Lawson then put pressure on the Austrian driver over the closing laps but the order remained the same as they took the chequered flag, their cars nose to tail when they crossed the finish line.
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"I was getting told every lap no risk Liam, don't take any risks, but honestly I was in the zone," said Lawson at the podium.
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"It's not often that you get in the zone like I was just in then; when you're just flat-out every single lap, qualie laps, just trying to get close.
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"If I am honest I was absolutely flat-out trying to get to Lucas but he didn't put a foot wrong, so congrats to him and his whole team, they did a good job today with the pole and then to convert it into a win, but good points for us to take the lead in the championship. Now we have to maintain this for the last two rounds."
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Wittman held on to third place from the Audi of Kelvin van der Linde, who was the championship leader heading onto Assen, with Lawson's AlphaTauri AF Corse teammate Alex Albon in fifth.
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Heading into the last four races of the season, Lawson leads the DTM Championship with 175 points to Wittmann's 165, with van der Linde on 160 and Maxillian Gotz (Mercedes) with 155 points.
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The penultimate round will be at Hockenheimring in two weeks time, but in the meantime Lawson heads to Russia next weekend for his next round in the FIA Formula 2 championship in Sochi.
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DTM Championship - Top 6 points
- Lawson 175
- Wittman 165
- K van der Linde 160
- Gotz 155
- Philip Ellis (Mercedes) 109
- Albon 104.