What actually is meant by shares submitted?

For all I know, a block is mined when the header hash is lower or equal to the current difficulty. That's true on a single miner.

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What about in a pool mining where there's multiple miner mining a single block? How does the pool know that each miner is actually hashing billions of times?

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To verify it, the miner could send all mining blobs and it's hash, and the server would verify that all this blobs, matches with the hash submitted together, confirming that they're doing honest work, but thats just more work for the pool.