"It's going to hurt for a while of course. We can hold our heads up high. It's been a fantastic journey, we got further than anyone else thought we would have," he added.
Playing in their first World Cup semi-final since 1990, England took a fifth-minute lead through Kieran Trippier's superb free-kick.
"It's been great to get to this stage and we know we've done everyone proud but we wanted to go on and win it," Kane added.
"We thought we were just good enough, we thought we could have done that. But we've fallen just a bit short. It hurts. I don't know what else to say.
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