Derivation of an identity in HMM model Problem
In HMM model, the forward and backward probability is defined as
$$
\begin{aligned}
\alpha_t(i) &= \Pr(o_1,\cdots, o_t, i_t=q_i\vert \lambda)\\
\beta_t(i) &= \Pr(o_{t+1},\cdots, o_T\vert i_t=q_i,\lambda)
\end{aligned}
$$
where $o_i$ is observation, $i_t$ is state and $\lambda$ is HMM model parameter.
One commonly used identity is 
$$
\alpha_t(i)\beta_t(i)=\Pr(i_t=q_i, o_1,\cdots,o_t,\cdots, o_T\vert \lambda)
$$
However, I am not sure how this is derived. Specifically, I do not know how to combine two events $o_1,\cdots,o_t$ and $o_{t+1},\cdots,o_T$ into one $o_1,\cdots, o_T$.
Could anyone help me, thank you in advance.
Edit
Sorry I did not specify what is $\lambda$, the book I read define it as a triplet
$$
\lambda = (\mathbf{A}, \mathbf{B},\mathbf{\pi})
$$
where $\mathbf{A}$ is the state transition matrix, $\mathbf{B}$ is the observation matrix and $\mathbf{\pi}$ is the initial state probability vector. More specifically,
$$
\begin{aligned}
\mathbf{A}_{ij}&=\Pr(i_{t+1}=q_j\vert i_t=q_i)\\
\mathbf{B}_{jk}&=\Pr(o_t=v_k\vert i_t=q_i)\\
\mathbf{\pi}_i&=\Pr(i_1=q_i)
\end{aligned}
$$
 A: In this answer, we’ll assume that $\lambda$ is a collection of deterministic parameters that specify the HMM (but this needs to be clarified in the question). First, notice that
\begin{align}
\beta_t(i) &= \Pr(o_{t+1},\cdots, o_T\vert i_t=q_i,\lambda) \\
&= \Pr(o_{t+1},\cdots, o_T\vert i_t=q_i, o_1,\cdots,o_t, \lambda)
\end{align}
because the observations from time $t+1$ to $T$ are independent from past observations. This is an assumption in the HMM model, which can be seen when looking at the graph structure of states and observations, where no connections are drawn between the observations, so that the observations depend only on states and not on each other.
Also, recall that
$$\Pr(x,y|\lambda) = \Pr(y|x, \lambda)\Pr(x|\lambda),$$
for any random variables $x$, $y$, and $\lambda$, which is a basic property of conditional probability  (although $\lambda$ is assumed not to be a random variable in our setting). Applying this property, we have
\begin{align}
\Pr(i_t=q_i, o_1,\cdots,o_t,\cdots, o_T\vert \lambda) &= \Pr(o_{t+1},\cdots, o_T\vert i_t=q_i, o_1,\cdots,o_t, \lambda)\Pr(i_t=q_i, o_1,\cdots, o_t \vert \lambda).
\end{align}
Substitute the above equation for $\beta_t(i)$ to get
$$\beta_t(i)\Pr(i_t=q_i, o_1,\cdots, o_t \vert \lambda).$$
Substitute your equation for $\alpha_t(i)$ to get $\beta_t(i)\alpha_t(i)$.
