Given the following definition of a pseudo-metric on the set $X$ :
A pseudo-metric on the set $X$ is a map $d:X \times X \to \Bbb R^+$ such that for all $x ,y \text{ and } z \in X :$
(PM1) $x=y \Rightarrow d(x,y)=0 $
(PM2) $d(x,y) =d(y,x)$
(PM3) $d(x,y) \le d(x,z) +d(z,y)$
We defined the following equivalence relation ~ on $X$ as $$ x \sim y \iff d(x,y)= 0$$
We then proceeded to define $\delta$ as follows. For any $x,y \in X$
$$ \delta([x],[y]) = d(x,y)$$
I have to show that $\delta $ is a metric on the set of equivalence classes on $X$
I am busy with (M3) and would like to know whether this is enough to show that $\delta $satifies M3:
$$\delta ([x],[y]) = d(x,y)$$ $$\leq d(x,z) +d (z,y) \text{ by PM3}$$ $$= \delta([x],[z]) + \delta ([y],[z])$$