A question about AP. 
How is the encircled step justifiable? According to my knowledge I can substitute m=any variable but how can I substitute m=2m-1, isn't it the same as assuming m=1? 

The question itself is confusing. What it means is that the condition is true for any $m,n$ you care to choose, rather than a specific $m,n$.
Given that it is true for any $m,n$ it is somewhat confusing to use the same symbols in the suggested answer, as has been done.
Maybe clearer to say that if it is true for any $m,n$ then we can pick the specific values $m=2r-1, n=2t-1$ and go from there.
It doesn't matter what you call your variables - you can give them what names you like. So having worked out an expression in $r,t$, which is true for any $r$ and $t$, it is true in particular for $r=m, t=n$.
The essential thing here is that the expression in $r$ and $t$ has been shown to be true for any choice of $r$ and $t$ (or any choice of interest) and does not depend on the original relation with $m$ and $n$.
