There are a ton of different formulas for integrals, and my favourite is one I often hear called the discreet sum integral, defined such that $$\int_a^bf(x)dx=\lim_{d\to0^+}\sum_{x=a/d}^{b/d}df(dx)$$ or, sometimes $$\int_a^bf(x)dx=\lim_{d\to\infty}\sum_{x=ad}^{bd}d^{-1}f(x/d)$$ This is the formula that's typically used for numerical integration (where instead of calculating the integral, you just approximate it by using a sufficently small/large value of $d$ and calculating discreetly, since that's way easier to program). However, I've never seen the main integral properties proven in this form.
Of course, the phrase "main integral property" is a bit vague, but I'm reffering to things such as $\int_a^bf(x)dx=-\int_b^af(x)dx$, $\int_a^af(x)dx=0$, $\int_a^bf(x)dx+\int_b^cf(x)dx=\int_a^cf(x)dx$, integration by parts, etcetera.
Is that because this form is a bit unwieldy, so it's hard to prove them? Or is it impossible? Or is proving these things actually pretty easy, and I've just missed it?
Thank you for your time!