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lay down
lie down

Lay down has several different meanings.

If you lay something down it can mean you officially establish a rule, or officially state the way in which something should be done.

For example:-

Please follow the rules laid down by the administrator.

If you lay down your weapons it means you stop fighting.

For example:-

They laid down their guns and surrendered.

If you lay wine down it means you are storing it in order to drink it in the future.

For example:-

I laid down this bottle in 1998, it should be perfect for drinking now.

Lie down means to move into a position in which your body is flat, usually in order to sleep or rest.

For example: "If you are tired lie down and have a rest."