<?xml version="1.0"?>
<div class="add">

<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>assume </b><a NAME="E1:72"/><i><font color="Green">E29</font></i>: 
(  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">f</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a> )
 ;<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>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">f</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <a href="complfld.html#K1">F_Complex</a>   st  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Olive">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">f</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a> &amp;  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  holds <br/> ( (  <a href="complex1.html#K3">Re</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 implies <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Olive">f</font>,<font color="Olive">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><font color="Olive">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Olive">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> ) &amp; (  <a href="complex1.html#K3">Re</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Olive">f</font>,<font color="Olive">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><font color="Olive">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Olive">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> ) &amp; (  <a href="complex1.html#K3">Re</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> 0 implies <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Olive">f</font>,<font color="Olive">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><font color="Olive">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Olive">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> ) );<br/>
<a NAME="E2:72"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> , <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="E1:72_1"/><i><font color="Green">E34</font></i>: 
 <a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a>
 ;<br/><b>assume </b><a NAME="E2:72_1"/><i><font color="Green">E36</font></i>: 
 <a href="complex1.html#K3">Re</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><a NAME="E3:72_1"/>
<font color="Maroon">z</font> <a href="polynom5.html#R1">is_a_root_of</a>  <a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E8:72_1"><i><font color="Green">E39</font></i></a></i>;<br/><a NAME="E4:72_1"/><b>then </b><i><font color="Green">E37</font></i>: 
 <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="ref" href="hurwitz.html#T1" target="_self">Th1</a></i>;<br/><b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/><a NAME="E6:72_1"/><i><font color="Green">E38</font></i>: 
(  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K10">-</a> <font color="Maroon">y'</font> &amp;  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K10">-</a> <font color="Maroon">z'</font> )
 <b>by </b><i><a class="ref" href="complfld.html#T4">COMPLFLD:4</a></i>;<br/><a NAME="E7:72_1"/>
( <font color="Maroon">y</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">y</font> &amp;  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="complfld.html#K2">*'</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="hurwitz.html#E6:72_1"><i><font color="Green">E38</font></i></a>, </i>;<br/><a NAME="E8:72_1"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">y</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">y</font> &amp;  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">y'</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">z'</font> <a href="complex1.html#K15">*'</a> </span>)</span> )
 <b>by </b><i>, <a class="ref" href="complfld.html#T5">COMPLFLD:5</a></i>;<br/><a NAME="E9:72_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>;<br/><a NAME="E10:72_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y'</font> <a href="complex1.html#K11">-</a> <font color="Maroon">z'</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y'</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">z'</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i>, , </i>;<br/><a NAME="E11:72_1"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y'</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">z'</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">y'</font> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">z'</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>;<br/><b>hence </b><a NAME="E12:72_1"/>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i>, <a class="ref" href="hurwitz.html#T8" target="_self">Th8</a>, <a class="ref" href="complfld.html#T5">COMPLFLD:5</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> , <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> , <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="E1:72_2"/><i><font color="Green">E43</font></i>: 
(  <a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a> &amp; <font color="Maroon">a</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/><b>assume </b><a NAME="E2:72_2"/><i><font color="Green">E44</font></i>: 
 <a href="complex1.html#K3">Re</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">e'</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/><a NAME="E4:72_2"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i>, , </i>;<br/><a NAME="E5:72_2"/><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T133">COMPLEX1:133</a>, , <a class="ref" href="complfld.html#T9">COMPLFLD:9</a></i>;<br/><i><font color="Green">E61</font></i>: <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span></span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> = 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p4">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#T31">POLYNOM5:31</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a'</font> <a href="complex1.html#K9">*</a> <font color="Maroon">e'</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
;<br/><b>reconsider </b><font color="Maroon">r'</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/><span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> = 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T139">COMPLEX1:139</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a'</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="complex1.html#K9">*</a> <font color="Maroon">r'</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#T31">POLYNOM5:31</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E9:72_2"/>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span></span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="ref" href="hurwitz.html#T8" target="_self">Th8</a>, , , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> , <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="E1:72_3"/><i><font color="Green">E63</font></i>: 
 <a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a>
 ;<br/><b>assume </b><a NAME="E2:72_3"/><i><font color="Green">E90</font></i>: 
 <a href="complex1.html#K3">Re</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E3:72_3"/>
<font color="Maroon">z</font> <a href="polynom5.html#R1">is_a_root_of</a>  <a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font>
