At the end of part II of Appen. A (aren't I a dweeb to start a sentence like that?), it says that Éomer renewed the Oath of Eorl...
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Often he fulfilled it. For though Sauron had passed, the hatreds and evils that he bred had not died, and the King of the West had many enemies to subdue before the White Tree could grew in peace. And whenever King Elessar went with war King Éomer went with him; and beyond the Sea of Rhûn and on the far fields of the South the thunder of the calvary of the Mark was heard, and the White Horse upon Green flew in many winds until Éomer grew old.
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So maybe not "vanquishing", but beating some kind of battles against them.