A metric cone $C$ is a nonempty metric space (whose distance is denoted $d$) together with a map $\cdot\colon \mathbf{R}\times C \mapsto C$ satisfying these axioms:

$a\cdot(b\cdot x) = (ab)\cdot x$ for all reals $a$ and $b$, and all $x$ in $C$,

$d(a\cdot x;a\cdot y) = \vert a\vert d(x;y)$ for all real $a$ and all $x$ and $y$ in $C$,

$d(a\cdot x;b\cdot x) = \vert a-b\vert d(x;0)$ for all reals $a$ and $b$, and all $x$ in $C$, where $0$ denote $0\cdot x$ for any $x \in C$ (this definition is independent of the choice of $x$),

$d((a+a')\cdot x;(b+b')\cdot y) \leq d(a\cdot x;b\cdot y) + d((a'\cdot x;b'\cdot y)$ for all reals $a$, $b$, $a'$, and $b'$, and all $x$ and $y$ in $C$.

Those structures arise in my work about metric vector spaces. In some situations, I manage to prove the three first axioms, but not the fourth. So my question is:

Is the fourth axiom independent from the three other ones?

I have looked for a counterexample, and for a demonstration of the last axiom from the previous, but I failed in both attempts.