Deriving the Matrix Inversion Lemma for RLS Equations vs the Woodbury Derivation
It is not clear what are you asking but I will try answer both things.

Deriving the Matrix Inversion Lemma

The Matrix Inversion Lemma goes as:

$$ {\left( A + U C V \right)}^{-1} = {A}^{-1} - {A}^{-1} U {\left( {C}^{-1} + V {A}^{-1} U \right)}^{-1} V {A}^{-1} $$

Deriving it is by utilizing these useful identities:
$$\begin{align}
U + U C V {A}^{-1} U &amp; = U C \left( {C}^{-1} + V {A}^{-1} U \right) = \left(A + U C V \right) A^{-1} U \label{auxEqn001}\tag{1} \\
{\left( A + U C V \right)}^{-1} U C &amp; = {A}^{-1} U {\left( {C}^{-1} + V {A}^{-1} U \right)} ^{-1} \label{auxEqn002}\tag{2}
\end{align}$$

Where $ \ref{auxEqn002} $ comes from $ \ref{auxEqn001} $ by multiplying the 2 last terms by $ {\left(A + U C V \right)}^{-1} $ on the left and $ {\left( {C}^{-1} + V {A}^{-1} U \right)}^{-1} $ on the right.

Then it is straight forward: 
$$\begin{align}
{A}^{-1} &amp;= {\left( A + U C V \right)}^{-1} \left( A + U C V \right) {A}^{-1} &amp;&amp; \text{As $ {\left( A + U C V \right)}^{-1} \left( A + UCV \right) = I $} \\
&amp; = \left( A + U C V \right)^{-1} \left( I + UCVA^{-1} \right) \\
&amp; = \left( A + U C V \right)^{-1} + \left( A + U C V \right)^{-1} U C V {A}^{-1} &amp;&amp; \text{Multiplying the terms} \\ 
&amp; = \left( A + U C V \right)^{-1} + {A}^{-1} U {\left( {C}^{-1} + V {A}^{-1} U \right)}^{-1} V {A}^{-1} &amp;&amp; \text{Utilizing \ref{auxEqn002}} \\
&amp; \Rightarrow {\left( A + U C V \right)}^{-1} = {A}^{-1} - {A}^{-1} U {\left( {C}^{-1} + V {A}^{-1} U \right)}^{-1} VA^{-1}
\end{align}$$

The nice thing is we don't need the Matrix Inversion Lemma (Woodbury Matrix Identity) for the Sequential Form of the Linear Least Squares but we can do with a special case of it called Sherman Morrison Formula:

$$ {\left( A + u {v}^{T} \right)}^{-1} = {A}^{-1} - \frac{ {A}^{-1} u {v}^{T} {A}^{-1} }{ 1 + {v}^{T} {A}^{-1} u } $$

The Sherman Morrison Formula is the MIL where $ C = I $, $ U = u $ and $ V = {v}^{T} $.

Deriving the Sequential form of the Linear Least Squares Estimator

In Sequential Form of the Least Squares Estimator for Linear Least Squares Model I derived the sequential form. To add something new I will extend the derivation to the Weighted Linear Least Squares Estimator.