<?xml version="1.0"?>
<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">b</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="E1:74"/><i><font color="Green">E29</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> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span>
 ;<br/>

<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="E2:74"/><i><font color="Green">E30</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
 ;<br/>

<b>set </b><font color="Maroon">g</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span>;<br/>
<a NAME="E3:74"/><i><font color="Green">E31</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <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>
 
<b>by </b><i><a class="ref" href="hurwitz.html#T7" target="_self">Th7</a>, <a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E4:74"/><b>then </b><i><font color="Green">E34</font></i>: 
<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> 
 
<b>by </b><i><a class="ref" href="complfld.html#T9">COMPLFLD:9</a>, <a class="ref" href="complex1.html#T133">COMPLEX1:133</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_1"/>
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  or <font color="Maroon">b</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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1"><b>suppose </b></a><a NAME="E1:74_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T27">POLYNOM5:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="polynom3.html#T29">POLYNOM3:29</a></i>
;<br/><b>hence </b><a NAME="E3:74_1_1"/>
( <font color="Maroon">f</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a> iff <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2"><b>suppose </b></a><a NAME="E1:74_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 ;<br/><div class="add"><a NAME="E2:74_1_2"/><b>then </b><i><font color="Green">E37</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span>
 <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/><div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:74_1_2_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/><b>assume </b><a NAME="E1:74_1_2_1_1"/>
<font color="Maroon">z</font> <a href="polynom5.html#R1">is_a_root_of</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span>
 ;<br/><b>then </b><i><font color="Green">E41</font></i>:  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  = 
 <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="polynom5.html#D6">POLYNOM5:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom4.html#T24">POLYNOM4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T31">POLYNOM5:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <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">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T31">POLYNOM5:31</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:74_1_2_1_1_2"/><i><font color="Green">E42</font></i>: 
 <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><b>reconsider </b><font color="Maroon">e1</font> =  <a href="polynom4.html#K2">eval</a> <font color="Maroon">f</font>,<font color="Maroon">z</font>, <font color="Maroon">a1</font> = <font color="Maroon">a</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">E43</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="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> <font color="Maroon">f</font>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="complex1.html#K9">*</a> <font color="Maroon">e1</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"><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> <font color="Maroon">f</font>,<font color="Maroon">z</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/><b>reconsider </b><font color="Maroon">e2</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">z</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</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">E44</font></i>: <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</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"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="complex1.html#K9">*</a> <font color="Maroon">e2</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"><font color="Maroon">b</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"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">z</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/><a NAME="E6:74_1_2_1_1_2"/><i><font color="Green">E45</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_1_1_2_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_1_2_1_1_2_3"/>
(  <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 or  <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_1_1_2_3_1"><b>suppose </b></a><a NAME="E1:74_1_2_1_1_2_3_1"/>
 <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:74_1_2_1_1_2_3_1"/><b>then </b>
( <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">z</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">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> &amp; <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">z</font></span>)</span></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="txt" href="hurwitz.html#E4:74_1_2"><i><font color="Green">E62</font></i></a>, , </i>;<br/><a NAME="E3:74_1_2_1_1_2_3_1"/><b>then </b>
<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> <font color="Maroon">f</font>,<font color="Maroon">z</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"><font color="Maroon">b</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"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="ref" href="hurwitz.html#T7" target="_self">Th7</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E4:74_1_2_1_1_2_3_1"/>
contradiction
 <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#T9" target="_self">Th9</a>, , <a class="ref" href="vectsp_1.html#T66">VECTSP_1:66</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_1_1_2_3_2"><b>suppose </b></a><a NAME="E1:74_1_2_1_1_2_3_2"/>
 <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:74_1_2_1_1_2_3_2"/><b>then </b><i><font color="Green">E60</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">f</font>,<font color="Maroon">z</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">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E4:74_1_2"><i><font color="Green">E62</font></i></a>, , </i>;<br/><a NAME="E3:74_1_2_1_1_2_3_2"/><b>then </b>
<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">z</font></span>)</span></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#T132">COMPLEX1:132</a></i>;<br/><a NAME="E4:74_1_2_1_1_2_3_2"/><b>then </b><i><font color="Green">E61</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="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> <font color="Maroon">f</font>,<font color="Maroon">z</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"><font color="Maroon">b</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> <font color="Maroon">f</font>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="ref" href="hurwitz.html#T7" target="_self">Th7</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E5:74_1_2_1_1_2_3_2"/>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</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> <font color="Maroon">f</font>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</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"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">z</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="txt" href="hurwitz.html#E14"><i><font color="Green">Lemma76</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E6:74_1_2_1_1_2_3_2"/>
