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<p>The American President (Michael Douglas) is a widower, who because of his position is reluctant to resume dating. Annette Bening is a lobbyist. The President invites Ms. Bening to a state dinner for the President of France. The problem is that nobody speaks French and the President (of France) and his wife are bored to tears. Ms. Bening notices the problem and immediately engages the President of France and his wife in a lively conversation thereby saving Michael Douglas from an embarrassing political faux pas. That’s when he puts his hand on her shoulder and says “My date”.</p>
<p>So it is sort of shorthand for “This person is with me and I am proud of what they just did”.</p>