The non incompatible transactions will be present in both forks, and they will get validated and fixed whatever fork wins. This includes most of the transactions that are not related to the double spending.
In the case of a double spending, each transaction can live in only one of the forks, and the decedents [1] transactions can be validated only in the same fork. When a fork wins, all the transactions that are incompatible can’t be added and they disappear.
[1] Sopuse that X sends the same bitcoins to A and B. Then A send these bitcoins to A1, that sends these bitcoins to A2. During the same time B send these bitcoins to B1, that sends these bitcoins to B2. Then you have two forks, one with the A, A1 and A2 transactions and another with the B, B1 and B2 transactions. Only these transactions have problems and one half of them will disappear, all the other transactions will apear in both forks.
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The non incompatible transactions will be present in both forks, and they will get validated and fixed whatever fork wins. This includes most of the transactions that are not related to the double spending.
In the case of a double spending, each transaction can live in only one of the forks, and the decedents [1] transactions can be validated only in the same fork. When a fork wins, all the transactions that are incompatible can’t be added and they disappear.
[1] Sopuse that X sends the same bitcoins to A and B. Then A send these bitcoins to A1, that sends these bitcoins to A2. During the same time B send these bitcoins to B1, that sends these bitcoins to B2. Then you have two forks, one with the A, A1 and A2 transactions and another with the B, B1 and B2 transactions. Only these transactions have problems and one half of them will disappear, all the other transactions will apear in both forks.