Division of two Elliptic curve points in KZG polynomial commitment scheme!

I have some issue to understand the verify round of the KZG polynomial commitment scheme. The following diagram is associated to the scheme. I appreciate any help.

To verify, the verifier should compute the pairing of $e(g^{f(\tau)-f(u)}, g)$ and $e(g^{\tau-u}, g^{q(\tau)})$.

However, to compute these pairings, verifier should compute $g^{f(\tau)-f(u)}$ and $g^{\tau-u}$ first. So, we see that $g^{f(\tau)-f(u)}=g^{f(\tau)}/g^{f(u)}$ and this is division of two points of the elliptic curve! However, the division of two elliptic curve points is not defined! We have the same issue with computing $g^{(\tau-u)}$ which is equal to $g^\tau/g^u$.

In this lecture, they use multiplicative notation for the pairing groups instead of additive notation. Thus, division is well-defined. Division is just the inverse of the group operation.

The choice of additive vs multiplicative notation for a group is purely a semantic choice.

I'll translate some of the items to additive notation. The lowercase letters and symbols will be elements of the scalar field of the pairing group $\mathbb{G}$, while the capital $G$ will denote the generator of the group.

• The commitment $\mathsf{com}_f = f(\tau) G$.
• The element $vG$
• The element $(\tau-u)G = \tau G - uG$
• The subtraction will be $(f(\tau)-v)G = f(\tau)G - vG$.
• The proof $\pi=q(\tau)G$
• The pairing check $e\left(\mathsf{com}_f-vG,G\right)=e\left((\tau-u)G, \pi\right)$