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

<b>assume </b><a NAME="E1:59"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">h</font>
 ;<br/>

<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="E2:59"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">x</font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">g</font> or <font color="Maroon">x</font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">h</font> )
 ;<br/>

<a NAME="E3:59"/><i><font color="Green">E31</font></i>: 
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">g</font>,<font color="Maroon">x</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">h</font>,<font color="Maroon">x</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="59_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1"/>
( <font color="Maroon">x</font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">g</font> or <font color="Maroon">x</font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">h</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1"><b>suppose </b></a><a NAME="E1:59_1_1"/>
<font color="Maroon">x</font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><a NAME="E2:59_1_1"/><b>then </b>
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">g</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="polynom5.html#D6">POLYNOM5:def 6</a></i>;<br/><a NAME="E3:59_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="txt" href="hurwitz.html#E5"><i><font color="Green">Lemma52</font></i></a>, <a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>;<br/><b>hence </b><a NAME="E4:59_1_1"/>
<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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2"><b>suppose </b></a><a NAME="E1:59_1_2"/>
<font color="Maroon">x</font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">h</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2"/><b>then </b>
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">h</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="polynom5.html#D6">POLYNOM5:def 6</a></i>;<br/><a NAME="E3:59_1_2"/><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="txt" href="hurwitz.html#E5"><i><font color="Green">Lemma52</font></i></a>, <a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>;<br/><b>hence </b><a NAME="E4:59_1_2"/>
<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/></div><b>end;</b></div></div><b>end;</b></div>

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