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Fabian Cancellara (Trek-Segafredo) missed out, however - his time only good enough to break into the provisional top five as his chances of a stage win in his final Tour de France faded.
Amid reinforced security, the race rolled on and thousands of people lined the 23-mile route in Ardeche to celebrate the Tour and pay homage to the dozens killed and injured by a truck which drove through beachfront crowds celebrating Bastille Day. "The race must continue". "Cycling gets a little less important on a day like this", said Tom Dumoulin after winning the 13th stage time trial Friday. But the French spirit is just as immutable and in a statement released before the race, Tour director Christian Prudhomme said the race must go on. "We think, after agreeing with authorities, that the race must continue".
But the carnage did not deter fans from showing up in numbers at the starting line in the small town of Bourg-Saint-Andeol.
Released last today with the time trial conducted in reverse order, Team Sky's two-time victor and defending champion clocked a finishing time of 51.18 to trail Dumoulin by one minute and three seconds.
After briefly racing up the mountain on foot before changing bikes, Froome lost nearly two minutes - enough to cost him the yellow jersey - until the race jury intervened and he came out with an increased lead, 47 seconds over Yates.
Time trial specialist Dumoulin won with a time of 50 minutes and 15 seconds over the rolling 37.5km stage, with Froome one minute and three seconds off that pace.
Froome gained around two minutes on Adam Yates (Orica-BikeExchange) and Nairo Quintana (Movistar) who drop to third and fourth overall.
The Tour de France resumed as planned on Friday but under heightened security as riders observed a moment of silence in tribute to the victims of the terrorist attack in Nice, France.
In a mark of respect for the victims of the Nice attack, the usual podium presentation and celebration was replaced by Froome, Dumoulin, best young rider Yates, Green Jersey holder Peter Sagan and King of the Mountains frontrunner Thomas de Gendt standing on stage for a minute's silence.
Later, Froome and Dumoulin spoke with outward emotion. "It's where I do most of my training", said Froome, who wore a black armband to his news conference.
"It's pretty clear today", said Froome, who lives near Nice in Monaco, "everyone's thoughts are with those affected down in Nice, and it's hard for us to be talking about the race".
There was a minute's silence after the stage, as the yellow, green and polka-dot jersey holders plus the stage victor gathered on the podium with bouquets while wearing black armbands. Our thoughts are with everyone, and the families in Nice. "I suffered more on the flat because of the wind". On Thursday we had the problems on Mont Ventoux; but this is only a bike race, and that was put into perspective by the events in Nice. "I'm happy with the decision", said Yates, who was initially given the race leadership according to preliminary results.