 <b>by </b><i/>;<br/><a NAME="E4:72_3"/><b>then </b><i><font color="Green">E91</font></i>: 
 <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="ref" href="hurwitz.html#T1" target="_self">Th1</a></i>;<br/><b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/><a NAME="E6:72_3"/><i><font color="Green">E92</font></i>: 
(  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K10">-</a> <font color="Maroon">y'</font> &amp;  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K10">-</a> <font color="Maroon">z'</font> )
 <b>by </b><i><a class="ref" href="complfld.html#T4">COMPLFLD:4</a></i>;<br/><a NAME="E7:72_3"/>
( <font color="Maroon">y</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">y</font> &amp;  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="complfld.html#K2">*'</a> </span>)</span> )
 <b>by </b><i>, </i>;<br/><a NAME="E8:72_3"/><b>then </b><i><font color="Green">E93</font></i>: 
( <font color="Maroon">y</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">y</font> &amp;  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">y'</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">z'</font> <a href="complex1.html#K15">*'</a> </span>)</span> )
 <b>by </b><i>, <a class="ref" href="complfld.html#T5">COMPLFLD:5</a></i>;<br/><a NAME="E9:72_3"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>;<br/><a NAME="E10:72_3"/><b>then </b><i><font color="Green">E94</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y'</font> <a href="complex1.html#K11">-</a> <font color="Maroon">z'</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y'</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">z'</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i>, <a class="ref" href="hurwitz.html#T8" target="_self">Th8</a>, </i>;<br/><a NAME="E11:72_3"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y'</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">z'</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">y'</font> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">z'</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>;<br/><b>hence </b><a NAME="E12:72_3"/>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i>, <a class="txt" href="hurwitz.html#E12"><i><font color="Green">Lemma72</font></i></a>, <a class="ref" href="complfld.html#T5">COMPLFLD:5</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E95</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> , <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> , <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="E1:72_4"/><i><font color="Green">E96</font></i>: 
(  <a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a> &amp; <font color="Maroon">a</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/><b>assume </b><a NAME="E2:72_4"/><i><font color="Green">E97</font></i>: 
 <a href="complex1.html#K3">Re</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">e'</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/><a NAME="E4:72_4"/><i><font color="Green">E99</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i>, , <a class="ref" href="hurwitz.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E5:72_4"/><i><font color="Green">E137</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T133">COMPLEX1:133</a>, , <a class="ref" href="complfld.html#T9">COMPLFLD:9</a></i>;<br/><i><font color="Green">E138</font></i>: <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span></span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> = 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p4">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#T31">POLYNOM5:31</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a'</font> <a href="complex1.html#K9">*</a> <font color="Maroon">e'</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
;<br/><b>reconsider </b><font color="Maroon">r'</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/><span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> = 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T139">COMPLEX1:139</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a'</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="complex1.html#K9">*</a> <font color="Maroon">r'</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#T31">POLYNOM5:31</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E9:72_4"/>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span></span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i>, <a class="txt" href="hurwitz.html#E12"><i><font color="Green">Lemma72</font></i></a>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E139</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> , <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="E1:72_5"/><i><font color="Green">E140</font></i>: 
 <a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a>
 ;<br/><b>assume </b><a NAME="E2:72_5"/><i><font color="Green">E141</font></i>: 
 <a href="complex1.html#K3">Re</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E3:72_5"/>
<font color="Maroon">z</font> <a href="polynom5.html#R1">is_a_root_of</a>  <a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font>
 <b>by </b><i/>;<br/><a NAME="E4:72_5"/><b>then </b><i><font color="Green">E142</font></i>: 
 <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="ref" href="hurwitz.html#T8" target="_self">Th8</a>, <a class="ref" href="hurwitz.html#T1" target="_self">Th1</a></i>;<br/><b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/><a NAME="E6:72_5"/><i><font color="Green">E143</font></i>: 
(  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K10">-</a> <font color="Maroon">y'</font> &amp;  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K10">-</a> <font color="Maroon">z'</font> )
 <b>by </b><i><a class="ref" href="complfld.html#T4">COMPLFLD:4</a></i>;<br/><a NAME="E7:72_5"/>
( <font color="Maroon">y</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">y</font> &amp;  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="complfld.html#K2">*'</a> </span>)</span> )
 <b>by </b><i>, </i>;<br/><a NAME="E8:72_5"/><b>then </b><i><font color="Green">E144</font></i>: 
( <font color="Maroon">y</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">y</font> &amp;  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">y'</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">z'</font> <a href="complex1.html#K15">*'</a> </span>)</span> )
 <b>by </b><i>, <a class="ref" href="complfld.html#T5">COMPLFLD:5</a></i>;<br/><a NAME="E9:72_5"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>;<br/><a NAME="E10:72_5"/><b>then </b><i><font color="Green">E145</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y'</font> <a href="complex1.html#K11">-</a> <font color="Maroon">z'</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y'</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">z'</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i>, <a class="txt" href="hurwitz.html#E12"><i><font color="Green">Lemma72</font></i></a>, </i>;<br/><a NAME="E11:72_5"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y'</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">z'</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">y'</font> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">z'</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>;<br/><b>hence </b><a NAME="E12:72_5"/>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i>, <a class="ref" href="hurwitz.html#T9" target="_self">Th9</a>, <a class="ref" href="complfld.html#T5">COMPLFLD:5</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E146</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> , <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> , <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="E1:72_6"/><i><font color="Green">E147</font></i>: 
