A problem on a proof in a graph theory textbook

I met with a problem in the proof of Theorem 1.3 in page 14 of "Combinatorics and Graph Theory", 2nd Edition, John Harris, Jeffry L. Hirst, Michael Mossinghoff. The proof is excerpted as follows:

My problem comes from the claim $v'=v_i=w_i$ underlined with a red line. Why $v_i$ and $w_i$ must have the same subscript $i$? Thank you.

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2 Answers

Suppose $v'=v_i=w_j$ with $i\ne j$. We may assume that $i<j$. Then the path $v=v_0,v_1,\ldots,v_i=w_j,w_{j+1},w_{j+2},\ldots,w_{2t}=x'$ is a shorter path from $v$ to $x'$, which is a contradiction, as $P_2$ was supposed to be minimal.

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I see. "shortest" is the key. Thank you very much! –  Zhou Heng Aug 7 '13 at 5:55

Clearly $v = v_{0} = w_{0}$ is common in $P_{1}$ and $P_{2}$. Now starting from $0$ there must be a first integer $j$ where the paths $P_{1}$, $P_{2}$ diverge and the vertices before $j$ i.e. $v_{j - 1}$ and $w_{j - 1}$ must be same so that $i = j - 1$ and clearly both $v$ and $w$ should have same subscript $i$ because the starting index for common vertex in $P_{1}$, $P_{2}$ is same.

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