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# How is point addition for points of elliptic curve in $\mathbb{F}_p$ carried out technically?

From a very basic introduction text to elliptic curve cryptography point arithmetic is derived from "standard analysis": The (negative) sum of $$P_1$$ and $$P_2$$ is defined as the Point $$P_3$$, which is on the line connecting $$P_1$$ and $$P_2$$:

From that it is derived

In real numbers I would understand that completely. But typically, ECC is carried out within a finite (prime) field $$\mathbb{F}_p$$.

How shall I understand the multiplication/division in the addition formula $$m \cdot (x_1-x_3)$$ since m is a "fraction"? Normally I would expect that $$m$$ is a "fractional number" and in general not in $$\mathbb{F}_p$$. So how is $$m$$ obtained, because "division" is not an operation in a field - there is only an multiplicative inverse element:

Does it mean, I have to calculate

$$y_3 = (y_2-y_1)(x_2-x_1)^{-1}(x_1-x_3) - y_1$$

Since the inverse is well defined and exists, $$y_3$$ is also well defined. Is my assumption right or what else is the "fraction" $$m$$? Does it mean, the derivation gives sense, because formally each operation which is carried out in $$\mathbb{R}$$ I can in parallel be done also within $$\mathbb{F}_p$$, because both are fields and the notation $$a/b$$ means implicitly $$a b^{-1}$$?