The numbers inserted in $L$ must increase strictly down each column.
This accords with the intuition that as we pass down the coding tree, we find out more and more detailed information about the ordering actually represented.
We show that ...... by reverse induction on $i$, starting at $i=n$ and working down to $i=0$.
Thus, everything comes down to proving the existence of $M$.
For size 1 this makes no difference, but for sizes 2 and 3 it considerably cuts down the number of cases to be considered.
Important analytic differences appear when one writes down precisely what is meant by ......
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