contradiction
 <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#T9" target="_self">Th9</a>, , <a class="ref" href="vectsp_1.html#T66">VECTSP_1:66</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:74_1_2_1_1"/>
 <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:74_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a>
 <b>by </b><i><a class="ref" href="hurwitz.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><i><font color="Green">E62</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="hurwitz.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hurwitz.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hurwitz.html#E6"><i><font color="Green">Lemma53</font></i></a></i>
;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</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="E6:74_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a>  <a href="hidden.html#R2">in</a>  <a href="numbers.html#K2">COMPLEX</a> 
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D2">XCMPLX_0:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon">zz</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a>  as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/><b>set </b><font color="Maroon">a1</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font>;<br/><b>set </b><font color="Maroon">b1</font> =  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">a1'</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font>, <font color="Maroon">b1'</font> =  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span> 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/><div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><a NAME="E1:74_1_2_3"/><i><font color="Green">E90</font></i>: 
(  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a> </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="square_1.html#K5">^2</a> </span>)</span> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="complex1.html#T126">COMPLEX1:126</a></i>;<br/><a NAME="E2:74_1_2_3"/><i><font color="Green">E91</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="complex1.html#K9">*</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="complex1.html#T29">COMPLEX1:29</a></i>;<br/><a NAME="E3:74_1_2_3"/>
( <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="ref" href="square_1.html#T72">SQUARE_1:72</a></i>;<br/><a NAME="E4:74_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a> </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="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E5:74_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="hurwitz.html#T9" target="_self">Th9</a>, , <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E10:74_1_2"/><b>then </b><i><font color="Green">E92</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="complex1.html#K9">*</a> <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">b'</font></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a>  &amp; <span class="p1">(<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">a'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">a'</font></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a>  )
 ;<br/><i><font color="Green">E93</font></i>: <span class="p1">(<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">a'</font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K12">"</a> </span>)</span></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K15">*'</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">a'</font></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K12">"</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> 1
<b>by </b><i><a class="txt" href="hurwitz.html#E12"><i><font color="Green">Lemma72</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">a'</font></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K12">"</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K12">"</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="complfld.html#T9">COMPLFLD:9</a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">a'</font></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K12">"</a> </span>)</span> <a href="complex1.html#K9">*</a> <font color="Maroon">b'</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">b'</font></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K12">"</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">a'</font></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">b'</font></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K12">"</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="complfld.html#T9">COMPLFLD:9</a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T26">POLYNOM3:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T30">POLYNOM3:30</a></i>
;<br/><b>then </b><i><font color="Green">E94</font></i>: <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T29">POLYNOM3:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T26">POLYNOM3:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T30">POLYNOM3:30</a></i>
;<br/><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a>  = 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T28">POLYNOM5:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="polynom3.html#T29">POLYNOM3:29</a></i>
;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p0">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<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="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <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="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p0">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<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="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T26">POLYNOM3:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<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="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="complfld.html#K2">*'</a> </span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="complfld.html#K2">*'</a> </span>)</span></span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="txt" href="hurwitz.html#E1"><i><font color="Green">Lemma23</font></i></a></i>
<br/>.= 
<span class="p1">(<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"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="complfld.html#K2">*'</a> </span>)</span></span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i/>
;<br/><b>then </b><i><font color="Green">E95</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p2">(<span class="default"><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="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="complfld.html#K2">*'</a> </span>)</span></span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><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="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T26">POLYNOM3:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="complfld.html#K2">*'</a> </span>)</span></span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T30">POLYNOM3:30</a></i>
