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It can't prove the mechanism because there is no way to tell what else is happening. The rate law is taken only from the slow steps, and doesn't account for what is happening in the faster steps.
So, the reaction mechanism describes the sequence of elementary reactions that must occur to go from reactants to products. Reaction intermediates are formed in one step and then consumed in a later step of the reaction mechanism (can be used in faster steps that are not shown), which kinetic information is unable to prove because only the slowest step in the mechanism is used to determine the rate.
Reaction mechanisms are found experimentally, and are very similar to putting together a puzzle. You know through experimental data that a mechanism will create the final reaction, but you need to (through trial and error) figure out how the initial reaction gets to its final form. Creating a mechanism is like finding the best fit that explains your results.
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