PARIS, May 6 (Xinhua) -- Right winger Nicolas Sarkozy claimed victory in the French presidential run-off on Sunday, defeating Segolene Royal of the left-wing Socialist Party.

    "I should (be) the president of all the French people and should speak on behalf of all the French," Sarkozy told thousands of jubilant supporters at the headquarters of his ruling Union of a Popular Movement.

    "This is not the victory of one France over another. It is the victory of democracy, and of French values," said Sarkozy, who tried to woo support from those who had not voted for him.

    He said his first priority will be to do everything to ensure that "the French people continue to speak to each other, to understand each other, and to work together."

    According to early projections released by French TV channels, Sarkozy won 53 percent of the vote against 47 percent for Royal, who had been bidding to become France's first woman president.