Prince William said Tedworth House made him and his brother, Prince Harry, "very, very proud" as he declared the recovery centre officially open.
He said: "It is an enormous pleasure to be here at Tedworth House. This place, and what Help for Heroes and its partners have done here, makes Harry and me very, very proud."
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The threat level has been raised to critical by MI5's Counter-Terrorism Analysis Centre.
Prime Minister Theresa May announced the decision from Downing Street.
A man, apparently armed with a small axe and a large stick, was detained close to the event and police cleared the area as a precaution.