Proof of Possession: Does a Schnorr Signature done with the sum of 2 Elliptic Curve private keys prove possession of the 2 individual keys?

G is the Generator of a Discrete Log Hard Elliptic Curve Group.

2 Private keys x1 & x2, corresponding Public Keys P1 = x1G & P2 = x2G.

Now P = P1 + P2 is also a public key with corresponding private key x = x1 + x2.

If I sign (Schnorr Signature) with x, does it only prove possession of the private key corresponding to P or does it also prove possession of the 2 individual public keys x1 & x2? Or if not proof of possession of both x1 & x2, does it atleast prove something more than just x?

I am looking up Monero Documents & they seem to do this (MLSAG) & it's kind of confusing me.