If [tex]a\equiv b\mod n[/tex], then there is some integer [tex]N[/tex] such that [tex]a=Nn+b[/tex]. Because [tex]m|n[/tex], it follows that [tex]m|Nn[/tex], which means there is some integer [tex]M[/tex] such that [tex]a=Mm+b[/tex], which is equivalent to the statement that [tex]a\equiv b\mod m[/tex].
