deletions | additions       diff --git a/Now_we_need_to_show__.tex b/Now_we_need_to_show__.tex  index 4f7bef2..7015036 100644  --- a/Now_we_need_to_show__.tex  +++ b/Now_we_need_to_show__.tex 
...
Now we need to show only that  $$ \vec{x}(t) - \vec{x}(t-\Delta t) = \vec{v}(t-\Delta t)\Delta t + \frac{1}{2} \vec{a}(t-\Delta t) \Delta t^2 $$ We can shift this equation $\Delta t$ forward in time:  $$ \vec{x}(t+\Delta t) - \vec{x}(t) = \vec{v}(t)\Delta t + \frac{1}{2} \vec{a}(t) \Delta t^2 $$