Theorem: Sum of Riemann-Stieltjes Integrals on Adjacent Intervals/Whole to Part

Let $a < c < b$ be real numbers.
Let $f, \alpha$ be real functions that are bounded on $\closedint a b$
Suppose that $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $\closedint a b$, and also on one of the two intervals $\closedint a c$ and $\closedint c b$.
Then, $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on the other interval as well, and:
:$\ds \int_a^c f \rd \alpha + \int_c^b f \rd \alpha = \int_a^b f \rd \alpha$

Proof:
Let $\closedint p q$ be the interval from among $\closedint a c$ and $\closedint c b$ on which we know $f$ is Riemann-Stieltjes integrable with respect to $\alpha$.
Let $\closedint u v$ be the other interval.
Then, we want to prove that $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $\closedint u v$, and:
:$\ds \int_u^v f \rd \alpha = \int_a^b f \rd \alpha - \int_p^q f \rd \alpha$
Let $\epsilon > 0$ be arbitrary.
By definition of the Riemann-Stieltjes integral, let $P'_\epsilon$ be a subdivision of $\closedint a b$ and $P' '_\epsilon$ be a subdivision of $\closedint p q$ such that:
:For every $P'$ finer than $P'_\epsilon$, $\ds \size {\map S {P', f, \alpha} - \int_a^b f \rd \alpha} < \frac \epsilon 2$
:For every $P' '$ finer than $P' '_\epsilon$, $\ds \size {\map S {P' ', f, \alpha} - \int_p^q f \rd \alpha} < \frac \epsilon 2$
Define $P_\epsilon := \paren {P'_\epsilon \cup \set {u, v}} \cap \closedint u v$.
Then, $P_\epsilon$ is a subdivision of $\closedint u v$.
Let $P = \set {x_0, \dotsc, x_n}$ be a subdivision of $\closedint u v$ that is finer than $P_\epsilon$.
Define:
:$P' = \set {y_0, \dotsc, y_m} := P \cup P'_\epsilon \cup P' '_\epsilon$
:$P' ' = \set {z_0, \dotsc, z_r} := P' \cap \closedint p q$
Since $P'_\epsilon \subseteq P' \subseteq \closedint a b$:
:$P'$ is a subdivision of $\closedint a b$ that is finer than $P'_\epsilon$
Since $P' '_\epsilon \subseteq P' ' \subseteq \closedint p q$:
:$P' '$ is a subdivision of $\closedint p q$ that is finer than $P' '_\epsilon$
We have:
{{begin-eqn}}
{{eqn | l = \map S {P, f, \alpha}
      | r = \sum_{k \mathop = 1}^n \map f {t_k} \paren {\map \alpha {x_k} - \map \alpha {x_{k - 1} } }
      | c = {{Defof|Riemann-Stieltjes Sum}}
}}
{{eqn | r = \sum_{k \mathop = 1}^m \map f {t'_k} \paren {\map \alpha {y_k} - \map \alpha {y_{k - 1} } } - \sum_{k \mathop = 1}^r \map f {z_k} \paren {\map \alpha {z_k} - \map \alpha {z_{k - 1} } }
      | c = where $t'_k = \begin{cases} t_j & : y_k = x_j \\ z_j & : y_k = z_j \end{cases}\quad$ and $j \ge 1$
}}
{{eqn | r = \map S {P', f, \alpha} - \map S {P' ', f, \alpha}
      | c = {{Defof|Riemann-Stieltjes Sum}}
}}
{{end-eqn}}
Therefore:
{{begin-eqn}}
{{eqn | l = \size {\map S {P, f, \alpha} - \paren {\int_a^b f \rd \alpha - \int_p^q f \rd \alpha} }
      | r = \size {\map S {P', f, \alpha} - \int_a^b f \rd \alpha + \int_p^q f \rd \alpha - \map S {P' ', f, \alpha} }
      | c = Above
}}
{{eqn | o = \le
      | r = \size {\map S {P', f, \alpha} - \int_a^b f \rd \alpha} + \size {\int_p^q f \rd \alpha - \map S {P' ', f, \alpha} }
      | c = Triangle Inequality for Real Numbers
}}
{{eqn | r = \size {\map S {P', f, \alpha} - \int_a^b f \rd \alpha} + \size {\map S {P' ', f, \alpha} - \int_p^q f \rd \alpha}
      | c = Absolute Value of Negative
}}
{{eqn | o = <
      | r = \frac \epsilon 2 + \frac \epsilon 2
      | c = Definitions of $P'_\epsilon$ and $P' '_\epsilon$
}}
{{eqn | r = \epsilon
}}
{{end-eqn}}
As $P$ finer than $P_\epsilon$ and $\epsilon > 0$ were arbitrary, it follows from the definition of the Riemann-Stieltjes integral that:
:$\ds \int_u^v f \rd \alpha = \int_a^b f \rd \alpha - \int_p^q f \rd \alpha$
{{qed}}