It is problem 2.17 in the book Gröbner Bases in Commutative Algebra, by Ene and Herzog.
Let $f,g \in S$ such that $\textrm{in}_{<}(f)$ and $\textrm{in}_{<}(g)$ are relatively prime and let $u$ and $v$ be any monomials in $S=K[x_{1}, \dots, x_{n}]$. Then $S(uf,vg)$ reduces to $0$ with respect to $uf$ and $vg$.
I know the result that if $\textrm{in}_{<}(f)$ and $\textrm{in}_{<}(g)$ are relatively prime, then $S(f,g)$ reduces to $0$ with respect to $f$ and $g$. Also I know the partial answer when $u=v$. (It is a trivial corollary of former statement.) However, I don't know how to approach in another case. Any answer, hint or suggestion would be appreciated.
EDIT: Now I've got an answer.
Let $f = m+f_{1}, g = w + g_{1}$ where $\textrm{in}_{<}(f) = m, \textrm{in}_{<}(g) = w$. Then, $$\textrm{lcm}(mu,wv) = mwp$$ for some monomial $p = \textrm{lcm}(u,v)$ since $\gcd(m,w) = 1$. Hence, $$uf = mu + uf_{0}, vg = wv+vg_{0}, \textrm{in}_{<}(uf) = mu, \textrm{in}_{<}(vg) = wv, \textrm{lcm}(\textrm{in}_{<}(uf),\textrm{in}_{<}(vg)) = \textrm{lcm}(mu,wv) = mwp,$$ hence \begin{align*}S(uf,vg) &= \frac{\textrm{lcm}(\textrm{in}_{<}(uf),\textrm{in}_{<}(vg))}{mu}uf - \frac{\textrm{lcm}(\textrm{in}_{<}(uf),\textrm{in}_{<}(vg))}{wv}vg \\ & = \frac{wp}{u}uf-\frac{mp}{v}vg = p(wf -mg) \\ & = p(gf-g_{1}f - fg+f_{1}g) =p(f_{1}g - g_{1}f). \end{align*} Now we claim that $S(uf,vg)$ is reduces to 0 with respect to $pg, pf$. Suppose that $$\textrm{in}_{<}(pf_{1}g) = \textrm{in}_{<}(pg_{1}f).$$ Then, $$\textrm{in}_{<}(f_{1}g) = \textrm{in}_{<}(g_{1}f),$$ which gives contradiction by the argument on the proof of the proposition 2.15 in the book. So we may now assume without loss of generality that $\textrm{in}_{<}(pf_{1}g) > \textrm{in}_{<}(pg_{1}f) \implies \textrm{in}_{<}(f_{1}g) > \textrm{in}_{<}(g_{1}f).$ Then, by the lemma 2.4, $$\textrm{in}_{<}(S(uf,vg)) = \textrm{in}_{<}(f_{1}pg) > \textrm{in}_{<}(g_{1}pf).$$ Hence, it is standard expression. Since this argument holds with respect to $uf$, $ug$, and $vf$,$vg$, and $uf$,$vg$, our problem is solved. (Note that for $u,v$ cases, we can recall $p = lcm(u,v)$, so just rewrite it with respect to $u,v$. However, the first argument that $\textrm{in}_{<}(pf_{1}g) \neq \textrm{in}_{<}(pg_{1}f)$ holds in any representation of such terms, thus this representation only gives the rewritten second conditions, therefore it holds for any $u,v$.)