<?xml version="1.0"?>
<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E31">ZZ1d</font></span>: <a NAME="T29"><span class="comment"><font color="firebrick">:: FIELD_14:29</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">F</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a><br/>  for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being   non  <a href="ratfunc1.html#NV1" title="RATFUNC1:NV.1">zero</a>  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Olive" title="b1">F</font>  st (  for <font color="Olive" title="b4">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">F</font>  st <font color="Olive" title="b4">a</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Olive" title="b2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Olive" title="b3">q</font> holds <br/> <a href="uproots.html#K6" title="UPROOTS:func.6">multiplicity</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Olive" title="b3">q</font></span>)</span>,<font color="Olive" title="b4">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) holds <br/><span class="p1">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <font color="Olive" title="b3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </div></div>
