[proofplan]
The shelling order decomposes the face set of $K$ into disjoint intervals $[R_j,F_j]$, one interval for each facet. We count faces by cardinality inside these intervals: a facet whose restriction face has size $i$ contributes exactly $\binom{d-i}{r-i}$ faces of cardinality $r$. Substituting this count into the defining generating-function relation for the $h$-vector gives a polynomial whose coefficient of $t^{d-i}$ is precisely the number of restriction faces of size $i$.
[/proofplan]
[step:Decompose the face set into shelling intervals]
For each $j \in \{1,\dots,m\}$, define the interval
\begin{align*}
[R_j,F_j] := \{G \in K : R_j \subseteq G \subseteq F_j\}.
\end{align*}
By the defining property of the restriction face in a shelling, the faces introduced at the $j$-th stage are exactly the faces in $[R_j,F_j]$. Therefore the intervals $[R_j,F_j]$ are pairwise disjoint and their union is the whole face set of $K$:
\begin{align*}
K = \bigsqcup_{j=1}^{m} [R_j,F_j].
\end{align*}
[guided]
Let $K_j$ denote the subcomplex generated by the first $j$ facets, so $K_j$ has facets $F_1,\dots,F_j$. The restriction face $R_j$ is defined so that the faces of $F_j$ that were not already present in $K_{j-1}$ are exactly those faces $G$ with $R_j \subseteq G \subseteq F_j$.
Thus, at stage $j$, the new faces are precisely the interval
\begin{align*}
[R_j,F_j] := \{G \in K : R_j \subseteq G \subseteq F_j\}.
\end{align*}
Every face of $K$ first appears at a unique stage of the shelling, namely the smallest $j$ such that the face is contained in $F_j$. Hence every face lies in exactly one interval $[R_j,F_j]$. This gives the disjoint union
\begin{align*}
K = \bigsqcup_{j=1}^{m} [R_j,F_j].
\end{align*}
This interval decomposition is the mechanism that lets us convert shelling data into face-number data.
[/guided]
[/step]
[step:Count faces of each cardinality in one interval]
For $r \in \{0,\dots,d\}$, let $f_{r-1}$ denote the number of faces of $K$ with cardinality $r$, so $f_{r-1}$ is the usual face number in dimension $r-1$. Since $K$ is pure of dimension $d-1$, every facet $F_j$ has cardinality $d$.
Fix $j \in \{1,\dots,m\}$ and set $i := |R_j|$. A face $G \in [R_j,F_j]$ has cardinality $r$ exactly when it is obtained by adjoining $r-i$ vertices from $F_j \setminus R_j$ to $R_j$. Since $|F_j \setminus R_j| = d-i$, the number of such faces is
\begin{align*}
\binom{d-i}{r-i},
\end{align*}
with the convention that this binomial coefficient is $0$ when $r<i$.
Define, for each $i \in \{0,\dots,d\}$,
\begin{align*}
a_i := \#\{j \in \{1,\dots,m\} : |R_j| = i\}.
\end{align*}
Using the disjoint interval decomposition, we obtain for every $r \in \{0,\dots,d\}$:
\begin{align*}
f_{r-1} = \sum_{i=0}^{d} a_i \binom{d-i}{r-i}.
\end{align*}
[/step]
[step:Insert the interval count into the defining relation for the $h$-vector]
The usual $h$-vector $h(K)=(h_0,\dots,h_d)$ is defined by the identity
\begin{align*}
\sum_{r=0}^{d} f_{r-1}(t-1)^{d-r} = \sum_{i=0}^{d} h_i t^{d-i}.
\end{align*}
Substituting the formula for $f_{r-1}$ gives
\begin{align*}
\sum_{r=0}^{d} f_{r-1}(t-1)^{d-r} = \sum_{r=0}^{d} \sum_{i=0}^{d} a_i \binom{d-i}{r-i}(t-1)^{d-r}.
\end{align*}
Because all sums are finite, we may interchange the order of summation:
\begin{align*}
\sum_{r=0}^{d} f_{r-1}(t-1)^{d-r} = \sum_{i=0}^{d} a_i \sum_{r=i}^{d} \binom{d-i}{r-i}(t-1)^{d-r}.
\end{align*}
For fixed $i$, set $s := r-i$. Then $s$ ranges from $0$ to $d-i$, and the inner sum becomes
\begin{align*}
\sum_{s=0}^{d-i} \binom{d-i}{s}(t-1)^{d-i-s}.
\end{align*}
By the [binomial theorem](/theorems/750), this equals
\begin{align*}
((t-1)+1)^{d-i} = t^{d-i}.
\end{align*}
Therefore
\begin{align*}
\sum_{r=0}^{d} f_{r-1}(t-1)^{d-r} = \sum_{i=0}^{d} a_i t^{d-i}.
\end{align*}
[/step]
[step:Compare coefficients to identify the entries of the $h$-vector]
The defining identity for the $h$-vector gives
\begin{align*}
\sum_{r=0}^{d} f_{r-1}(t-1)^{d-r} = \sum_{i=0}^{d} h_i t^{d-i}.
\end{align*}
The interval count gives
\begin{align*}
\sum_{r=0}^{d} f_{r-1}(t-1)^{d-r} = \sum_{i=0}^{d} a_i t^{d-i}.
\end{align*}
Hence
\begin{align*}
\sum_{i=0}^{d} h_i t^{d-i} = \sum_{i=0}^{d} a_i t^{d-i}.
\end{align*}
Equality of polynomials implies equality of coefficients, so for every $i \in \{0,\dots,d\}$,
\begin{align*}
h_i = a_i = \#\{j \in \{1,\dots,m\} : |R_j| = i\}.
\end{align*}
This is exactly the claimed shelling interpretation of the $h$-vector.
[/step]
