If [tex]n\equiv9\mod14[/tex], then there is some integer [tex]k[/tex] for which [tex]n=k+\dfrac9{14}[/tex] where [tex]k[/tex] would be a multiple of 14, i.e. [tex]k\equiv0\mod14[/tex].
Because [tex]k[/tex] is a multiple of 14, it's also a multiple of 7, so [tex]k\equiv0\mod7[/tex].
This then means that [tex]n\equiv9\equiv2\mod7[/tex], so the remainder upon dividing by 7 is 2.