(  <a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a> &amp; <font color="Maroon">a</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/><b>assume </b><a NAME="E2:72_6"/><i><font color="Green">E148</font></i>: 
 <a href="complex1.html#K3">Re</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">e'</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/><i><font color="Green">E151</font></i>: <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span></span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> = 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p4">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#T31">POLYNOM5:31</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a'</font> <a href="complex1.html#K9">*</a> <font color="Maroon">e'</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
;<br/><b>reconsider </b><font color="Maroon">r'</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/><span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> = 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T139">COMPLEX1:139</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a'</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="complex1.html#K9">*</a> <font color="Maroon">r'</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#T31">POLYNOM5:31</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E7:72_6"/>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span></span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i>, , , <a class="txt" href="hurwitz.html#E12"><i><font color="Green">Lemma72</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E152</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:72_7"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_7_1"><b>now </b></a><div class="add"><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>assume </b><a NAME="E1:72_7_1"/><i><font color="Green">E154</font></i>: 
(  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">f</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a> &amp;  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<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/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:72_7_1_1"/>
( <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_7_1_1_1"><b>suppose </b></a><a NAME="E1:72_7_1_1_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> , <font color="Maroon">z2</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E3:72_7_1_1_1"/><i><font color="Green">E155</font></i>: 
( <font color="Maroon">z1</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>  &amp; <font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">z1</font> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z2</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:72_7_1_1_1"/><i><font color="Green">E156</font></i>: 
 <a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z2</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a>
 <b>by </b><i><a class="ref" href="hurwitz.html#T9" target="_self">Th9</a>, , </i>;<br/><b>thus </b><a NAME="E5:72_7_1_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 implies <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> ) &amp; (  <a href="complex1.html#K3">Re</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> ) &amp; (  <a href="complex1.html#K3">Re</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 implies <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> ) )
 <b>by </b><i><a class="ref" href="hurwitz.html#T9" target="_self">Th9</a>, , , <a class="ref" href="hurwitz.html#T8" target="_self">Th8</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_7_1_1_2"><b>suppose </b></a><a NAME="E1:72_7_1_1_2"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:72_7_1_1_2"/><i><font color="Green">E158</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:72_7_1_1_2"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E4:72_7_1_1_2"/><b>then </b>
<font color="Maroon">f</font> is <a href="polynom5.html#V1">with_roots</a>
 <b>by </b><i><a class="ref" href="polynom5.html#T75">POLYNOM5:75</a></i>;<br/><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E6:72_7_1_1_2"/><i><font color="Green">E159</font></i>: 
<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#D7">POLYNOM5:def 7</a></i>;<br/><b>consider </b><font color="Maroon">f1</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E8:72_7_1_1_2"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">f1</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E14"><i><font color="Green">Lemma76</font></i></a>, <a class="txt" href="hurwitz.html#E4:72"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E9:72_7_1_1_2"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom3.html#K12">0_.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i>, <a class="ref" href="polynom4.html#T6">POLYNOM4:6</a>, </i>;<br/><a NAME="E10:72_7_1_1_2"/><i><font color="Green">E175</font></i>: 
 <a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom3.html#K12">0_.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i>, <a class="ref" href="hurwitz.html#T11" target="_self">Th11</a>, <a class="ref" href="polynom3.html#T35">POLYNOM3:35</a></i>;<br/><a NAME="E11:72_7_1_1_2"/>
<font color="Maroon">f1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom3.html#K12">0_.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i>, <a class="ref" href="hurwitz.html#T11" target="_self">Th11</a>, <a class="ref" href="polynom3.html#T35">POLYNOM3:35</a></i>;<br/><b>then </b><i><font color="Green">E176</font></i>:  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">f1</font></span>)</span>
<b>by </b><i><a class="ref" href="hurwitz.html#T11" target="_self">Th11</a>, , </i>
<br/>.= 
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">f1</font></span>)</span>
<b>by </b><i/>
;<br/><a NAME="E13:72_7_1_1_2"/><i><font color="Green">E177</font></i>: 
( <font color="Maroon">f1</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a> &amp;  <a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a> )
 <b>by </b><i>, <a class="ref" href="hurwitz.html#T11" target="_self">Th11</a>, </i>;<br/><a NAME="E14:72_7_1_1_2"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="hurwitz.html#T9" target="_self">Th9</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><i><font color="Green">E179</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_7_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_7_1_1_2_2"/><i><font color="Green">E180</font></i>: 
