Suppose $k$ is a field. Suppose $f_1,\cdots,f_r\in k[x_1,\cdots,x_n]$ is a regular sequence, and $g_1,\cdots,g_r\in k[y_1,\cdots,y_m]$ is a sequence, where $x_i$ and $y_j$ are different symbols.
It seems like that $f_1+g_1,\cdots,f_r+g_r$ is still a regular sequence, but is there any reference for it?
One could add many other assupmtions, for instance $f_i$ are homogeneous polynomials and $LT(f_i)>LT(f_{i+1})$ where $LT$ is the leading term, etc.