;<br/><a NAME="E17:74_1_2"/><i><font color="Green">E96</font></i>: 
<span class="p1">(<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">a'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>;<br/><a NAME="E18:74_1_2"/>
<font color="Maroon">b'</font> <a href="complex1.html#K12">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> 
 <b>by </b><i>, <a class="ref" href="complfld.html#T7">COMPLFLD:7</a></i>;<br/><a NAME="E19:74_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<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">a'</font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K12">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="hurwitz.html#T11" target="_self">Th11</a>, <a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>;<br/><a NAME="E20:74_1_2"/><b>then </b><i><font color="Green">E97</font></i>: 
<span class="p1">(<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">a'</font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K12">"</a> </span>)</span></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<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> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complfld.html#K2">*'</a> </span>)</span>
 <b>by </b><i><a class="ref" href="complfld.html#T5">COMPLFLD:5</a></i>;<br/><i><font color="Green">E99</font></i>: <span class="p1">(<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> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complfld.html#K2">*'</a> </span>)</span> = 
<span class="p1">(<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> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="complfld.html#K2">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="complfld.html#K2">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
;<br/><a NAME="E22:74_1_2"/><i><font color="Green">E137</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">a'</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">a'</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">b'</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">a'</font></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i>, <a class="ref" href="hurwitz.html#T7" target="_self">Th7</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E23:74_1_2"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">a'</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">b'</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">b'</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">b'</font></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i>, <a class="ref" href="hurwitz.html#T7" target="_self">Th7</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E24:74_1_2"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">a'</font></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">b'</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">b'</font></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="square_1.html#D3">SQUARE_1:def 3</a>, </i>;<br/><a NAME="E25:74_1_2"/><b>then </b><i><font color="Green">E138</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">a'</font></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">b'</font></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="square_1.html#D3">SQUARE_1:def 3</a></i>;<br/><a NAME="E26:74_1_2"/><b>then </b><i><font color="Green">E139</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">a'</font></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">b'</font></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T203">XCMPLX_1:203</a></i>;<br/><a NAME="E27:74_1_2"/>
<font color="Maroon">b'</font> <a href="complex1.html#K12">"</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T203">XCMPLX_1:203</a>, , <a class="ref" href="complfld.html#T9">COMPLFLD:9</a></i>;<br/><a NAME="E28:74_1_2"/><b>then </b><i><font color="Green">E140</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="complfld.html#K2">*'</a> </span>)</span></span>)</span> <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> 
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a>, <a class="ref" href="hurwitz.html#T9" target="_self">Th9</a>, , , <a class="ref" href="complfld.html#T9">COMPLFLD:9</a></i>;<br/><a NAME="E29:74_1_2"/>
<font color="Maroon">b'</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font>
 <b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>;<br/><b>then </b><i><font color="Green">E141</font></i>: <font color="Maroon">b1'</font> = 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complfld.html#T4">COMPLFLD:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">b'</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span>

;<br/><div><i><font color="Green">E142</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_12"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:74_1_2_12"/>
<font color="Maroon">b1'</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:74_1_2_12"/><b>then </b>
 <a href="complex1.html#K10">-</a> <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="hurwitz.html#T12" target="_self">Th12</a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><b>hence </b><a NAME="E3:74_1_2_12"/>
contradiction
 <b>by </b><i>, <a class="ref" href="complfld.html#T9">COMPLFLD:9</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E143</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_13"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:74_1_2_13"/>
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 ;<br/><a NAME="E2:74_1_2_13"/><b>then </b>
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i><a class="ref" href="rlvect_1.html#T25">RLVECT_1:25</a></i>;<br/><b>hence </b><a NAME="E3:74_1_2_13"/>
contradiction
 <b>by </b><i>, <a class="ref" href="complfld.html#T9">COMPLFLD:9</a>, <a class="ref" href="rlvect_1.html#T30">RLVECT_1:30</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E144</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_14"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:74_1_2_14"/><i><font color="Green">E145</font></i>: 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a>  <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 ;<br/><a NAME="E2:74_1_2_14"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i>, <a class="ref" href="complfld.html#T9">COMPLFLD:9</a>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>;<br/><a NAME="E3:74_1_2_14"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i>, <a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>;<br/><b>hence </b><a NAME="E4:74_1_2_14"/>
contradiction
 <b>by </b><i><a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</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="74_1_2_15"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:74_1_2_15"/>
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 ;<br/><a NAME="E2:74_1_2_15"/><b>then </b>
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i><a class="ref" href="rlvect_1.html#T25">RLVECT_1:25</a></i>;<br/><b>hence </b><a NAME="E3:74_1_2_15"/>
