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<b>let </b><font color="Maroon" title="c1">f</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/>

<b>assume </b><a NAME="E1:75"/><i><font color="Green" title="E65">A1</font></i>: 
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ;<br/>

<b>let </b><font color="Maroon" title="c2">rho</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/>

<b>assume </b><a NAME="E2:75"/><i><font color="Green" title="E66">A2</font></i>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">rho</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 ;<br/>

<b>assume </b><a NAME="E3:75"/><i><font color="Green" title="E67">A3</font></i>: 
<font color="Maroon" title="c1">f</font> is <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>
 ;<br/>

<a NAME="E4:75"/><b>then </b><i><font color="Green" title="E68">A4</font></i>: 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">rho</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c2">rho</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 
<b>by </b><i><a class="txt" href="hurwitz.html#E1:75"><i><font color="Green" title="E65">A1</font></i></a>, <a class="txt" href="hurwitz.html#E2:75"><i><font color="Green" title="E66">A2</font></i></a>, <a class="ref" href="hurwitz.html#T50" target="_self" title="HURWITZ:th.50">Th50</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">ef</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">rho</font>, <font color="Maroon" title="c4">ef1</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c2">rho</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  ;<br/>
<a NAME="E6:75"/><i><font color="Green" title="E69">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">ef1</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ef</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span></span>)</span> is <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>
 
<b>by </b><i><a class="txt" href="hurwitz.html#E1:75"><i><font color="Green" title="E65">A1</font></i></a>, <a class="txt" href="hurwitz.html#E3:75"><i><font color="Green" title="E67">A3</font></i></a>, <a class="txt" href="hurwitz.html#E4:75"><i><font color="Green" title="E68">A4</font></i></a>, <a class="ref" href="hurwitz.html#T51" target="_self" title="HURWITZ:th.51">Th51</a></i>;<br/>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">ef1</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">ef</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">rho</font> = 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">ef1</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c2">rho</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">ef</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span></span>)</span>,<font color="Maroon" title="c2">rho</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom4.html#T24" title="POLYNOM4:th.24">POLYNOM4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">ef1</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">rho</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">ef</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span></span>)</span>,<font color="Maroon" title="c2">rho</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T31" title="POLYNOM5:th.31">POLYNOM5:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">ef1</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">rho</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ef</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c2">rho</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T31" title="POLYNOM5:th.31">POLYNOM5:31</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
;<br/>
<a NAME="E8:75"/><b>then </b>
<font color="Maroon" title="c2">rho</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">ef1</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ef</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom5.html#D6" title="POLYNOM5:def.6">POLYNOM5:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">s</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> <b> such that </b><br/><a NAME="E10:75"/><i><font color="Green" title="E70">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">ef1</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ef</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c2">rho</font></span>)</span> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <font color="Maroon" title="c5">s</font>
 <b>by </b><i><a class="ref" href="hurwitz.html#T33" target="_self" title="HURWITZ:th.33">Th33</a></i>;<br/>
<a NAME="E11:75"/><i><font color="Green" title="E71">A7</font></i>: 
<font color="Maroon" title="c5">s</font> is <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>
 
<b>by </b><i><a class="txt" href="hurwitz.html#E6:75"><i><font color="Green" title="E69">A5</font></i></a>, <a class="txt" href="hurwitz.html#E10:75"><i><font color="Green" title="E70">A6</font></i></a>, <a class="ref" href="hurwitz.html#T41" target="_self" title="HURWITZ:th.41">Th41</a></i>;<br/>
<a NAME="E12:75"/><i><font color="Green" title="E72">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">ef1</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ef</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c2">rho</font></span>)</span></span>)</span> <a href="polynom3.html#K9" title="POLYNOM3:func.9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">0_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="hurwitz.html#E10:75"><i><font color="Green" title="E70">A6</font></i></a>, <a class="ref" href="polynom3.html#T29" title="POLYNOM3:th.29">POLYNOM3:29</a></i>;<br/>
<a NAME="E13:75"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c2">rho</font></span>)</span>
 
<b>by </b><i><a class="ref" href="hurwitz.html#T27" target="_self" title="HURWITZ:th.27">Th27</a></i>;<br/>
<a NAME="E14:75"/><b>then </b>
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">0_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c2">rho</font></span>)</span>
 
<b>by </b><i><a class="ref" href="hurwitz.html#T20" target="_self" title="HURWITZ:th.20">Th20</a></i>;<br/>
<b>hence </b><a NAME="E15:75"/>
<span class="p1">(<span class="default"><a href="hurwitz.html#K8" title="HURWITZ:func.8">F*</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">rho</font></span>)</span> <a href="hurwitz.html#K5" title="HURWITZ:func.5">div</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c2">rho</font></span>)</span> is <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>
 <b>by </b><i><a class="txt" href="hurwitz.html#E11:75"><i><font color="Green" title="E71">A7</font></i></a>, <a class="txt" href="hurwitz.html#E12:75"><i><font color="Green" title="E72">A8</font></i></a>, <a class="ref" href="hurwitz.html#D5" target="_self" title="HURWITZ:def.5">Def5</a></i>;<br/>


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