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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E58">pf1</font></span>: <a NAME="T54"><span class="comment"><font color="firebrick">:: RING_5:54</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">B</font> being   non  <a href="termord.html#V1" title="TERMORD:attr.1">zero</a>  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font><br/>  for <font color="Olive" title="b3">p</font> being    <a href="ring_5.html#M2" title="RING_5:mode.2">Ppoly</a> of <font color="Olive" title="b1">R</font>,<font color="Olive" title="b2">B</font><br/>  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">R</font> holds <br/> ( <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Olive" title="b4">a</font>)</span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">`^</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b4">a</font></span>)</span> <a href="ring_4.html#R1" title="RING_4:pred.1">divides</a> <font color="Olive" title="b3">p</font> &amp;  not <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Olive" title="b4">a</font>)</span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">`^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">B</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b4">a</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="ring_4.html#R1" title="RING_4:pred.1">divides</a> <font color="Olive" title="b3">p</font> )</div></div>
