Proper and comoving distance in embedded $\Lambda$CDM geometry I have found this image on a Wikipedia page, i want to know if it safe to say that the orange line rappresent the proper distance between the brown and yellow lines, and the red line it's the comoving distance.
How i can proove more rigourosly knowing that the manifold it's described by this equation:
\begin{align} r & = a(t) R \\ \phi & = x / R \\ z & = \int \sqrt{c^2 - a'(t)^2 R^2} \, dt \end{align}
Two views of an isometric embedding of part of the visible universe over most of its history, showing how a light ray (red line) can travel an effective distance of 28 billion light years (orange line) in just 13 billion years of cosmological time. 
other POV
 A: The orange line is a set of events all at the same cosmic time. The length of this line is indeed the proper distance between the brown and yellow lines, if we define "proper distance" to mean the distance on the spacelike surface of given cosmic time, and this is what is usually meant by the term "proper distance" in cosmology. (Another name for it is "ruler distance".)
The red line is a representation on the diagram of a null geodesic of the spacetime. The total integrated invariant interval along this line is therefore zero. The ends of the red line are at different comoving coordinate locations. The difference of those coordinate values provides information about the spatial separation of the emission and reception events, but to convert that information into a distance you have to do something a little artificial. You have to chose how to talk about distances between events at different times in a dynamic spacetime. There is no single prescription for that. You can say that a long time ago, when the light was emitted by the quasar, the proper distance between the Milky Way and the quasar was a lot smaller than it is in the present, and that proper distance can be obtained by using the metric at the emission time (you integrate along the spatial hypersurface, i.e. one of the lines of latitude (purple) on the diagram). Another way to give information about the distance here is to appeal to concepts such as "luminosity distance".
Sometimes people refer to emission events in the distant past and talk about distances to their comoving coordinate location now, in the present. That is a little misleading, it seems to me, but it is often going on when people talk about the size of "the observable universe".
