Eq. (2) vs. Eq. (3): $D_{ds} = D_s - \frac{1+z_d}{1+z_s} D_d$ in flat ΛCDM

Let $\Omega_m \ge 0$ and $0 \le z_{\rm lens} \le z_s$, and let $c$, $H_0$ be arbitrary reals. In flat ΛCDM the angular diameter distance between lens and source (Eq. (2)) is expressed through the observer distances (Eq. (3)) by

where $D_d = D(z_{\rm lens})$ and $D_s = D(z_s)$.

This is the additivity of comoving distance in a spatially flat universe; it shows that Eqs. (2) and (3) describe one and the same distance measure, so that $D_{\Delta t}$ in Eq. (1) is a function of the two observer distances alone. Here $E(z)=\sqrt{\Omega_m(1+z)^3+1-\Omega_m}$ is the flat-ΛCDM expansion rate.