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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> ;<br/><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="E1:58"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 ;<br/>

<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">f</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</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="E1:58_1_1"/>
<font color="Maroon">x</font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">z</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font>
 ;<br/><b>then </b> <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">z</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="polynom5.html#D6">POLYNOM5:def 6</a></i>
<br/>.= 
<font color="Maroon">z</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T31">POLYNOM5:31</a></i>
;<br/><a NAME="E3:58_1_1"/><b>then </b>
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i><a class="ref" href="hurwitz.html#T3" target="_self">Th3</a>, <a class="ref" href="vectsp_1.html#T44">VECTSP_1:44</a></i>;<br/><a NAME="E4:58_1_1"/><b>then </b>
<font color="Maroon">x</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="E5:58_1_1"/>
 <a href="complex1.html#K3">Re</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <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="E3:58_1"/>
<font color="Maroon">z</font> <a href="polynom5.html#K3">*</a> <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><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">z</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</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="E1:58_2_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">f</font>
 ;<br/> <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">x</font> = 
<font color="Maroon">z</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T31">POLYNOM5:31</a></i>
<br/>.= 
<font color="Maroon">z</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>
<b>by </b><i><a class="txt" href="hurwitz.html#E5"><i><font color="Green">Lemma52</font></i></a>, <a class="ref" href="polynom5.html#D6">POLYNOM5:def 6</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="vectsp_1.html#T36">VECTSP_1:36</a></i>
;<br/><a NAME="E3:58_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">z</font> <a href="polynom5.html#K3">*</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="E4:58_2_1"/>
 <a href="complex1.html#K3">Re</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <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="E3:58_2"/>
<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><b>end;</b></div>
<b>hence </b><a NAME="E4:58"/>
( <font color="Maroon">f</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a> iff <font color="Maroon">z</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a> )
 <b>by </b><i/>;<br/>


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