realisations of each binary system under the assumption that the adopted errors on the effective temperatures (2.3 per cent), the surface gravities (0.07 dex) and the observed magnitudes (Table \[phot\]) of the component stars are normally distributed. We also noted that the SDSS magnitudes (Table \[phot\]) have an absolute precision of 2 per cent [@adelman_mccarthy08].

The progenitors of the HFMWDs {#disc1}
=============================

In wide double-degenerate systems
---------------------------------

[ccccccc]{} Component & & &\
SDSS & $M$$_{\rm init}$ (M$_{\odot}$) & System age (Myr) & $M$$_{\rm init}$ (M$_{\odot}$) & System age (Myr) & $M$$_{\rm init}$ (M$_{\odot}$) & System age (Myr)\

&3.06$^{+0.66}_{-0.58}$ & 599$^{+357}_{-181}$ & 2.76$^{+0.50}_{-0.47}$ & 753$^{+390}_{-217}$ & 2.76$^{+0.37}_{-0.36}$ & 754$^{+259}_{-167}$\
\
& 5.30$^{+0.86}_{-0.75}$ & 426$^{+67}_{-44}$ & 5.50$^{+0.62}_{-0.59}$ & 416$^{+55}_{-38}$ &5.07$^{+0.41}_{-0.42}$ & 437$^{+52}_{-37}$\
\[progcool1\]

From their observed positions on the sky and the distance moduli in Table \[temps\], we determine that the components of System 1 and System 2 have minimum separations of a$\sim$650AU and a$\sim$750AU, respectively. Even assuming that the original orbital separations of these binaries were substantially smaller [e.g. @valls88], the Roche lobes of their components have likely always been much larger than the dimensions of an asymptotic giant branch (AGB) star [r$\simless$4-5AU; @ibenlivio93]. Thus this work doubles the number of 