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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="E1:71"/><i><font color="Green" title="E62">A1</font></i>: 
<font color="Maroon" title="c1">f</font> is <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">x</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="E2:71"/><i><font color="Green" title="E63">A2</font></i>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c3">z</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_1"/>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> 
 ;<br/><a NAME="E2:71_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="ref" href="polynom5.html#D6" title="POLYNOM5:def.6">POLYNOM5:def 6</a></i>;<br/><b>hence </b><a NAME="E3:71_1"/>
contradiction
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:71"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="hurwitz.html#E2:71"><i><font color="Green" title="E63">A2</font></i></a>, <a class="ref" href="hurwitz.html#D8" target="_self" title="HURWITZ:def.8">Def8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:71"/>
0 <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> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T133" title="COMPLEX1:th.133">COMPLEX1:133</a>, <a class="ref" href="complfld.html#T9" title="COMPLFLD:th.9">COMPLFLD:9</a></i>;<br/>


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