We have shown how to obtain Szemeredi-Trotter from the cutting lemma and the weak Sz-T (homework) that bound the number of incidences by O(m.n^1/2 +Â n). We set r=m^2/3 / n^1/3 and then apply cutting lemma, which, as noted by Pradeep, can be used only when 1<=r<=n. (If r is out of this range, then the weak Sz-T (or its dual) already implies Sz-T.) Inside each of the resulting O(r^2) cells we use the weak Sz-T bound. Summing these up we get at most O(m^2/3.n^2/3) incidences. This also works for the points on the boundary (but not in the vertices!) of the cells, just imagine they are included in both the cells they are incident to. Finally, for the points in the vertices denote their degree by d_i. Each of them can be incident to at most d_i O(n/r) lines. Summing these up and using Euler's formula we get O(nr)=O(m^2/3.n^2/3) incidences again.