## Online Material

## Appendix A: The lensing magnification factor

Although and most likely do not depend on the magnification factor, since [CII], CO, and FIR are emitted from the same regions of the galactic gaseous disk, the inferred intrinsic luminosities, and in particular , do depend on the magnification factor. Therefore, especially for what concerns the versus diagram, it is important to address the issue of the magnification lensing factor in detail.

Guilloteau et al. (1999) provided only a rough estimate of the magnification factor (
).
Lehár et al. (2000) performed a more detailed lensing analysis by using different mass models for the lensing galaxy along with HST images (although they do not directly provide the associated magnification factors). The positions and the flux ratio of the two quasar images are reproduced by a singular-isothermal-ellipsoid model (SIE). A constant M/L model, where the lensing mass distribution matches the light distribution of the observed galaxy, plus an external shear
(due to galaxies surrounding the lens) also provides a good fit to the data. Details on the best-fit
parameters describing the lensing models can be found in Table 5 of Lehár et al. (2000). Based on the image constraints, it is impossible to differentiate between the SIE and the M/L models. We used the public software *GRAVLENS* and its application *LENSMODEL* (Keeton 2001) to repeat the fits by Lehár et al. (2000) and to reconstruct their lens models. These were used to estimate the
magnification factors at the positions of the two quasar images. The lens is a typical
early-type galaxy, whose light distribution can be described by means of a De Vaucouleurs
profile. The two models predict significantly different magnification factors. Assuming
a point source, the SIE and the
models give total magnification factors
and ,
respectively. The large difference is caused by the substantially
different slopes of the two density profiles, with the SIE much steeper than the De Vaucouleurs profile. Without an external shear, the SIE model may be overestimating the lens convergence and therefore the magnification. However, since no correlation is expected between the external shear and the environment of the lens galaxy, it is hard to estimate the possible bias (see the discussion in Lehár et al. 2000). In the
model, even a modest external shear strongly perturbs the lens. In general, steeper profiles are favored by observations of strong lensing systems (e.g. Treu & Koopmans 2004; Rusin et al. 2003). Therefore we can assume that the magnification factor predicted by the
model is a lower limit to the true value.

By using the result of the fit of the quasar double image, we attempted to estimate how
much the magnification factor changes for an extended source. This is important in our case since the [CII] and the FIR emissions come from an extended region of the gaseous disk of the quasar host galaxy. For instance, in the case of J1148+5251 at *z*=6.41, high angular resolution images show that [CII] is distributed in a region of about 1.5 kpc in diameter around the quasar (Walter et al. 2009). By replacing, on the source plane, the point-like source (the quasar nucleus observed in the optical images) with a circular disk (the [CII]-FIR emitting region)
of radius 0.2'' (equivalent to 1 h^{-1} kpc at *z*=4.43) centered on the location
of the un-lensed quasar (from the modeling of its point images), and mapping it on the lens plane, we find that the total magnification factors change by 0.5 for both the SIE and the
models. The magnification factor does not change significantly when varying the size of
the [CII]-FIR disk as long as it is less than 2 kpc in radius. However, we note that the morphology of the extended images is very different for the two models. In particular, the SIE model produces an Einstein ring-like image, while the
model produces an extended asymmetric arc and a counter image. Therefore, high angular resolution observations of the [CII] and FIR emission would allow us to determine which of the two lens models is more appropriate.

In summary, the magnification factor ranges between and , mostly depending on the lens model. In the paper we assume a magnification factor (the mean value of the two extreme magnification factors in ) as a reference, although we discuss the implications of the wide range of possible magnification factors. For what concerns the intrinsic far infrared luminosity, the observed value inferred by Priddey & McMahon (2001) (and corrected for our assumed cosmology) translates into an intrinsic, magnification-corrected, far-infrared luminosity of , where the errors reflect the range in possible magnification factors.