Euclidean Algorithm 
Proof
Let us prove that:
if \(a\trianglelefteq b\) and \(b\trianglelefteq c\) then \(a\trianglelefteq c\).
If \(a\trianglelefteq b\) then there exists \(k_{1}\) such that \(b=k_{1}a\). On the other hand, if \(b\trianglelefteq c\) then there exists \(k_{2}\) such that \(c=k_{2}b\).
Hence, \(c=k_{2}b=k_{2}\left(k_{1}a\right)=\left(k_{1}k_{2}\right)a\), which means, \(a\trianglelefteq c\).
