How use multiple keys to sign the same document?
Teach the verifier what set of keys must sign a document for it to be valid.  When the verifier receives a document, it must have a valid signature by every one of those keys affixed to it.

Then before a verifier will accept a signed document, the possessor of every one of the signing keys must sign it.

That's it.  That's enough for the security properties you've described.

It's not the most efficient scheme.  There are more efficient schemes, e.g. in an earlier answer I wrote to a related question, but this will satisfy the security properties you asked for and can be built out of any ordinary public-key signature scheme.