<?xml version="1.0"?>
<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E31">Th31</font></span>: <a NAME="T31"><span class="comment"><font color="firebrick">:: UNIROOTS:31</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">n</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <br/>  for <font color="Olive" title="b2">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <br/>  for <font color="Olive" title="b3">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K17" title="SIN_COS:func.17">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> holds <br/> <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Olive" title="b1">n</font></span>)</span></div></div>