contradiction
 <b>by </b><i><a class="ref" href="hurwitz.html#T13" target="_self">Th13</a>, <a class="ref" href="rlvect_1.html#T30">RLVECT_1:30</a></i>;<br/></div><b>end;</b></div><a NAME="E35:74_1_2"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>;<br/><a NAME="E36:74_1_2"/><b>then </b><i><font color="Green">E148</font></i>: 
<font color="Maroon">b1'</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="complfld.html#T4">COMPLFLD:4</a></i>;<br/> <a href="complex1.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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">a'</font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K12">"</a> </span>)</span></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span> <a href="complex1.html#K12">"</a> </span>)</span> = 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">a'</font></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">b'</font></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K12">"</a> </span>)</span> <a href="complex1.html#K12">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hurwitz.html#T9" target="_self">Th9</a>, <a class="ref" href="xcmplx_1.html#T205">XCMPLX_1:205</a></i>
<br/>.= 
<font color="Maroon">b1'</font>
<b>by </b><i>, <a class="ref" href="complfld.html#T4">COMPLFLD:4</a></i>
;<br/><a NAME="E38:74_1_2"/><b>then </b><i><font color="Green">E151</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">b1'</font></span>)</span> <a href="complex1.html#K12">"</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a'</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K12">"</a> </span>)</span> <a href="complex1.html#K9">*</a> <font color="Maroon">a'</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E39:74_1_2"/>
<font color="Maroon">b'</font> <a href="complex1.html#K12">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> 
 <b>by </b><i><a class="ref" href="complfld.html#T7">COMPLFLD:7</a>, </i>;<br/><a NAME="E40:74_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K12">"</a> </span>)</span> <a href="complex1.html#K9">*</a> <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>;<br/><a NAME="E41:74_1_2"/><b>then </b><i><font color="Green">E152</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K12">"</a> </span>)</span> <a href="complex1.html#K9">*</a> <font color="Maroon">a'</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<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="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>;<br/><a NAME="E42:74_1_2"/><i><font color="Green">E153</font></i>: 
 <a href="complex1.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="complfld.html#K2">*'</a> </span>)</span>
 <b>by </b><i><a class="ref" href="complfld.html#T4">COMPLFLD:4</a></i>;<br/><a NAME="E43:74_1_2"/>
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K10">-</a> <font color="Maroon">b1'</font>
 <b>by </b><i><a class="ref" href="complfld.html#T4">COMPLFLD:4</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">b1'</font></span>)</span> <a href="complex1.html#K12">"</a>  = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> 
<b>by </b><i><a class="ref" href="complfld.html#T7">COMPLFLD:7</a>, </i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="complfld.html#T9">COMPLFLD:9</a>, </i>
;<br/><a NAME="E45:74_1_2"/><b>then </b><i><font color="Green">E154</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="complfld.html#K2">*'</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E11:74_1_2"><i><font color="Green">E93</font></i></a>, , <a class="txt" href="hurwitz.html#E10:74_1_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="complfld.html#T3">COMPLFLD:3</a></i>;<br/><i><font color="Green">E155</font></i>: <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a>  = 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="hurwitz.html#T13" target="_self">Th13</a>, </i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T73">VECTSP_1:73</a>, , <a class="ref" href="complfld.html#T9">COMPLFLD:9</a></i>
;<br/><a NAME="E47:74_1_2"/>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b1'</font> <a href="complex1.html#K12">"</a> 
 <b>by </b><i>, <a class="ref" href="complfld.html#T9">COMPLFLD:9</a>, <a class="ref" href="complfld.html#T7">COMPLFLD:7</a></i>;<br/><a NAME="E48:74_1_2"/><b>then </b>
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">b1'</font> <a href="complex1.html#K12">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="complfld.html#T4">COMPLFLD:4</a></i>;<br/><a NAME="E49:74_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">b1'</font> <a href="complex1.html#K12">"</a> </span>)</span></span>)</span> <a href="complex1.html#K12">"</a> 
 <b>by </b><i>, <a class="ref" href="complfld.html#T7">COMPLFLD:7</a></i>;<br/><b>then </b><i><font color="Green">E156</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">b1'</font> <a href="complex1.html#K12">"</a> </span>)</span></span>)</span> <a href="complex1.html#K12">"</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a'</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K12">"</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">b'</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><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="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a'</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K12">"</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="hurwitz.html#E16:74_1_2"><i><font color="Green">E95</font></i></a>, <a class="ref" href="complfld.html#T4">COMPLFLD:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K12">"</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="complex1.html#K15">*'</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="complex1.html#K15">*'</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="complfld.html#T9">COMPLFLD:9</a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
<font color="Maroon">a1'</font>
<b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>
;<br/><a NAME="E51:74_1_2"/>
<font color="Maroon">b'</font> <a href="complex1.html#K12">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> 
 <b>by </b><i><a class="ref" href="complfld.html#T7">COMPLFLD:7</a>, </i>;<br/><a NAME="E52:74_1_2"/><b>then </b><i><font color="Green">E157</font></i>: 
<span class="p1">(<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="p1">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="complex1.html#K12">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>;<br/><a NAME="E53:74_1_2"/>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b1'</font> <a href="complex1.html#K12">"</a> 
 <b>by </b><i>, <a class="ref" href="complfld.html#T9">COMPLFLD:9</a>, <a class="ref" href="complfld.html#T7">COMPLFLD:7</a></i>;<br/><a NAME="E54:74_1_2"/><b>then </b>
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">b1'</font> <a href="complex1.html#K12">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="complfld.html#T4">COMPLFLD:4</a></i>;<br/><a NAME="E55:74_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">b1'</font> <a href="complex1.html#K12">"</a> </span>)</span></span>)</span> <a href="complex1.html#K12">"</a> 
