Why does this derivation/proof of KVL from KCL and Tellegens theorem hold?

Question

I have come across a strange derivation of Kirchhoff's voltage law. You assume that Tellegen's theorem and Kirchhoff's current law both hold. (Bold letters signify a vector.)
The proof:
Let L be an arbitrary loop in the graph G. Consider the i obtained by assigning zero current to all branches of G except for those of loop L; depending on whether the reference direction of branch j in loop L agrees with that of loop L, we assign each branch current to be either 1A or -1A. The resulting i satisfies KCL at all nodes of G. Tellegen's theorem gives: $$ \sum_{j=1}^{b}v_{j}i_{j}= \sum_{overBranchesInLoopL}\pm v_{j}=0 $$ thus the algebraic sum of the branch voltages around loop L is zero, i.e., KVL holds for loop L. Since L is arbitrary, we have shown that KVL holds for all loops of G.*

The Problem I have with this proof is that he just chooses that the current is zero in all branches, but the branches of L. Can anyone elaborate on why one is allowed to do this?

This proof was obtained from the book Linear and Nonlinear Circuits by Leon O. Chua, Charles A. Desoer, Ernest S. Kuh on page 30.

Answer 1

At the beginning of the referenced section, the authors state the assumption on which the derivation is based (emphasis mine):

2. If, for all $\mathbf{i}$ satisfying KCL, $\mathbf{v}^T\mathbf{i} = 0$, then $\mathbf{v}$ satisfies KVL.

The choice of currents that is used in the derivation is just a case of $\mathbf{i}$ satisfying KCL: the currents in the branches external to the loop don't matter because the assumption is that $\mathbf{v}^T\mathbf{i} = 0$ holds true for any choice of currents satisfying KCL.