Extended Kalman Filter formulation

For a nonlinear system,
$$
begin{align}
&{boldsymbol x}(k+1)={boldsymbol f}({boldsymbol x}(k),{boldsymbol u}(k),{boldsymbol w}(k))
&{boldsymbol y}(k)={boldsymbol h}({boldsymbol x}(k),{boldsymbol v}(k))
end{align}
$$

I would like to find the Extended Kalman Filter (EKF), having a look at wikipedia, I have reached

$$
begin{align}
& hat {boldsymbol x}(k+1|k)={boldsymbol f}(hat {boldsymbol x}(k|k),{boldsymbol u}(k|k))
& {boldsymbol P}(k+1|k)={boldsymbol F}(k){boldsymbol P}(k|k){boldsymbol F}^T(k)+{boldsymbol L}(k) {boldsymbol Q}(k) {boldsymbol L}^T(k)
end{align}
$$

and

$$
begin{align}
&{boldsymbol S}(k+1)={boldsymbol H}(k+1){boldsymbol P}(k+1|k){boldsymbol H}^T(k+1)+{boldsymbol M}(k+1){boldsymbol R}(k+1){boldsymbol M}^T(k+1)
& {boldsymbol K}(k+1)={boldsymbol P}(k+1|k) {boldsymbol H}^T (k+1){boldsymbol S}^{-1}(k+1)
& hat{boldsymbol x}(k+1|k+1)=hat{boldsymbol x}(k+1|k)+{boldsymbol K}(k+1)[y(k+1)-{boldsymbol h}(hat {boldsymbol x}(k+1|k))]
& {boldsymbol P}(k+1|k+1)k=[{boldsymbol I}-{boldsymbol K}(k+1){boldsymbol H}(k+1)]{boldsymbol P}(k+1|k)
end{align}
$$

where

$$
begin{align}
& {boldsymbol F}(k)=frac{partial boldsymbol f}{partial boldsymbol x}|_{hat {boldsymbol x}(k|k),{boldsymbol u}(k)}
& {boldsymbol H}(k+1)=frac{partial boldsymbol h}{partial boldsymbol x}|_{hat {boldsymbol x}(k+1|k)}
&{boldsymbol L}(k)=frac{partialboldsymbol f}{partialboldsymbol w}|_{hat {boldsymbol x}(k|k),{boldsymbol u}(k)}
&{boldsymbol M}(k+1)=frac{partialboldsymbol h}{partialboldsymbol v}|_{hat {boldsymbol x}(k+1|{color{red}{k+1}})}
end{align}
$$

Is this obtained formulation correct?

My main concerns are about the whole timing indexes especially the last one.