 <b>by </b><i>, <a class="ref" href="complfld.html#T7">COMPLFLD:7</a></i>;<br/><a NAME="E56:74_1_2"/><b>then </b><i><font color="Green">E158</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E17:74_1_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="hurwitz.html#E20:74_1_2"><i><font color="Green">E97</font></i></a>, <a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>;<br/><i><font color="Green">E159</font></i>: <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="polynom5.html#T28">POLYNOM5:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="hurwitz.html#D1" target="_self">Def1</a>, <a class="txt" href="hurwitz.html#E13:74_1_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span> <a href="polynom3.html#K9">+</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p0">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<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#E13:74_1_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="polynom3.html#T29">POLYNOM3:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p0">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><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="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p4">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p0">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><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="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<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> <font color="Maroon">zz</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<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> <font color="Maroon">zz</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="complfld.html#T9">COMPLFLD:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<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> <font color="Maroon">zz</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#T73">VECTSP_1:73</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<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> <font color="Maroon">zz</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/><i><font color="Green">E172</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"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> = 
<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="complfld.html#K2">*'</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span></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"><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="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a> </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"><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="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T139">COMPLEX1:139</a></i>
;<br/><a NAME="E59:74_1_2"/>
<font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>;<br/><b>then </b><i><font color="Green">E173</font></i>: <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span> = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><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="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="complfld.html#T4">COMPLFLD:4</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a> </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/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a> </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/><a NAME="E61:74_1_2"/><i><font color="Green">E175</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="xcmplx_0.html#K3">*</a> <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> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</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"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="ref" href="hurwitz.html#T7" target="_self">Th7</a>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E62:74_1_2"/>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</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"><font color="Maroon">a</font></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"><font color="Maroon">b</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"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="ref" href="hurwitz.html#T7" target="_self">Th7</a>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E63:74_1_2"/><b>then </b>
<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"><font color="Maroon">a</font></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"><font color="Maroon">b</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"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E64:74_1_2"/><b>then </b>
<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="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</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"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="ref" href="square_1.html#D3">SQUARE_1:def 3</a></i>;<br/><a NAME="E65:74_1_2"/><b>then </b>
<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="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="square_1.html#D3">SQUARE_1:def 3</a></i>;<br/><a NAME="E66:74_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E67:74_1_2"/><b>then </b><i><font color="Green">E176</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/><a NAME="E68:74_1_2"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">b</font></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span></span><a href="complex1.html#K17">.|</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></i>;<br/><a NAME="E69:74_1_2"/><b>then </b><i><font color="Green">E177</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"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</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="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="ref" href="hurwitz.html#T14" target="_self">Th14</a>, <a class="txt" href="hurwitz.html#E20"><i><font color="Green">Lemma100</font></i></a>, <a class="ref" href="hurwitz.html#T7" target="_self">Th7</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><div><i><font color="Green">E178</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_23"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:74_1_2_23"/>
<font color="Maroon">b1'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 ;<br/><a NAME="E2:74_1_2_23"/><b>then </b>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i><a class="ref" href="vectsp_1.html#T41">VECTSP_1:41</a></i>;<br/><a NAME="E3:74_1_2_23"/><b>then </b>
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i>, <a class="ref" href="complfld.html#T9">COMPLFLD:9</a>, <a class="ref" href="vectsp_1.html#T44">VECTSP_1:44</a></i>;<br/><a NAME="E4:74_1_2_23"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T30">RLVECT_1:30</a></i>;<br/><b>hence </b><a NAME="E5:74_1_2_23"/>
contradiction
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#T25">RLVECT_1:25</a></i>;<br/></div><b>end;</b></div><a NAME="E71:74_1_2"/>
<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> <font color="Maroon">zz</font></span>)</span></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#T132">COMPLEX1:132</a></i>;<br/><a NAME="E72:74_1_2"/><b>then </b><i><font color="Green">E179</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</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;  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span> <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>  )
