left(\mathbf{T} - \mathbf{\Sigma}_{yy}\right) \mathbf{\Sigma}_{yy}^{-1} \mathbf{\Sigma}_{yx} \\ \mathbb{E}\left[{\mathbf{y}|\mathbf{x}=\mathbf{s}}\right] = \mathbf{\mu}_{\mathbf{y}} + \mathbf{\Sigma}_{yx} \mathbf{\Sigma}_{xx}^{-1}( \mathbf{\mu}_{\mathbf{s}} - \mathbf{\mu}_{\mathbf{x}} ) \\ \text{Var}(\mathbf{y}|\mathbf{x}=\mathbf{s}) = \mathbf{\Sigma}_{yy} + \mathbf{\Sigma}_{yx} \mathbf{\Sigma}_{xx}^{-1}\left(\mathbf{S} - \mathbf{\Sigma}_{xx}\right) \mathbf{\Sigma}_{xx}^{-1} \mathbf{\Sigma}_{xy}. $$