A real is called properly n-generic if it is n-generic but not n+1-generic. We show that every 1-generic real computes a properly 1-generic real. On the other hand, if m > n ≥ 2 then an m-generic real cannot compute a properly n-generic real.
"Every 1-generic computes a properly 1-generic." J. Symbolic Logic 71 (4) 1385 - 1393, December 2006. https://doi.org/10.2178/jsl/1164060461