# to prove two lines intersect in a unique point

I just read an answer proving two lines intersect in a unique point in which any other pairs of line will not intersect. Actually I did't quite get what it means.Here is the answer described like this:

Now I don't know clear about why "y coordinate uniquely identifies j and k." Does it mean we can't find another pair of numbers such as l and m suffices that $n^j + n^k = n^l + n^m$.If so, how can I prove it.Thanks for any help. And thanks for Buie's suggestion, I'd like to mention that all the lines go through points $(n^{2j}, 0)$ and $(n^{2j} - n^j, 1)$ or $(n^{2j} - n^j - n^{-n}, 1)$.Both n and j is integer and that 1<= j <= n.

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@CameronBuie: Thank you, I've added the necessary conditions. –  tuan long Sep 3 '13 at 14:43