deletions | additions       diff --git a/begin_align_p_i_k__.tex b/begin_align_p_i_k__.tex  index 78b918e..7a9fc4f 100644  --- a/begin_align_p_i_k__.tex  +++ b/begin_align_p_i_k__.tex 
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\begin{align*}  p(i,k)&= p(i,j) + |jk|\\  &\leq t\cdot |ij| + |jk| \qquad\text{from (1)}\\  &\leq t\cdot |ik| \qquad\text{from \qquad\qquad\text{from  above lemma} \end{align*}