Let $S_1=k[x_1,\ldots,x_n]$ and $S_2=k[y_1,\ldots,y_m]$ be two polynomial ringsover a field $k$ and $I\subset S_1$ and $J\subset S_2$ be two ideals. Let $S=k[x_1,\ldots,x_n,y_1,\ldots,y_m].$

Question Can we say $in_<(I+J)S=(in_<I+in_<J)S$ (all monomial orders are degree reverse lex in repective rings)?

(where $in_<I$ is the ideal generated by $\{in_<f:f\in I\}$)