Is the inheritance of the children from the first wife from the apartment permissible (halal) or forbidden (haram), if the second wife is the one who paid for the apartment?

If the apartment is registered in the father's name, then the children have a right to inherit from it, unless there is proof that it belongs to the second wife or unless the father's acknowledgment of that is established. The mere payment of the apartment's price by the wife does not necessarily mean it is her property; it could have been a loan. If the second wife proves her ownership of the apartment, then the children have no right to it. Otherwise, it is considered part of the father's inheritance. This issue can only be resolved through the Sharia (definition) court.