I've read in a bunch of texts that the S-polynomial of two polynomials with relatively prime leading terms always reduces to zero, i.e. the remainder of the division of $S(f,g)$ by a set of polynomials containing $f$ and $g$ is zero.

Can someone give me a hint about the proof? I've read in an answer to [this post](https://math.stackexchange.com/questions/1753588/showing-the-polynomials-form-a-gr%C3%B6bner-basis) that whenever $\gcd(\mathrm{LT}(f),\mathrm{LT}(g))=1$ we can write $S(f,g)=-(g-\mathrm{LT}(g))f+(f-\mathrm{LT}(f))g$, but I don't know why this should help: why can we say that that form of $S(f,g)$ as a polynomial combination of $f$ and $g$ comes from a division? I know that not all the expressions of a polynomial $f\in k[x\_1,\dots, x\_n]$ as $f=p\_1q\_1+\cdots+p\_sq\_s+r$ come from a division of $f$ by $\{q\_1,\dots, q\_s\}$...