  We prove a new structural lemma for partial Boolean functions $f$, which we
call the seed lemma for DNF. Using the lemma, we give the first subexponential
algorithm for proper learning of DNF in Angluin's Equivalence Query (EQ) model.
The algorithm has time and query complexity $2^{(\tilde{O}{\sqrt{n}})}$, which
is optimal. We also give a new result on certificates for DNF-size, a simple
algorithm for properly PAC-learning DNF, and new results on EQ-learning $\log
n$-term DNF and decision trees.
