Compact interval/Real function/Riemann integrable on partition/Fact

Let ${}I=[a,b]$ be a compact interval, and let

$f\colon I\longrightarrow \mathbb {R}$

be a function. Then the following statements are equivalent.

1. The function ${}f$ is Riemann-integrable.
2. There exists a partition ${}a=a_{0}<a_{1}<\cdots <a_{n}=b$, such that the restrictions ${}f_{i}:=f|_{[a_{i-1},a_{i}]}$ are Riemann-integrable.
3. For every partition ${}a=a_{0}<a_{1}<\cdots <a_{n}=b$, the restrictions ${}f_{i}:=f|_{[a_{i-1},a_{i}]}$ are Riemann-integrable.

In this situation, the equation

${}\int _{a}^{b}f(t)\,dt=\sum _{i=1}^{n}\int _{a_{i-1}}^{a_{i}}f_{i}(t)\,dt\,$

holds.

Write a proof