The gemara Yoma on page 55a quotes a mishna which has R' Yehuda saying that there were no boxes of money for people to donate for kinei chova offerings because the kohanim could confuse that box with donations for nedava offerings. Abaye suggests writing on the boxes which box is for which kind of offering, but the gemara retorts that R' Yehuda does not hold such inscriptions are useful. The gemara then seems to try to prove this assertion by quoting our mishna and going through basically the same reasoning: R' Yehuda says there is only one stand and the gemara asks why. Evidently because the stands could be mixed up. So why not write which stand is which? Because R' Yehuda doesn't hold this works.
But how is this a proof? The reasoning seems exactly the same as the original point we are tying to prove. If this is a sufficient proof, why did the equivalently structured statement need to be proven at all?