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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E33">Th31</font></span>: <a NAME="T31"><span class="comment"><font color="firebrick">:: RINGDER1:31</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">R</font> being   <a href="vectsp_2.html#NM2" title="VECTSP_2:NM.2">domRing</a><br/>  for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Olive" title="b1">R</font>  st <font color="Olive" title="b2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="field_1.html#NK3" title="FIELD_1:NK.3">anpoly</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Olive" title="b1">R</font></span>)</span>,1) holds <br/> for <font color="Olive" title="b4">i</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>  holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Olive" title="b3">q</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">q</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b4">i</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Olive" title="b3">q</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">R</font> )</div></div>
