By your lights we should have to prove astrology or necromancy are not sciences, otherwise they should be taught in science class.
The assertion has been made that ID "is not science." By your own admission, this is a statement that cannot be proven since it is a negative. That means it must be left as an open question. Don't tell me about bantering sematics. The logic and meaning are clear as a bell.
To define science as only capable of treating "natural" phenomena is to set up a non-scientific standard and invite semantic bantering. What is the scientific definition of "natural?"