Very nice flowchart. However that doesn't answer my question.
I think i explain it clear, but let me try again.. The question:
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Is it possible to keep one chain prior the fork of bitcoin and bitcoin cash then two chains after the fork, one for each coin?
Yes is possible to have one chain before the fork, because there was only one coin. And yes, after the fork we have two chains, one for the old coin and other for the new coin. But the chains should look equal on the first blocks but they will look different after the fork. and you can see that on a block explorer: