\[\sqrt{14 - 2\sqrt{13}} - \sqrt{17 - 4\sqrt{13}} =\]
\[= \sqrt{\left( \sqrt{13} - 1 \right)^{2}} - \sqrt{\left( \sqrt{13} - 2 \right)^{2}} =\]
\[= \sqrt{13} - 1 - \left( \sqrt{13} - 2 \right) =\]
\[= \underset{> 0}{\overset{\sqrt{13} - 1}{︸}} - \underset{> 0}{\overset{\sqrt{13} + 2}{︸}} = 1 \Longrightarrow\]
\[\Longrightarrow целое\ число \Longrightarrow ч.т.д.\]