Δ and E are just functions whose inputs and outputs are other functions. Δ = E - 1 is just an equality of functions, just like sin^(2) + cos^(2) = 1 which you are probably more familiar with.
>Autor basically takes f(x) common and after that cancels f(x). How is that possible?
if f and g are functions, the equation f = g is the same as saying: for all input values x, f(x) = g(x). the same is true here, except the inputs to Δ and E are functions, not numbers. Δ = E - 1 means: for all input functions f, Δ(f) = (E-1)(f), which in turn means: for all functions f and inputs x, Δ(f)(x) = (E-1)(f)(x).