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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a> <a href="vectsp_1.html#V9">Field-like</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>, <font color="Maroon">s</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:51"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 ;<br/>

<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">s</font> <a href="hurwitz.html#R1" target="_self">divides</a> <font color="Maroon">p</font>
 ;<br/><b>consider </b><font color="Maroon">t</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:51_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">p</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="hurwitz.html#K5" target="_self">div</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K9">+</a> <font color="Maroon">t</font> &amp;  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="ref" href="hurwitz.html#T1" target="_self">Th1</a>, </i>;<br/><font color="Maroon">p</font> <a href="hurwitz.html#K6" target="_self">mod</a> <font color="Maroon">s</font> = 
<font color="Maroon">t</font> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="hurwitz.html#K5" target="_self">div</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="hurwitz.html#K5" target="_self">div</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hurwitz.html#T3" target="_self">Th3</a>, <a class="ref" href="polynom3.html#T26">POLYNOM3:26</a></i>
<br/>.= 
<font color="Maroon">t</font> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T30">POLYNOM3:30</a></i>
<br/>.= 
<font color="Maroon">t</font>
<b>by </b><i><a class="ref" href="polynom3.html#T29">POLYNOM3:29</a></i>
;<br/><a NAME="E5:51_1"/><b>then </b>
<font color="Maroon">t</font> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:51_1"/><b>then </b>
<font color="Maroon">p</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hurwitz.html#K5" target="_self">div</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="hurwitz.html#T3" target="_self">Th3</a>, <a class="ref" href="polynom3.html#T29">POLYNOM3:29</a></i>;<br/><b>hence </b><a NAME="E7:51_1"/>
 ex <font color="Olive">t</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> st <font color="Olive">t</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">p</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">t</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font><b> such that </b><a NAME="E1:51_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">t</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:51_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">p</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="polynom3.html#T29">POLYNOM3:29</a></i>;<br/><a NAME="E3:51_2"/>
<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="hurwitz.html#T1" target="_self">Th1</a>, <a class="txt" href="hurwitz.html#E1:51"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E4:51_2"/><b>then </b>
 <a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/><a NAME="E5:51_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hurwitz.html#K5" target="_self">div</a> <font color="Maroon">s</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="hurwitz.html#T3" target="_self">Th3</a>, </i>;<br/><a NAME="E6:51_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hurwitz.html#K6" target="_self">mod</a> <font color="Maroon">s</font> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="polynom3.html#T30">POLYNOM3:30</a></i>;<br/><b>hence </b><a NAME="E7:51_2"/>
<font color="Maroon">s</font> <a href="hurwitz.html#R1" target="_self">divides</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:51"/>
( <font color="Maroon">s</font> <a href="hurwitz.html#R1" target="_self">divides</a> <font color="Maroon">p</font> iff  ex <font color="Olive">t</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> st <font color="Olive">t</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="hurwitz.html#T2" target="_self">Th2</a></i>;<br/>


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