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Austrians Sebastian Proedl and Arnautovic scored midway through the first half as Bremen stormed to a 2-0 lead against a toothless Hoffenheim.
With home fans chanting "We want to see you fight", and jeering them off the pitch at halftime, the hosts looked transformed after the break.
Article continues below"This is a bitter defeat because we wanted a lot from this game," Babbel told reporters. "The first half was extremely weak and after the third goal my boys looked a bit broken. Now we are focusing on our game at Hamburg SV on Friday."
Their next opponents drew 1-1 at VfL Wolfsburg with Hamburg leading through Maximilian Beister's first-half goal and the hosts drawing level through a well-timed header by defender Simon Kjaer.
Wolfsburg, who hit the post through Vierinha and could have snatched all three points in the dying seconds when Srdjan Lakic's point-blank header was saved by Rene Adler, moved up to 16 points, a four-point buffer from the relegation zone. Hamburg are ninth on 21.
