Delete some numbers in $x_1+x_2+...+x_n=y_1+y_2+...+y_m<mn$

Question

Let $$x_1+x_2+...+x_n=y_1+y_2+...+y_m<mn,$$ where $x_i,y_i -$ positive integers.

Prove that you can delete some terms (but not all) in the equation and equality remains true.

My work so far:

Induction on $n + m$. Base $n + m = 4 -$ is obvious.

Let $x_1\ge x_2\ge...x_n$ and $y_1\ge y_2\ge...y_n$ and $x_1\ge y_1$.

1) $x_1=y_1 \Rightarrow x_2+...+x_n=y_2+...+y_m$.

2) $x_1>y_1 \Rightarrow (x_1-y_1)+x_2+...+x_n=y_2+...+y_m$.

Here I need help.