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WASHINGTON — Venus Williams wanted to send a message — to herself and to others — about coming back from a long layoff, about competing in a sport at age 45, about never giving up. Yes, there was something special about just being back on a tennis court Tuesday night.

There also was this: She really, really wanted to win.

And Williams did just that, becoming the second-oldest woman to win a tour-level singles match in professional tennis, delivering some of her familiar big serves...