Flux rule for motional emf

The flux rule for motional emf can be stated as:

d$\phi$/dt = -$\epsilon$

I have some questions regarding this. 1) Should I find this rule intuitive?
And secondly a proof of this rule is given in my book, which I have attached. I don't really understand the idea of the proof - is the idea to proof the above rule or to just proof that if it works, it will do so for loops, which "do not even maintain a fixed shape". Because the words seem to suggest the later but on the other hand that would mean that the above rule is generally unproved in my book.

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The flux rule for motional emf can be stated as:

d$\phi$/dt = -$\epsilon$

I have some questions regarding this. 1) Should I find this rule intuitive?[..]
Just answering 1: You may find it intuitive by means of Lenz's law, which is a qualitative version of the above. It has similarity to Newton's third law, in the sense of action is reaction.

okay but Lenz' law is just a more qualitative statement of the flux rule - but perhaps there isn't more to it.
What about 2)? That one was more of a puzzle to me.

okay but Lenz' law is just a more qualitative statement of the flux rule - but perhaps there isn't more to it.
What about 2)? That one was more of a puzzle to me.
It seems to prove the flux rule from their definition of flux plus the Lorentz force law. Both should appear earlier in the text.