My composite control has a private System.Web.UI.Button member, which is
exposed by a read-only public property "Button". In design-time, when I'm
using the composite control, I see the "Button" property and can expand it
in Events tab of Properties window, but when I double click on any nested
event, designer generates appropriate event handler but doesn't bind to it
in the "InitializeComponent" method.

Thanks in advance!