You sort of made a 'general' statement --- which you have now "clarified". That's fine.
How-ever: In your EN-'sample', the new corporation still cannot have the "same"/"identical" name --- RE-gardless of owner-ship. There MUST be some "variation" --- arms length, and all of that! And BOTH must be appropriately RE-gistered! And BOTH must keep separate 'financial' RE-cords!