Reminder of how we defined the relation:
(x,y,r) R (a,b,s) if and only if \(\sqrt{(x-a)^2 + (y-b)^2} \le s- r\)
Let (x,y,r) and (a,b,s) be elements of \(C = \mathbb{R} \times \mathbb{R} \times \mathbb{R}^+\). Suppose that (x,y,r) R (a,b,s) and (a,b,s) R (x,y,r).
Applying the definition of R, we get that \(\sqrt{(x-a)^2 + (y-b)^2} \le s- r\) and \(\sqrt{(a-x)^2 + (b-y)^2} \le r- s\).
Notice that the lefthand sides of these equations can't be negative. So both s-r and r-s must be non-negative. But this can be true only when s-r = 0, i.e. r=s.
So now we know that \(\sqrt{(x-a)^2 + (y-b)^2} = 0\). So \((x-a)^2 + (y-b)^2 = 0\). So \((x-a)^2 = (y-b)^2 = 0\). So \(x-a = y-b = 0\). So x=a and y=b.
Since r=s, x=a, and y=b, (x,y,r) = (a,b,s), which is what we needed to show.