Cross-chain double spending - possible?

[Slava79]

What if we have some valuable cryptocurrency (say, B), and a fork C was created:

We have addresses A1 and A2 on both blockchains.

Address A1 has high balance on both B and C (immediately after fork). Address A1 pays to address A2  big amount on chain C (fork).

Does it mean this transaction can be replayed on chain B as both public/private keys are the same?

Edit: question closed, I just read how replay attack works )