Lemma 5.3.3 (Summation by Parts).label Let $[a,b]\suf\R$, $E,F,H$ be TVSs over $K\in \curl{\R,\com}$, $f:[a,b]\to E,G:[a,b]\to F$, $(P,c)\in\scr{P}_{t}([a,b])$ and suppose there exists a continuous bilinear map $E\ti F\ni (x,y)\mto xy\in H$ then
\begin{align*}S(P,c,f,G)+S(P',c',G,f)=f(b)G(b)-f(a)G(a)\end{align*}
for any $P'=\curl{y_j}_{j=0}^{n+1}=[a,c_{1},...,c_{n},b]$ and $c'=\curl{d_j}_{j=1}^{n+1}=[x_{0},...,x_{n}]$.
Proof. We denote $c_{0}=a$ and $c_{n+1}=b$ then reindexing gives
\begin{align*}S(P,c,f,G)&=\sums{n}{j=1}f(c_{j})[G(x_{j})-G(x_{j-1})]\\&=\sums{n}{j=1}f(c_{j})G(x_{j})-\sums{n}{j=1}f(c_{j})G(x_{j-1})\\&=f(c_{n})G(x_{n})-f(c_{0})G(x_{0})+\sums{n-1}{j=0}f(c_{j})G(x_{j})-\sums{n}{j=1}f(c_{j})G(x_{j-1})\\&=f(c_{n})G(x_{n})-f(c_{0})G(x_{0})+\sums{n}{j=1}f(c_{j-1})G(x_{j-1})-\sums{n}{j=1}f(c_{j})G(x_{j-1})\\&=f(c_{n})G(x_{n})-f(c_{0})G(x_{0})-\sums{n}{j=1}G(x_{j-1})[f(c_{j})-f(c_{j-1})]\\&=f(c_{n+1})G(x_{n})-f(c_{0})G(x_{0})-\sums{n+1}{j=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*}
$\square$