to simplify the algebra, let:

a = d(x,y)

b = d(x,z)

c = d(z,y), and we are given that a ≤ b+c, and also that a,b,c are all non-negative.

then:

p(x,y) = a/(1+a)

p(x,z) = b/(1+b)

p(z,y) = c/(1+c)

what we want to prove is:

,

that is:

or, equivalently, that:

which is to say:

so:

since

, it follows that:

, from which the desired result follows.