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But straight after half-time, new Lyon signing Nicolas Tagliafico made up for giving away the penalty by putting the home side back ahead.
Tete then scored for a 3-1 lead in the 49th minute and rammed home the fourth in the 76th minute to increase his season's tally to three.
It could have been five in the last minute but teenage substitute Rayan Cherki crashed his effort against the woodwork.
Troyes have lost all three of their games in Ligue 1 this season and have slipped to the bottom of the French league table.