It seems like you are trying to do something that an interface isn't meant for. An interface is simply a contract and, when a class implements that contract, it is promising that it will fulfill it in some way or another, but the interface itself doesn't care since it doesn't concern itself with implementation details. If you want to pass behavior on to children, you need to subclass.
However, if you really are trying to create an interface that "carries functions" over from a parent interface, that is definitely supported and easily accomplished. All you need to do is create a child interface to inherit your parent.
Example:
interface IParentInterface
{
int FirstProperty {get;set;}
void OnChange();
}
interface IChildInterface: IParentInterface
{
string SecondProperty {get;set;}
}
class InterfaceInheritanceGoodness: IChildInterface
{
public int FirstProperty { get; set; }
public string SecondProperty { get; set; }
public void OnChange()
{
throw new NotImplementedException();
}
}
And, interfaces support multiple inheritance... Have fun!