First, fix point in time $T_0,...,T_n$ whereas $T_1,...,T_n$ are the coupon dates and $T_0$ is interpreted as the emission date of the bond. At time $T_i$, $i = 1,...,n$ the owner of the bond receives $c_i$. If the coupon rate $r_i$ is set to the spot **LIBOR** rate $L(T_{i−1}, T_i)$ ,then
\begin{align}
c_i=(T_i-T_{i−1})L(T_{i−1}, T_i)K
\end{align}
We now go on to compute the value of this bond at some time $t < T_0$, in the case when the coupon dates are equally spaced, with $T_i−T_{i−1}=\delta$, and to this end we study the individual coupon $c_i$. Without loss of generality we may assume that $K = 1$, and inserting the definition of the LIBOR rate
\begin{align}
c_i=\frac{1}{P(T_{i−1}, T_i)}-1
\end{align}
The value at $t$, of the term $−1$ , is of course equal to $-P(t, T_i)$ and it remains to compute the value of the term$\frac{1}{P(T_{i−1}, T_i)}$ which is paid out at $T_i$.This is, however, easily done through the following argument.

 1. Buy, at time $t$, one $T_{i−1}$-bond. This will cost $P(t, T_{i−1})$.
 2. At time $T_{i−1}$ you will receive the amount $1$.
 3. Invest this unit amount in $T_{i−1}$-bond. This will give you exactly 
$\frac{1}{P(T_{i−1}, T_i)}$ bonds.
 4. At  $\,T_i$ the bonds will mature, each at the face value $1$. Thus, at time $T_i$, you will obtain the amount $\frac{1}{P(T_{i−1}, T_i)}$

Thus the value at $t$, of obtaining  $\frac{1}{P(T_{i−1}, T_i)}$ at $T_i$, is given by $P(t, T_{i-1})$, and the value at t of the coupon
$c_i$ is $P(t, T_{i−1}) − P(t, T_i)$. then, by summing up all values of $c_i$ and the value of the notional amount (i.e. 1) at $T_n$, 
 we have
$$P(t)=P(t,T_{n})+\sum_{i=1}^{n}P(t,T_{i-1})-P(t,T_{i})=p(t,T_0)$$