2 tiny subscript edit

Let the pair $( S, d \, )$ be a metric space, i.e.
$d\!: S^2 \rightarrow R$, where for any three distinct elements $k$, $p$, $q$ $\in S$:
$d[ \, p, q \, ] = d[ \, q, p \, ] > 0$,
$d[ \, p, q \, ] + d[ \, q, k \, ] \geq d[ \, p, k \, ]$, and
$d[ \, p, p \, ] = 0$.

My question concerns a (any) set $X$, and

a (any) map $x\!: S^2 \rightarrow X$
which satisfies that there exists an element $z_{X}$ $\in X$ such that
for each element $p$ $\in S$: $x[ \, p, p \, ] = z_{X}$ and
for any two distinct elements $p$, $q$ $\in S$: $x[ \, p, q \, ] \ne z_{X}$, and

a map $r_d\!: (X \verb|\| z_{X})^2 \rightarrow R$ which is defined such that
for any two not necessarily distinct elements $a$, $b$ $\in (X \verb|\| z_{X})$
and for any four not necessarily all distinct elements $j$, $k$, $p$, $q$ $\in S$ such that
$x[ \, j, k \, ] = a$ and $x[ \, p, q \, ] = b$