deletions | additions       diff --git a/begin_align_p_i_k__.tex b/begin_align_p_i_k__.tex  index 7a9fc4f..d78a5e2 100644  --- a/begin_align_p_i_k__.tex  +++ b/begin_align_p_i_k__.tex 
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Given (2), we can apply the lemma we proved in section (a). That is:  \begin{align*}  p(i,k)&= p(i,j) + |jk|\\ |j,k|\\  &\leq t\cdot |ij| |i,j|  + |jk| |j,k|  \qquad\text{from (1)}\\ &\leq t\cdot |ik| |i,k|  \qquad\qquad\text{from above (2) &  lemma} \end{align*}  \end{align*} and our proof is complete $\square$.