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The all-Russian final came down to the wire as Egorian equalised at 14-14 and then landed one last hit on Velikaya.
"It's the biggest win in one's life. It's something that everybody wants to achieve. It was the most difficult fight because Sofya has as much right to this medal as I do. Sport is sport and there can only be one winner,” said Olympic champion Egorian.
The 22-year-old world No.5 had the upper hand on her more experienced 31-year-old team-mate Velikaya, who had to be happy with her second consecutive silver medal after the London 2012 Olympic Games, following a fourth place in Beijing.
World No.1, defending European and world champion Velikaya had managed to survive two scares in the table of 16 as well as in the semifinals when she defeated 15-14 both her rivals, only to miss out on the Olympic gold medal in the decisive bout.
"I'm quite pleased, but I'm not totally satisfied because I got a silver medal,” Velikaya said. "In the beginning after I lost I felt annoyed and sad but now I'm happy that Russia has two medals and that I got the silver. I wasn't moving enough. Everybody has their own strategy and tactics, and I made some mistakes."
Like Velikaya, world No.2 Olga Kharlan from the Ukraine had to be happy with her second consecutive bronze medal after London 2012, defeating 20-year-old French sensation Manon Brunet 15-10.
Kharlan was not completely happy with her performance, with nerves taking over before the final match, but said: "I'm very happy that I got this medal. I can't say it's deserved, but every medal is a gold medal in the Olympics."