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

<div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">f</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">g</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</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="E1:60_1_1"/>
<font color="Maroon">z</font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:60_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">f</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="hurwitz.html#T3" target="_self">Th3</a></i>;<br/><b>hence </b><a NAME="E3:60_1_1"/>
 <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="txt" href="hurwitz.html#E5"><i><font color="Green">Lemma52</font></i></a>, <a class="ref" href="hurwitz.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:60_1"/>
<font color="Maroon">f</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a>
 <b>by </b><i><a class="ref" href="hurwitz.html#T1" target="_self">Th1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</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="E1:60_1_2"/>
<font color="Maroon">z</font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">g</font>
 ;<br/><a NAME="E2:60_1_2"/><b>then </b>
<font color="Maroon">z</font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">f</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="hurwitz.html#T3" target="_self">Th3</a></i>;<br/><b>hence </b><a NAME="E3:60_1_2"/>
 <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="txt" href="hurwitz.html#E5"><i><font color="Green">Lemma52</font></i></a>, <a class="ref" href="hurwitz.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:60_1"/>
<font color="Maroon">g</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a>
 <b>by </b><i><a class="ref" href="hurwitz.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_2"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">f</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a> &amp; <font color="Maroon">g</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</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="E1:60_2_1"/>
<font color="Maroon">z</font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">f</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">g</font>
 ;<br/><b>then </b><i><font color="Green">E34</font></i>:  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  = 
 <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="polynom5.html#D6">POLYNOM5:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">z</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">g</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom4.html#T27">POLYNOM4:27</a></i>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_2_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_2_1_2"/>
(  <a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  or  <a href="polynom4.html#K2">eval</a> <font color="Maroon">g</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  )
 </span><b>by </b><i><a class="txt" href="hurwitz.html#E6"><i><font color="Green">Lemma53</font></i></a>, <a class="ref" href="vectsp_1.html#T44">VECTSP_1:44</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_2_1_2_1"><b>suppose </b></a><a NAME="E1:60_2_1_2_1"/>
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 ;<br/><div class="add"><a NAME="E2:60_2_1_2_1"/><b>then </b>
<font color="Maroon">z</font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="polynom5.html#D6">POLYNOM5:def 6</a></i>;<br/><b>hence </b><a NAME="E3:60_2_1_2_1"/>
 <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="txt" href="hurwitz.html#E5"><i><font color="Green">Lemma52</font></i></a>, <a class="ref" href="hurwitz.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_2_1_2_2"><b>suppose </b></a><a NAME="E1:60_2_1_2_2"/>
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">g</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 ;<br/><div class="add"><a NAME="E2:60_2_1_2_2"/><b>then </b>
<font color="Maroon">z</font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="polynom5.html#D6">POLYNOM5:def 6</a></i>;<br/><b>hence </b><a NAME="E3:60_2_1_2_2"/>
 <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="txt" href="hurwitz.html#E5"><i><font color="Green">Lemma52</font></i></a>, <a class="ref" href="hurwitz.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:60_2"/>
<font color="Maroon">f</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">g</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a>
 <b>by </b><i><a class="ref" href="hurwitz.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:60"/>
( <font color="Maroon">f</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">g</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a> iff ( <font color="Maroon">f</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a> &amp; <font color="Maroon">g</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a> ) )
 <b>by </b><i/>;<br/>


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