The Germany international was unveiled at Arsenal on Thursday, following his €47 million (£42m) move on deadline day.
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"Ozil decided to leave," he told the club's official channel. "He is a good player who has given us so much so we wish him well.
"There are players who have responded with strength and will to fight for a place, but not everyone reacted the same.
"[Angel] Di Maria responded well and wanted to fight."
The Blancos travel to newly-promoted Villarreal in the Primera Liga on Saturday where €100 million (£85m) recruit Bale could be handed his debut.