 <a href="complex1.html#K3">Re</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><b>reconsider </b><font color="Maroon">r'</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">x</font>, <font color="Maroon">e'</font> =  <a href="polynom4.html#K2">eval</a> <font color="Maroon">f1</font>,<font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">r1'</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font>, <font color="Maroon">e1'</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/><a NAME="E4:72_7_1_1_2_2"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">r'</font> <a href="complex1.html#K9">*</a> <font color="Maroon">e'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<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">f1</font>,<font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>;<br/><a NAME="E5:72_7_1_1_2_2"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">r1'</font> <a href="complex1.html#K9">*</a> <font color="Maroon">e1'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<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> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>;<br/><a NAME="E6:72_7_1_1_2_2"/><i><font color="Green">E185</font></i>: 
( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">e'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">e1'</font></span><a href="complex1.html#K17">.|</a></span> &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">r'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">r1'</font></span><a href="complex1.html#K17">.|</a></span> )
 <b>by </b><i>, , , <a class="txt" href="hurwitz.html#E12"><i><font color="Green">Lemma72</font></i></a>, , , <a class="ref" href="hurwitz.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E7:72_7_1_1_2_2"/><i><font color="Green">E186</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">e'</font></span><a href="complex1.html#K17">.|</a></span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">r'</font></span><a href="complex1.html#K17">.|</a></span> )
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E8:72_7_1_1_2_2"/><b>then </b><i><font color="Green">E187</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">r1'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">e'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">r1'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">e1'</font></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="hurwitz.html#T13" target="_self">Th13</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E9:72_7_1_1_2_2"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">r'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">e'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">r1'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">e'</font></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i>, <a class="ref" href="hurwitz.html#T13" target="_self">Th13</a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E10:72_7_1_1_2_2"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">r'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">e'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">r1'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">e1'</font></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E11:72_7_1_1_2_2"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r'</font> <a href="complex1.html#K9">*</a> <font color="Maroon">e'</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">r1'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">e1'</font></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>;<br/><a NAME="E12:72_7_1_1_2_2"/><b>then </b><i><font color="Green">E188</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p4">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">f1</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i>, , <a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>;<br/><a NAME="E13:72_7_1_1_2_2"/>
( <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<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">f1</font>,<font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">f1</font></span>)</span>,<font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<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> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span>,<font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="polynom4.html#T27">POLYNOM4:27</a></i>;<br/><b>hence </b><a NAME="E14:72_7_1_1_2_2"/>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i>, <a class="ref" href="hurwitz.html#T11" target="_self">Th11</a>, <a class="ref" href="hurwitz.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E189</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_7_1_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_7_1_1_2_3"/><i><font color="Green">E190</font></i>: 
 <a href="complex1.html#K3">Re</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>reconsider </b><font color="Maroon">r'</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">x</font>, <font color="Maroon">e'</font> =  <a href="polynom4.html#K2">eval</a> <font color="Maroon">f1</font>,<font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">r1'</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font>, <font color="Maroon">e1'</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/><a NAME="E4:72_7_1_1_2_3"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">r'</font> <a href="complex1.html#K9">*</a> <font color="Maroon">e'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<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">f1</font>,<font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>;<br/><a NAME="E5:72_7_1_1_2_3"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">r1'</font> <a href="complex1.html#K9">*</a> <font color="Maroon">e1'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<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> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>;<br/><a NAME="E6:72_7_1_1_2_3"/><i><font color="Green">E193</font></i>: 
( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">e'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">e1'</font></span><a href="complex1.html#K17">.|</a></span> &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">r'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">r1'</font></span><a href="complex1.html#K17">.|</a></span> )
 <b>by </b><i>, , , <a class="txt" href="hurwitz.html#E12"><i><font color="Green">Lemma72</font></i></a>, , , <a class="ref" href="hurwitz.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E7:72_7_1_1_2_3"/><i><font color="Green">E194</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">e1'</font></span><a href="complex1.html#K17">.|</a></span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">r1'</font></span><a href="complex1.html#K17">.|</a></span> )
