= Solution
The ordinary interpolation formulation is unambiguous for strictly increasing knots; repeated knots require specified confluent interpolation or a limiting convention. First take distinct knots and $k\ge2$, and for fixed $t$ write $F_t(u)=(u-t)_+^{k-1}$. The two degree-at-most-$k-1$ <Lagrange interpolation polynomials> $\ell_i(\cdot,t)$ and $\ell_{i+1}(\cdot,t)$ agree at the $k-1$ shared knots $t_{i+1},\ldots,t_{i+k-1}$. Their difference is therefore
$$
\ell_{i+1}(x,t)-\ell_i(x,t)=c_i(t)\prod_{r=1}^{k-1}(x-t_{i+r})=c_i(t)\omega_i(x).
$$
This remains valid if their difference has lower degree: the <coefficient> is then zero. The <leading coefficient> of a degree-at-most-$k-1$ interpolant is its order-$(k-1)$ <divided difference>. Comparing leading <coefficients> and using the divided-difference recursion gives
$$
\begin{aligned}
c_i(t)&=[t_{i+1},\ldots,t_{i+k}]F_t-[t_i,\ldots,t_{i+k-1}]F_t\\
&=(t_{i+k}-t_i)[t_i,\ldots,t_{i+k}]F_t=N_i(t).
\end{aligned}
$$
Consequently the <Lee interpolation identity> holds for every real $x,t$:
$$
\boxed{\omega_i(x)N_i(t)=\ell_{i+1}(x,t)-\ell_i(x,t).}
$$
No differentiability of the truncated power in its argument was needed for this distinct-node proof.
Sum the identity over $i=1,\ldots,n$. The right side telescopes, giving $\sum_i\omega_i(x)N_i(t)=\ell_{n+1}(x,t)-\ell_1(x,t)$. When $t_k<t<t_{n+1}$, all the nodes for $\ell_1$ lie below $t$, so its interpolation data are zero and $\ell_1\equiv0$. All the nodes for $\ell_{n+1}$ lie above $t$, where $F_t(u)=(u-t)^{k-1}$ is already a <polynomial> of the admissible degree. Uniqueness of <polynomial> interpolation gives $\ell_{n+1}(x,t)=(x-t)^{k-1}$. Therefore the <Marsden identity> is
$$
\boxed{(x-t)^{k-1}=\sum_{i=1}^n\omega_i(x)N_i(t),\qquad t_k<t<t_{n+1},\quad x\in\mathbb R.}
$$
For order one the empty product is one, the interpolants are constants and a consistent half-open interval convention for $(u-t)_+^0$ gives the same telescoping proof. Nondecreasing repeated-knot sequences can be treated by the appropriate confluent or limiting convention wherever that definition is valid; merely specifying repeated data values does not uniquely determine the interpolation <polynomials>. The basic open knot interval in the <Marsden identity> avoids endpoint convention ambiguity.
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