The HM that you get after the fourth gym in Pokémon is surf. This is a move that can be learned by various Pokémon in the game.
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You beat Chuck, the 4th gym leader, in Cianwood city. After you beat him, leave the gym. His wife will come up to you and give the HM to you.
The 6th gym (Steel type) does not unlock any new HM , instead the 16th Gym (Blue's Gym) unlocks Rock Climb
u get to use certain HM's once u beat the gym leaders
To make fantina (hearthormes gym Leader) appear you have to get the hm surf before you challenge the gym leader. To get the hm you have to go to celistec town and talk to a old lady and she says to battle a galactic guy and you have to go in the ruin and she will go in to and she will give you the hm and the gym leader will appear.
After you beat the 4th, grass, gym you can go to the middle house in the bottom right of the city and grab a vulpix from the lady in there. If you check the Vulpix, it has HM 04, strength, as a held item