= Solution
Fix $t$ and let $F_t(u)=(u-t)_+^{k-1}$. Initially assume distinct <spline knots>, so the ordinary <polynomial interpolation> definitions are unambiguous. The two interpolants $\ell_i(\cdot,t)$ and $\ell_{i+1}(\cdot,t)$ agree with $F_t$ at their $k-1$ shared nodes $t_{i+1},\ldots,t_{i+k-1}$. Their difference has degree at most $k-1$ and all these roots, hence
$$
\ell_{i+1}(x,t)-\ell_i(x,t)=c_i(t)\prod_{r=1}^{k-1}(x-t_{i+r})=c_i(t)\omega_i(x).
$$
For $k=1$ the product is empty and the same statement concerns constants. The <leading coefficient> of an interpolating polynomial is its highest-order <divided difference>, so the <divided difference> recurrence gives
$$
\begin{aligned}
c_i(t)&=[t_{i+1},\ldots,t_{i+k}]F_t-[t_i,\ldots,t_{i+k-1}]F_t\\
&=(t_{i+k}-t_i)[t_i,\ldots,t_{i+k}]F_t=N_i(t).
\end{aligned}
$$
This proves <Lee's formula>:
$$
\boxed{\ell_{i+1}(x,t)-\ell_i(x,t)=\omega_i(x)N_i(t).}
$$
Summing over $i$ telescopes to $\ell_{n+1}(x,t)-\ell_1(x,t)$. If $t_k<t<t_{n+1}$, the first interpolation nodes $t_1,\ldots,t_k$ are all smaller than $t$, so their truncated-power data vanish and $\ell_1=0$. The last nodes $t_{n+1},\ldots,t_{n+k}$ are larger than $t$, so their data are values of the polynomial $(u-t)^{k-1}$ and uniqueness gives $\ell_{n+1}(x,t)=(x-t)^{k-1}$. Thus \b[the telescoped identity is]
$$
\boxed{(x-t)^{k-1}=\sum_{i=1}^n\omega_i(x)N_i(t),\qquad t_k<t<t_{n+1},}
$$
the <Marsden identity>, valid for every real $x$.
If repeated <spline knots> are allowed, ordinary value interpolation at repeated nodes does not specify $\ell_i$ uniquely. The standard extension uses confluent <divided differences> and <Hermite interpolation>, or limits of distinct knots: the same polynomial identity then persists wherever these quantities are defined. At a maximally repeated knot where the truncated power lacks the required derivative, the appropriate one-sided spline convention must be fixed; an unspecified classical derivative is not part of the proof. For distinct knots, the printed identity requires no such qualification.
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