Some "Theorem" on the generalized superfunction

[Leo.W]

Ok, I'm really confused now. I think there is a problem and maybe I know where it lies.
We have to reach a consensus on the definitions.
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We can make the thread more readable and short (less pages).

Pardon me, I didn't realized that when I simply pushed the reply button    I'll remove it really soon.
Your reply is really constructive and meaningful.
I never read about algebraic formal definition about multivalued functions, so that in my work I always handle it as ''a set in a set'', which can be a little confusing. Could you please give me some references? I'll be very grateful.
The law 5 is indeed strongly-stated, but it has an analogy to the generalization of iteration. I think, in spite of this generalization can cause some monsterous behavior, we can treat it as the methods to move between infinitely many branch cuts or solutions, and sometimes it does help, for instance, consider the case I mentioned before, f(z)=z+sqrt(z) and g(z)=z^2, we can generalize many different branch cuts of f, each one should satisfy most properties which f satisfies. 
I only have a informal statement about these multivalued iterations, it's ''choose an appropriate branch cut/value'' to satisfy the equation, and this can be observed through many constructions of zeta. The case f(z)=z^2+v is the one I pay most attention to, in which I use the multivalued property to make the whole plane suitable to the equation in some sense.
Also, the multivalued iterations of sine function and exponential function is also tested, you can see supersine and superlog has infinitely many branches, and it's still unproved that you can't choose a branch to fit in the equation in a certain domain. The iteration world is still so wide to explore!
Your set definition is gorgeous and already applicable in the various field of single-valued functional iterations, it's very helpful! I'm concerned about more accuate statements of the definition of multivalued functional iterations and can't wait any longer
I also have other laws which is unproved nowadays, someday I'll post those and your help would be really appreciated!
Regards, Leo