{{WLOG}}, assume $a < b$. First let $a < c < b$. Let $P_1$ and $P_2$ be any subdivisions of $\closedint a c$ and $\closedint c b$ respectively. Then $P = P_1 \cup P_2$ is a subdivision of $\closedint a b$. From the definitions of upper sum and lower sum: :$\map L {P_1} + \map L {P_2} = \map L P$ :$\map U {P_1} + \map U {P_2} = \map U P$ We consider the lower sum. The same conclusion can be obtained by investigating the upper sum. By definition of Darboux integral: :$\displaystyle \map L P \le \int_a^b \map f t \rd t$ Thus, given the subdivisions $P_1$ and $P_2$, we have: :$\displaystyle \map L {P_1} + \map L {P_2} \le \int_a^b \map f t \rd t$ and so: :$\displaystyle \map L {P_1} \le \int_a^b \map f t \rd t - \map L {P_2}$ So, for any subdivision $P_2$ of $\closedint c b$, $\displaystyle \int_a^b \map f t \rd t - \map L {P_2}$ is an upper bound of $\map L {P_1}$. Thus: :$\displaystyle \map {\sup_{P_1} } {\map L {P_1} } \le \int_a^b \map f t \rd t - \map L {P_2}$ where $\map {\sup_{P_1} } {\map L {P_1} }$ ranges over all subdivisions of $P_1$. Thus by definition of definite integral: :$\displaystyle \int_a^c \map f t \rd t \le \int_a^b \map f t \rd t - \map L {P_2}$ and so: :$\displaystyle \map L {P_2} \le \int_a^b \map f t \rd t - \int_a^c \map f t \rd t$ Similarly, we find that: :$\displaystyle \int_c^b \map f t \rd t \le \int_a^b \map f t \rd t - \int_a^c \map f t \rd t$ Therefore: :$\displaystyle \int_a^b \map f t \rd t \ge \int_a^c \map f t \rd t + \int_c^b \map f t \rd t$ Having established the above, we now need to put it into context. Let $P$ be ''any'' subdivision of $\closedint a b$, which may or may not include the point $c$. Let $Q = P \cup \set c$ be the subdivision of $\closedint a b$ obtained from $P$ by including with it, if necessary, the point $c$. It is easy to show that $\map L P \le \map L Q$. Let $Q_1$ be the subdivision of $\closedint a b$ which includes only the points of $Q$ that lie in $\closedint a c$. Let $Q_2$ be the subdivision of $\closedint a b$ which includes only the points of $Q$ that lie in $\closedint c b$. From the definition of lower sum: :$\map L Q = \map L {Q_1} + \map L {Q_2}$ We have: {{begin-eqn}} {{eqn | l = \map L P       | o = \le       | r = \map L Q       | c =  }} {{eqn | r = \map L {Q_1} + \map L {Q_2}       | c =  }} {{eqn | o = \le       | r = \int_a^c \map f t \rd t + \int_c^b \map f t \rd t       | c =  }} {{end-eqn}} So $\displaystyle \int_a^c \map f t \rd t + \int_c^b \map f t \rd t$ is an upper bound for $\map L P$, where $P$ is any subdivision of $\closedint a b$. Thus: :$\displaystyle \map {\sup_P} {\map L P} \le \int_a^c \map f t \rd t + \int_c^b \map f t \rd t$ Thus, by definition: :$\displaystyle \int_a^b \map f t \rd t \le \int_a^c \map f t \rd t + \int_c^b \map f t \rd t$ Combining this with the result: :$\displaystyle \int_a^b \map f t \rd t \ge \int_a^c \map f t \rd t + \int_c^b \map f t \rd t$ the result follows. {{qed|lemma}} {{Explain|"It is easy to show that $\map L P \le \map L Q$". This needs a separate theorem, but really requires the definitions for "finer" and "coarser" mesh, which leads us off into areas of topology that I need to lay some more groundwork for.}} Now suppose $a < b < c$. Then from the definition of Darboux integral: :$\displaystyle \int_c^b \map f t \rd t := -\int_b^c \map f t \rd t$ and it follows that: {{begin-eqn}} {{eqn | l = \int_a^b \map f t \rd t       | r = \int_a^c \map f t \rd t - \int_b^c \map f t \rd t        | c = main result }} {{eqn | r = \int_a^c \map f t \rd t + \int_c^b \map f t \rd t       | c =  }} {{end-eqn}} The case of $c < a < b$ is proved similarly. Finally, suppose $a = c < b$. Then: {{begin-eqn}} {{eqn | l = \int_a^b \map f t \rd t       | r = 0 + \int_c^b \map f t \rd t + \int_a^c \map f t \rd t       | c = as $a = c$ }} {{eqn | r = \int_a^c \map f t \rd t + \int_c^b \map f t \rd t       | c = Integral on Zero Interval, as $a = c$ }} {{end-eqn}} The case of $a < c = b$ is proved similarly. Hence the result, from Proof by Cases. {{qed}}