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E8:72_7_1_1_2_3"/><b>then </b><i><font color="Green">E195</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">r'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">e'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">r1'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">e'</font></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E9:72_7_1_1_2_3"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">r1'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">e'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">r1'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">e1'</font></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E10:72_7_1_1_2_3"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">r'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">e'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">r1'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">e1'</font></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E11:72_7_1_1_2_3"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r'</font> <a href="complex1.html#K9">*</a> <font color="Maroon">e'</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">r1'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">e1'</font></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>;<br/><a NAME="E12:72_7_1_1_2_3"/><b>then </b><i><font color="Green">E197</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p4">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">f1</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i>, <a class="ref" href="hurwitz.html#T13" target="_self">Th13</a>, <a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>;<br/><a NAME="E13:72_7_1_1_2_3"/>
( <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<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">f1</font>,<font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">f1</font></span>)</span>,<font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<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> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span>,<font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="polynom4.html#T27">POLYNOM4:27</a></i>;<br/><b>hence </b><a NAME="E14:72_7_1_1_2_3"/>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i>, <a class="ref" href="hurwitz.html#T11" target="_self">Th11</a>, <a class="ref" href="hurwitz.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_7_1_1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_7_1_1_2_4"/><i><font color="Green">E198</font></i>: 
 <a href="complex1.html#K3">Re</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:72_7_1_1_2_4"/><b>then </b><i><font color="Green">E199</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">f1</font>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E12"><i><font color="Green">Lemma72</font></i></a>, , , , <a class="ref" href="hurwitz.html#T12" target="_self">Th12</a></i>;<br/><b>reconsider </b><font color="Maroon">r'</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">x</font>, <font color="Maroon">e'</font> =  <a href="polynom4.html#K2">eval</a> <font color="Maroon">f1</font>,<font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">r1'</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font>, <font color="Maroon">e1'</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/><a NAME="E5:72_7_1_1_2_4"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">r'</font> <a href="complex1.html#K9">*</a> <font color="Maroon">e'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<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">f1</font>,<font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>;<br/><a NAME="E6:72_7_1_1_2_4"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">r1'</font> <a href="complex1.html#K9">*</a> <font color="Maroon">e1'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<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> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>;<br/><a NAME="E7:72_7_1_1_2_4"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">r'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">e'</font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">r1'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">e1'</font></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i>, , , </i>;<br/><a NAME="E8:72_7_1_1_2_4"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r'</font> <a href="complex1.html#K9">*</a> <font color="Maroon">e'</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">r1'</font></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">e1'</font></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>;<br/><a NAME="E9:72_7_1_1_2_4"/><b>then </b><i><font color="Green">E246</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p4">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">f1</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i>, <a class="ref" href="hurwitz.html#T13" target="_self">Th13</a>, <a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>;<br/><a NAME="E10:72_7_1_1_2_4"/>
( <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span>,<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">f1</font>,<font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">f1</font></span>)</span>,<font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<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> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hurwitz.html#K3" target="_self">rpoly</a> 1,<font color="Maroon">z</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span>,<font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="polynom4.html#T27">POLYNOM4:27</a></i>;<br/><b>hence </b><a NAME="E11:72_7_1_1_2_4"/>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i>, <a class="ref" href="hurwitz.html#T11" target="_self">Th11</a>, <a class="ref" href="hurwitz.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E18:72_7_1_1_2"/>
( (  <a href="complex1.html#K3">Re</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 implies <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> ) &amp; (  <a href="complex1.html#K3">Re</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> ) &amp; (  <a href="complex1.html#K3">Re</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 implies <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> ) )
 <b>by </b><i><a class="ref" href="hurwitz.html#D1" target="_self">Def1</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:72_7"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E10:72"/><i><font color="Green">E247</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="hurwitz.html#T7" target="_self">Th7</a>, );<br/></i>
<a NAME="E11:72"/>
<font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom3.html#K12">0_.</a> <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:72"/><b>then </b>
 <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">f</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="txt" href="hurwitz.html#E1:72"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E14:72"/><i><font color="Green">E248</font></i>: 
 <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/>
<b>thus </b><a NAME="E15:72"/>
( (  <a href="complex1.html#K3">Re</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 implies <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> ) &amp; (  <a href="complex1.html#K3">Re</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> ) &amp; (  <a href="complex1.html#K3">Re</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 implies <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> ) )
 <b>by </b><i>, <a class="txt" href="hurwitz.html#E12"><i><font color="Green">Lemma72</font></i></a>, </i>;<br/>


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