zero-knowledge proof of disjunctive statements (OR proofs)
You can use the CDS94-technique for that. Suppose you have two zero-knowledge proof systems $\sigma_1$ and $\sigma_2$, both of which consist of three messages: a commitment $com$, a (public-coin) challenge $ch$, and a response $r$. And suppose you want to prove the OR of both claims, meaning that you can generate a valid proof for (wlog.) $\sigma_1$ for any challenge even after generating the commitment, but in order to generate a valid proof for $\sigma_2$ you must get the challenge first and compute the commitment from there. So you have essentially one degree of freedom in the choice of $(ch_1, ch_2)$, i.e., one is fixed beforehand but you want to hide which one. The clue is to get a new challenge $ch'$ from the verifier and guarantee that $(ch_1, ch_2, ch')$ satisfies a suitable relation, for instance sum-to-zero. So the proof system for $\sigma_1$-OR-$\sigma_2$ looks like this:

preprocessing by the prover: compute $com_2, ch_2, rsp_2$ for a random $ch_2$
prover sends commitment: $com_1, com_2$
verifier sends challenge: $ch'$
prover computes $ch_1 \leftarrow ch' - ch_2$ and uses this to complete $\sigma_1$
prover sends response: $(ch_1, rsp_1), (ch_2, rsp_2)$
verifier verifies that $(com_1, ch_1, rsp_1)$ is valid, that $(com_2, ch_2, rsp_2)$ is valid, and that $ch_1 + ch_2 + ch' = 0$

You can use $\sum_i ch_i = ch'$ for OR-proofs consisting of any number of claims. However, in some cases you want to prove more specific facts such as &quot;$t$ out of these $n$ claims are true&quot;. In this case, you want to use Shamir's secret sharing and exchange the sum-to-zero relation for a polynomial of degree $n+1-t$: this guarantees that all $n-t$ degrees of freedom for choosing $ch_i$ must be used up by the false claims.
The CDS-94 technique applies to any zero-knowledge proof system that follows this three-pass public coin structure. Schnorr's protocol for proving discrete logarithm knowledge is just one particular case of that structure.
CDS94: Cramer, Ronald, Ivan Damgård, and Berry Schoenmakers. &quot;Proofs of partial knowledge and simplified design of witness hiding protocols.&quot; CRYPTO 1994.