Yes, a trade made under BSE or NSE will be cleared by clearing corporations opted by the broker. This is called interoperability wherein a stock bought or sold on any exchange will be settled or cleared through a clearing corporation opted by the broker. 

For instance, a trade can be cleared by ICCL or NCL irrespective of the exchange on which the trade was .

Note:

• Share.Market supports the interoperability of exchanges on its platform
• Share.Market settles or clears all its trades through NSE Clearing Limited