Medium run conclusions from Okun's Law and Expectations-Augmented Philips Curve In Blanchard's Macroeconomics, page 211, 5th edition, the author using the following three mathematical equalities


Okun's law : $u_t-u_{t-1}=-\beta(g_{yt}-\bar{g_y})$, where
$\bar{g_y}$ is defined as the output growth rate when the
unemployment rate is constant, $g_{yt}$ output growth rate from year
t-1 to t.
expectations-augmented Philips curve, 
and assuming a functional form  for the aggregate demand relation
that allows to deduce  $g_{yt}=g_{mt}-\pi_t$, where $g_{mt}$ is nominal
money growth rate


makes a reasoning on what happens in the medium-run when the central bank maintains a constant growth rate of nominal money, concluding that although changes in the growth rate of nominal money doesn't change unemployment rate, it does change the inflation rate one-for-one. One of the first assumptions of the reasoning is that in the medium run the unemployment rate is constant. But why can we assume this?

In previous chapters, we had assumed that output would change one-for-one with employment, and that the labour force was constant. Then, when the expected price level were equal to the actual price level, the economy would be in natural output level, and that would imply from previous two assumptions that unemployment rate would also be the natural one. However, in this chapter, when the author was deriving the Okun's law, we explicitly assumed that labour force was no longer constant, and that employment would respond less than one-for-one to output, and that changes in employment are not negatively reflected one-for-one in the unemployment. 
Also, in same page 211, it's the author's conclusion that the unemployment rate in the medium-run is the natural unemployment rate. So, we cannot assume what we're trying to conclude... 

Therefore, why can we assume that the unemployment rate must be constant in the medium run?

Any help would be appreciated.
At least in the 3d edition, Blanchard writes:


  "In the medium run, the unemployment rate must be constant (my emphasis).
  The unemployment rate cannot be decreasing or increasing forever."


This is an inherent characteristic of the definition of the "medium run": that in it, some core magnitudes are constant. In some models, levels are constant (eg. aggregate consumption, aggregate capital stock, etc). In other models, growth rates are constant, and the model "has a steady-sate in growth rates". This means that levels grow indefinitely of course, but it is not wise (or useful, or meaningful) to project economic concepts to "infinite future". 

Now for unemployment, it makes no sense to not be constant in equilibrium (at this simple level of exposition). In more sophisticated models, one could have the unemployment rate approaching asymptotically a constant (zero or not) without ever actually reaching it, but this is usually mathematical gadgetry with not much of economic intuition. (And in futuristic models, one could have unemployment approaching asymptotically unity, as machines and Artificial Intelligence take over all jobs). 

Can this constant-by-conception-in-the-medium-run unemployment rate $u^* =const.$ be something else than the "natural rate of unemployment" $u_n$?

The natural rate of unemployment is  defined through the Phillips Curve, as that rate of unemployment at which realized inflation equals expected inflation (see previous chapters). Note that it is not assumed a priori that the natural rate of unemployment will be strictly positive -it could very well be zero. It is actual data that tells us that inflation stabilizes at strictly positive rates of unemployment. Then it is constant when inflation is a constant, and Blanchard discusses why inflation will also be a constant in the medium run. So it obtains that in the medium run $u_t = u_n =const.$.  

Since the current unemployment rate stabilizes when inflation becomes constant, which happens in the medium run, the constants must coincide. If $u^* \neq u_n$ inflation will not be constant and we would violate another condition for the characterization of the medium run.

And Blanchard writes correctly


  "In the medium run the unemployment rate must be equal to the
  natural rate of unemployment". 


So it does not assume what he wants to prove. The natural rate of unemployment is defined separately and with its own logic to be a constant, from the medium-run unemployment rate which is constant based on other arguments than those related to the natural rate of unemployment. 

I hope this helps.