If a group of people participate in a mining pool and person 1 finds a valid proof of work on their machine what prevents them from submitting this proof of work for themselves and not giving it to the pool to share between the participants?
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The very last line of a block effectively contains:
Reward for Solving this Block: Add 12.5BTC to Account for [Mining Pool].
If the miner tried to change that line to read
Reward for Solving this Block: Add 12.5BTC to Account for [My Personal Account].
the altered line would change the block, which would change the hash, and their Proof of Work answer would no longer be valid.
So if a miner solves a block on behalf of the pool, they cannot submit that block for themselves; it will no longer be a valid solution.

abelenky
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This is correct in the essence, to be more specific though, the payout instruction is in the coinbase transaction. – Murch Jun 16 '17 at 23:25