Yes I saw the little mistake I made where I assumed . The reason is because I've only really been observing functions which meet that requirement.

I think I'm not doing anything inconsistent but instead we have to create the law, which is not dissimilar to division by zero:

or that the distributive law fails when f-multiplied by the identity.

I looked at that other thread too, very interesting. I had of hunch bo's proof but it's nice to see it proved.

but furthermore, this gives some very strange laws for multiplication:

which means for exponentiation:

which means f-multiplying a number to the power of another number we convert exponentiation to f-exponentiation:

which again is very very inconsistent. I must be doing something incorrect. I think we cannot give the distribution law, but that's not enough for me. I'd really like to know why. I'm absolutely puzzled.

I think I'm not doing anything inconsistent but instead we have to create the law, which is not dissimilar to division by zero:

or that the distributive law fails when f-multiplied by the identity.

I looked at that other thread too, very interesting. I had of hunch bo's proof but it's nice to see it proved.

but furthermore, this gives some very strange laws for multiplication:

which means for exponentiation:

which means f-multiplying a number to the power of another number we convert exponentiation to f-exponentiation:

which again is very very inconsistent. I must be doing something incorrect. I think we cannot give the distribution law, but that's not enough for me. I'd really like to know why. I'm absolutely puzzled.