<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/>

<b>assume </b><a NAME="E1:75"/><i><font color="Green">E29</font></i>: 
 <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<b>let </b><font color="Maroon">rho</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/>

<b>assume </b><a NAME="E2:75"/><i><font color="Green">E30</font></i>: 
 <a href="complex1.html#K3">Re</a> <font color="Maroon">rho</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/>

<b>assume </b><a NAME="E3:75"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">f</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a>
 ;<br/>

<a NAME="E4:75"/><b>then </b><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">rho</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">rho</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 
<b>by </b><i>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">ef</font> =  <a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">rho</font>, <font color="Maroon">ef1</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">rho</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  ;<br/>
<a NAME="E6:75"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">ef1</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">ef</font> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a>
 
<b>by </b><i>, , , </i>;<br/>
 <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ef1</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><font color="Maroon">ef</font> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">rho</font> = 
<span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2">(<span class="default"><font color="Maroon">ef1</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">rho</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2">(<span class="default"><font color="Maroon">ef</font> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span>,<font color="Maroon">rho</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom4.html#T24">POLYNOM4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">ef1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">rho</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2">(<span class="default"><font color="Maroon">ef</font> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span>,<font color="Maroon">rho</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T31">POLYNOM5:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">ef1</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">rho</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">ef</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">rho</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T31">POLYNOM5:31</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
;<br/>
<a NAME="E8:75"/><b>then </b>
<font color="Maroon">rho</font> <a href="polynom5.html#R1">is_a_root_of</a> <span class="p1">(<span class="default"><font color="Maroon">ef1</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">ef</font> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom5.html#D6">POLYNOM5:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">s</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E10:75"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">ef1</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">ef</font> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> <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">rho</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/>
<a NAME="E11:75"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">s</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a>
 
<b>by </b><i>, <a class="ref" href="hurwitz.html#T8" target="_self">Th8</a>, </i>;<br/>
<a NAME="E12:75"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">ef1</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">ef</font> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> <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">rho</font></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="hurwitz.html#T8" target="_self">Th8</a>, <a class="ref" href="polynom3.html#T29">POLYNOM3:29</a></i>;<br/>
<a NAME="E13:75"/>
 <a href="xcmplx_0.html#K4">-</a> 1 <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">rho</font></span>)</span>
 
<b>by </b><i><a class="txt" href="hurwitz.html#E6:75"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E14:75"/><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> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <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">rho</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E15:75"/>
<span class="p1">(<span class="default"><a href="hurwitz.html#K8" target="_self">F*</a> <font color="Maroon">f</font>,<font color="Maroon">rho</font></span>)</span> <a href="hurwitz.html#K5" target="_self">div</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">rho</font></span>)</span> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a>
 <b>by </b><i>, <a class="txt" href="hurwitz.html#E12"><i><font color="Green">Lemma72</font></i></a>, </i>;<br/>


</div>
