Lemma 11.3.3 (Summation by Parts, [Proposition 1.4, Lan93]). Let $[a, b] \subset \real$, $E_{1}, E_{2}, H$ be TVSs over $F \in \RC$, $E_{1} \times E_{2} \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, $f: [a, b] \to E_{1}$, $G: [a, b] \to E_{2}$, and $(P, c) \in \scp_{t}([a, b])$, then
\[S(P, c, f, G) + S(P', c', G, f) = f(b)G(b) - f(a)G(a)\]
where $P' = \seqfz[n+1]{y_j}= [a, c_{1}, \cdots, c_{n}, b]$ and $c' = \seqf[n+1]{d_j}= [x_{0}, \cdots, x_{n}]$.
Proof. Denote $c_{0} = a$ and $c_{n+1}= b$, then
\begin{align*}S(P, c, f, G)&= \sum_{j = 1}^{n} f(c_{j})[G(x_{j}) - G(x_{j - 1})] = \sum_{j = 1}^{n} f(c_{j})G(x_{j}) - \sum_{j = 1}^{n} f(c_{j})G(x_{j - 1}) \\&= f(c_{n})G(x_{n})- f(c_{0})G(x_{0}) + \sum_{j = 1}^{n} f(c_{j - 1})G(x_{j-1}) - \sum_{j = 1}^{n} f(c_{j})G(x_{j - 1}) \\&= f(c_{n})G(x_{n})- f(c_{0})G(x_{0}) - \sum_{j = 1}^{n} G(x_{j - 1})[f(c_{j}) - f(c_{j - 1})] \\&= f(c_{n+1})G(x_{n}) - f(c_{0})G(x_{0}) - \sum_{j = 1}^{n+1}G(x_{j - 1})[f(c_{j}) - f(c_{j - 1})] \\&= f(b)G(b) - f(a)G(a) - S(P', c', G, f)\end{align*}
$\square$