"It means more than extra", said Tsitsipas, the Greek player who kicked off his fine 2019 by reaching the semi-finals of the Australian Open.
The Swiss 20-time Grand Slam victor recovered quickly from being broken in the first game of the match but the Austrian fifth seed repeated the dose in the 11th game and took the opening set 7-5.
Federer recovered the early break but a couple of lazy strikes at 5-5 cost him his serve again before Thiem closed out the opener in confident fashion.
But 26-year-old Thiem was one of a rare band of players to have played Federer at least half a dozen times and he held a winning head-to-head, 4-2.
If Nadal is healthy, he arrives in London as the player of the year so far, a fact confirmed by his return to the top of the rankings this week.
The Serbian must reach the final for a chance to be year-end No 1 for a sixth time, which would match the record of Pete Sampras (1993-98). "That's how it is every week of the year for the last 20 years, so from that standpoint there is nothing new there".
Swiss tennis maestro Federer, who is bidding for his seventh ATP Finals title and first since 2011, struggled against Thiem's aggressive game to eventually suffer a 5-7, 5-7 defeat in a thrilling clash that lasted one hour and 40 minutes.
The Spaniard, who has never won the ATP Finals despite his trophy-laden career, dethroned Djokovic from the top of the rankings this week.
As an opening singles match in a tournament featuring the world's best eight players, it was a disappointing spectacle as the 23-year-old Berrettini succumbed all too easily.
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A former India A player, Gautam has had stints in the IPL for Royal Challengers Bangalore, Mumbai Indians and Delhi Daredevils. The coach allegedly manipulated a game between Bengaluru Blasters and Belagavi Panthers past year .
Nadal, takes on defending champion Alexander Zverev in the later match in Group Andre Agassi with major questions over his fitness since he withdrew injured from the Paris Masters at the semi-final stage.
But Federer lost 7-5 7-5 against fifth seed Thiem at London's O2 Arena.
The second set went with serve until Thiem broke at love for another 6-5 lead.
The pair have a spiky relationship.
Since that match, Medvedev had earned another four straight wins over Tsitsipas, including in the Shanghai semifinals in October.
World number four Medvedev is looking to add lustre to an impressive breakout year by winning the glitzy end-of-season event in London.
For Federer must regroup quickly, sharpen up, and prepare for the big-hitting Berrettini-and then another man to have scored heavily against him of late, Novak Djokovic.