= Solution
The <divided difference> is a finite linear combination of <spline knot> values, so it commutes with the finite integral. For every <spline knot> $t_j$ in the defining interval,
$$
\int_{t_i}^{t_{i+k}}(t_j-t)_+^{k-1}\,dt=\frac{(t_j-t_i)^k}{k}.
$$
It follows that
$$
\int_{t_i}^{t_{i+k}}M_i(t)\,dt=[t_i,\ldots,t_{i+k}](u-t_i)^k=1,
$$
because the order-$k$ <divided difference> of a monic degree-$k$ <polynomial> is its leading coefficient. This proves the <unit-integral normalization of a B-spline>.
For the <partition of unity>, let $N_{i,r}$ denote the partition-normalized order-$r$ <B-spline>. The <Cox-de Boor recurrence> is
$$
N_{i,r}(t)=\frac{t-t_i}{t_{i+r-1}-t_i}N_{i,r-1}(t)+\frac{t_{i+r}-t}{t_{i+r}-t_{i+1}}N_{i+1,r-1}(t).
$$
It follows by substituting the explicit <divided difference> formula into both sides and collecting each <truncated power function>. The starting case is $N_{i,1}=\mathbf1_{[t_i,t_{i+1})}$, with endpoint conventions irrelevant when $r\ge2$. The two weights are nonnegative wherever their respective lower-order <B-splines> are nonzero. Induction therefore gives $N_{i,r}\ge0$ and strict positivity in its open <support>.
Extend the finite <spline knot sequence> strictly in both directions, with no finite accumulation. At each point only finitely many <B-splines> are nonzero. For the full extended family the sum is one for order one. Summing the <Cox-de Boor recurrence>, the combined coefficient of $N_{j,r-1}$ is
$$
\frac{t-t_j}{t_{j+r-1}-t_j}+\frac{t_{j+r-1}-t}{t_{j+r-1}-t_j}=1.
$$
Induction gives a full <partition of unity> at every order. On $t_k<t<t_{n+1}$, all extended order-$k$ <functions> with index $i\le0$ or $i\ge n+1$ vanish by their supports. Hence
$$
\boxed{\sum_{i=1}^nN_i(t)=1\quad(t_k<t<t_{n+1}),\qquad\int M_i=1.}
$$
The same argument gives the useful global <subpartition of unity for B-splines>, $0\le\sum_{i=1}^nN_i(t)\le1$, including outside the basic <spline knot> interval.
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