The Spanish topflight side sparked hopes of a fairytale finish after taking the lead against the Premier League giants, but it was short-lived.
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The Blues responded with a dominant second-half performance to seal a commanding win and complete their European trophy collection.
"The sadness is for getting so close… but the pride I feel now is much greater!" Anthony wrote on Instagram.
"How proud I am to wear this shirt and build such a beautiful story. We made history, know that. To all the fans who followed us at every moment, I say with all the sincerity in the world: you made us stronger!"