How does Near prove that Light is Kira by seeing Kiyomi's name in Teru's death note? Both Teru Mikami and Light Yagami write Kiyomi's name in their death notes the only difference being the time of death they specify. But in what way does this help Near to prove that Light is Kira? 



This was a page from Teru Mikami's death note shown by Near to Light.

Does Near identify that Kiyomi died a minute earlier than the time specified by Teru?
Showing Kiyomi's name written in the notebook doesn't help Near proves that Light is Kira. Instead, it helps Near explains how he win by realizing the possibility of a fake notebook, which he previously not realize and would have lead to Light's win.

Here is the conversation on Anime:


  Near: "On the 26th, after the report featuring Takada's kidnapping, Mikami
  headed for the bank."
  
  Gevanni: "According to my investigation, Mikami only goes to the bank on the 25th of every month.  For someone as punctual as Teru, going to the bank twice in two days is pretty unusual. I followed him as he headed
  for the deposit box. I was able to get into the deposit box, and 
  inside, was the real notebook, with Takada's name written in it."
  
  Near: "Even though you could kill someone with a piece of the note, you
  weren't aware that Mikami had done something on his own. It was then
  that I realized the possibility of a fake notebook."


P.S.: The detailed (i.e., time specified) written by Light and Teru doesn't matter here since both match with how Kiyomi died.