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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> non <a href="vectsp_1.html#V10">degenerated</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>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>





<b>assume </b><a NAME="E1:53"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font> &amp; <font color="Maroon">z</font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">p</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">s</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:53"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">p</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:53"/><i><font color="Green">E31</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>
 
<b>by </b><i><a class="ref" href="hurwitz.html#T1" target="_self">Th1</a>, <a class="ref" href="hurwitz.html#T2" target="_self">Th2</a>, <a class="ref" href="polynom3.html#T35">POLYNOM3:35</a></i>;<br/>
<a NAME="E5:53"/><i><font color="Green">E34</font></i>: 
 <a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="hurwitz.html#T1" target="_self">Th1</a>, <a class="ref" href="hurwitz.html#T2" target="_self">Th2</a>, <a class="ref" href="polynom4.html#T5">POLYNOM4:5</a></i>;<br/>
<b>then </b><i><font color="Green">E36</font></i>:  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="hurwitz.html#T2" target="_self">Th2</a>, , </i>
<br/>.= 
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="txt" href="hurwitz.html#E6:53"><i><font color="Green">E36</font></i></a></i>
;<br/>
<a NAME="E7:53"/>
 ex <font color="Olive">t</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">p</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="polynom3.html#K15">*'</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <font color="Olive">t</font> &amp;  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Olive">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">t</font> =  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>;<br/>

<b>thus </b><a NAME="E1:53_2"/>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="polynom3.html#K15">*'</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <font color="Maroon">t</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>, <a class="ref" href="polynom3.html#T29">POLYNOM3:29</a></i>;<br/>

<a NAME="E2:53_2"/>
 <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> 1
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:53_2"/>
 <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> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E6:53"><i><font color="Green">E36</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:53"/>
 <a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hurwitz.html#K5" target="_self">div</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="hurwitz.html#T3" target="_self">Th3</a>, </i>;<br/>


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