we have the following map:
$d(x,y) = |x_2 - y_2|~~~~~~~~$ if $x_1 = y_1 $
and $= |x_2| + |y_2| + |x_1 - y_1| ~~~~~~~$ if $x_1 \neq y_1 $
where $x = (x_1,x_2)~~and~~y = (y_1,y_2) $
we have to prove that the above is a metric space on $R^2$,
I have already proved the first two axioms and the cases of the third where: $x_1 = y_1 = z_1$, and $ z_1 \neq x_1$ and $x1 \neq y_1 \neq z_1$,
what's left are the case where $ x_1 \neq y_1, z_1 = x_1$ and the other case where $x_1 \neq y_1, z_1 = y_1$
$z = (z_1,z_2)$ a third point I introduced.
I know i'm supposed to use the triangle inequality during the steps, but i'm totally stuck= at these last two cases.
I'd be thankful for any sort of help