Both Uses

demonstrate
in
American Sniper
(Edited)

In this case, because of when I was heading out, it happened that I passed a small group of protesters demonstrating against the war.
p. 134.6 *

demonstrating = joining with others in a public protest
We would establish strongholds in various parts of the city, demonstrating that we were in control—and essentially daring the enemy to attack.
p. 285.2 *

demonstrating = showing

Definitions:

(1) (demonstrate as in: It demonstrates my point.)
to show
The exact meaning of this sense of demonstrate can depend upon its context. For example:
- "I will demonstrate how to throw a Frisbee." -- show how to do something
- "I will demonstrate how much quicker the new computer is than the old one." -- show how something works
- "Her questioned demonstrated that she was listening and thinking deeply about what was said." -- showed to be true or proved
(2) (demonstrate as in: demonstrate to protest)
a public display supporting a cause -- usually joining with others in a political protest
(3) (meaning too rare to warrant focus)