Довести, що число [latex] \sqrt{19- \sqrt{3-8 \sqrt{35-8 \sqrt{19 } } } [/latex] цыле

[latex]\sqrt{19-\sqrt{3-8\sqrt{35-8\sqrt{19}}}}=\\\ \sqrt{19-\sqrt{3-8\sqrt{(\sqrt{19})^2-2*4*\sqrt{19}+4^2}}}=\\\ \sqrt{19-\sqrt{3-8\sqrt{(\sqrt{19}-4)^2}}}=\\\ \sqrt{19-\sqrt{3-8(\sqrt{19}-4)}}}=\sqrt{19-\sqrt{3-8\sqrt{19}+32}}=\\\ \sqrt{19-\sqrt{35-8\sqrt{19}}}=\sqrt{19-\sqrt{(\sqrt{19}-4)^2}}=\\\ \sqrt{19-\sqrt{19}+4}}=\sqrt{23-\sqrt{19}}[/latex]