Hmm, the symbology is wrong, in that it is customary to write time-domain functions in lowercase, and frequency-domain functions in uppercase. It should be y(t) = ... x(τ) ... (missing), and Y(s) = ... X(s) ... (correct). At least, that's the convention over here?
The first steps are just showing the identities of integration and differentiation, which can be proved with the full form of the Laplace transform integral.
The last step is not justified: the derivative is equal only when dt --> 0.
The notation is also incorrect again: the discrete-time function is usually shown with a subscript, or a square-brackets parameter. And the Z-domain function is again capitalized (discrete-time, lowercase), using z as the parameter.
What is needed, is to show an equivalence between differentiation in continuous-time functions, and differencing in discrete-time functions. To a suitable degree, it turns out they are, but this requires a different kind of proof, of course. You can't simply equate them, because a discrete-time function can't ever be equal to a smooth, continuous-time function.
As mentioned, there is a transformation between poles and zeroes in s+jw frequency space, and poles and zeroes in Z space. It's been so long I forget which one is easier to show and work from -- it may be that the transformation itself is the easier equivalence to prove, and then everything else follows from that.
Tim