Consider the definition of definite integral in terms of a Riemann sum, i.e. Let $[a,b]$ be some interval on the real line. We then define a partition $P$ of $[a,b]$ in terms of a set of subintervals of $[a,b]$ such that $P=\lbrace [x_{0},x_{1}],\ldots ,[x_{i-1},x_{i}], \ldots ,[x_{n-1},x_{n}]\rbrace$, where $a=x_{0}<x_{1}<\cdots <x_{i-1}<x_{i}<\cdots <x_{n-1}<x_{n}=b$. The width of each subinterval is then given by $\Delta x_{i}=x_{i}-x_{i-1}$, with $\text{Max}\,\lvert\Delta x_{i}\rvert =\lbrace\lvert x_{1}-x_{0}\rvert ,\ldots ,\lvert x_{i}-x_{i-1}\rvert ,\cdots ,\lvert x_{n}-x_{n-1}\rvert\rbrace$ the maximum possible width attainable for a subinterval. We then define the integral from $a$ to $b$ of a function $f(x)$ as follows, $$\int_{a}^{b}f(x)dx =\lim_{n\rightarrow\infty}\lim_{\text{Max}\,\lvert\Delta x_{i}\rvert\rightarrow 0}\sum_{i =1}^{n}f(\zeta_{i})\Delta x_{i}$$ where $\zeta_{i}\in [x_{i-1},x_{i}]$.
Given this definition of the definite integral, is it possible to prove the following properties: $$1.\qquad\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\ 2.\qquad\int_{a}^{a}f(x)dx =0\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\ 3.\qquad\int_{a}^{c}f(x)dx=\int_{a}^{b}f(x)dx +\int_{b}^{c}f(x)dx$$ or are these simply taken to be defining properties (although, looking at it $2.$ seems to follow from $1.$ in the case where $a=b$)?
In some notes I've read (for example, these ones: https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch11.pdf), properties $1.$ and $2.$ are simply given as definitions, whereas in other notes (for example, these ones: http://tutorial.math.lamar.edu/Classes/CalcI/ProofIntProp.aspx ) they are proven. In all cases, property $3.$ seems to be provable.