I am proving an integral property. Is the following manipulation valid in sums?
$\sum\limits_{i=1}^n x_i = \sum\limits_{i=1}^n y_i$
Then $\sum\limits_{i=1}^n x_i\cdot p_i = \sum\limits_{i=1}^n y_i\cdot p_i$
I am given $f$ and $g$ are integrable functions (I am using Darboux integration). So we know: $\sum\limits_{k=1}^n m(f,[t_{k-1}, t_k])\cdot(t_k - t_{k-1}) = \sum\limits_{k=1}^n M(f,[t_{k-1}, t_k])\cdot(t_k - t_{k-1})$
$\sum\limits_{k=1}^n m(g,[t_{k-1}, t_k])\cdot(t_k - t_{k-1}) = \sum\limits_{k=1}^n M(g,[t_{k-1}, t_k])\cdot(t_k - t_{k-1})$
So: $\sum\limits_{k=1}^n m(f,[t_{k-1}, t_k])\cdot m(g,[t_{k-1}, t_k])\cdot(t_k - t_{k-1})$
$= \sum\limits_{k=1}^n M(f,[t_{k-1}, t_k])\cdot m(g,[t_{k-1}, t_k])\cdot(t_k - t_{k-1})$
$= \sum\limits_{k=1}^n M(f,[t_{k-1}, t_k])\cdot M(g,[t_{k-1}, t_k])\cdot(t_k - t_{k-1})$
I just want to make sure the above sum manipulations are correct? Pretty much I am just substituting.