yes your rival will heal your Pokemon before you fight Cyrus.
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the top of mt cornet is spear pillar
Yes.
You have to go to the top of mt conoret, where you'll have a battle with your rival against commander Mars and Jupiter. After that you'll go to the distortion world where you have to beat Cyrus and fight Gratina. Once all that is completed, team Galactic will be gone from mt Coronet. To find Cyrus and the top of mt Coronet: http://www.mahalo.com/Pokemon-platinum-walkthrough-part-6/
There is a map of Mt. Coronet in the Sinnoh region.
Mt coronet is only in diamond or pearl or platinum not in firered.