Let $I$ be a non-zero ideal of the polynomial ring $S:=K[x_1,...,x_n]$ (where $K$ is a field) , which is equiped with a monomial ordering $<$, and $G:=\{g_1,...,g_t \}\subseteq I$. We want to show that: $$ \langle lt(g_1),..., lt(g_t) \rangle = in_<(I) \implies G \text{ is a Grobner basis for I}. $$
If we take one polynomial $0_K\neq f \in I$ then from the definition $lt(f) \in in_< (I)=\langle lt(g_1),...,lt(g_t) \rangle \implies \exists h_1,...,h_t\in S: lt(f)= h_1\cdot lt(g_1) + ... + h_t \cdot lt(g_t) $. It's clear that we need to show that $lt(g_i)|lt(f),$ for some $ i \in \{1,..., t\} $.
All the books I have searched, compare the RHS, LHS and immediately conclude that we have the division above. But, I can not understand this step. Could you please explain it?
Thanks in advance.