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If player $a$ has put player $b$'s king in check, but player $b$ has no legal move that will get him or her out of check, then player $a$ has won.

This situation is known as checkmate.

Player $a$ is said to have checkmated player $b$.

Also known as

The abbreviated mate is often heard, particularly in the triumphant announcement by the winning player, as he spanks down his piece:

"Check and mate."