\(d\left(x,y\right)=0\ \Leftrightarrow\ x=y\)
if x=y     \(\Rightarrow\left(x-y\right)^2\ =\left(x-x\right)^2=0\)
\(\Rightarrow\sqrt{\sum_{i=1}^d\left(x_{i-}y_i\right)^2\ \ }\ =0\ \ if\ \left(x=y\right)\)
iv. Triangle Inequality
\(d\left(x,z\right)\le d\left(x,y\right)+d\left(y,z\right)\)
To Prove:   \(\sqrt{\left(x-z\right)^2}\le\sqrt{\left(x-y\right)^2}+\sqrt{\left(y-z\right)^2}\)