Gil
---The claim that "[t]he only thing that is relevant is what they will exchange for" assumes exactly what must be proven, and is at best problematic, as the counter-example involving uncultivated land (or any other exchangeable which is not a product of labor) illustrates.
Paul ---- The point of my argument is that commodity bundle space constitutes a metric space, and that the metric on this space is homeomorphic to a single scalar dimension. Exchange enforces a projection of the multidimentional bundle space onto this underlying dimension such that the equivalence sets induced by exchange are mapped onto single points on this dimension.
Since the mapping is onto single points, the members of an equivalence set are equal under the mapping. It is in this sense that one is quite entitled to treat the equivalence relation as an equality relation, ( quite appart from the etymology ).
Whether things are the product of labour or not is irrelevant to the formal argument. One must first extablish the existence of a value dimension, and establish that it is real not complex or some higher manifold, before going onto determine what may cause it. That labour was the cause was pretty generally accepted when Marx was writing, and the empirical evidence for it is to my mind very convincing.