 <b>by </b><i><a class="ref" href="hurwitz.html#T15" target="_self">Th15</a>, <a class="ref" href="complfld.html#T9">COMPLFLD:9</a>, <a class="ref" href="complex1.html#T133">COMPLEX1:133</a></i>;<br/><a NAME="E73:74_1_2"/>
 <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p2">(<span class="default"><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="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p0">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E3:74"><i><font color="Green">E31</font></i></a>, <a class="txt" href="hurwitz.html#E21:74_1_2"><i><font color="Green">E99</font></i></a></i>;<br/><a NAME="E74:74_1_2"/><b>then </b>
 <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p2">(<span class="default"><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="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom4.html#T11">POLYNOM4:11</a></i>;<br/><a NAME="E75:74_1_2"/><b>then </b>
 <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="hurwitz.html#T16" target="_self">Th16</a>, <a class="ref" href="polynom5.html#T26">POLYNOM5:26</a></i>;<br/><a NAME="E76:74_1_2"/><b>then </b>
 <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="hurwitz.html#T15" target="_self">Th15</a>, <a class="ref" href="polynom5.html#T26">POLYNOM5:26</a></i>;<br/><a NAME="E77:74_1_2"/><b>then </b>
 <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E5"><i><font color="Green">Lemma52</font></i></a></i>;<br/><a NAME="E78:74_1_2"/><b>then </b><i><font color="Green">E180</font></i>: 
 <a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_24"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:74_1_2_24"/><i><font color="Green">E182</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_24_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/><b>assume </b><a NAME="E1:74_1_2_24_1"/>
<font color="Maroon">z</font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">f</font>
 ;<br/><b>then </b><i><font color="Green">E184</font></i>:  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  = 
 <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">z</font>
<b>by </b><i><a class="txt" href="hurwitz.html#E21:74_1_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="polynom5.html#D6">POLYNOM5:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2">(<span class="default"><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="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K6">-</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="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom4.html#T24">POLYNOM4:24</a></i>
<br/>.= 
<span class="p1">(<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> <font color="Maroon">zz</font></span>)</span> <a href="group_1.html#K10">*</a> <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> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</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="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T31">POLYNOM5:31</a></i>
<br/>.= 
<span class="p1">(<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> <font color="Maroon">zz</font></span>)</span> <a href="group_1.html#K10">*</a> <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> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <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"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T31">POLYNOM5:31</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_24_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:74_1_2_24_1_2"/><i><font color="Green">E185</font></i>: 
 <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><b>reconsider </b><font color="Maroon">e1</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">z</font>, <font color="Maroon">a</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</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">E186</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"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></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"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> = 
<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">e1</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"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</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"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">z</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/><b>reconsider </b><font color="Maroon">e2</font> =  <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"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">z</font>, <font color="Maroon">b</font> =  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span> 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">E187</font></i>: <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></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"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="complex1.html#K9">*</a> <font color="Maroon">e2</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"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></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"><span class="p0">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p0">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">z</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/><a NAME="E6:74_1_2_24_1_2"/><i><font color="Green">E188</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_24_1_2_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_1_2_24_1_2_3"/>
(  <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 or  <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_24_1_2_3_1"><b>suppose </b></a><a NAME="E1:74_1_2_24_1_2_3_1"/>
 <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:74_1_2_24_1_2_3_1"/><b>then </b>
( <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> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">z</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"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> &amp; <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> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">z</font></span>)</span></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="txt" href="hurwitz.html#E4:74_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="hurwitz.html#T17" target="_self">Th17</a>, <a class="ref" href="hurwitz.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E3:74_1_2_24_1_2_3_1"/><b>then </b>
<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> <font color="Maroon">zz</font></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"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">z</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="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></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"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E4:74_1_2_24_1_2_3_1"/>
contradiction
 <b>by </b><i><a class="ref" href="hurwitz.html#T19" target="_self">Th19</a>, <a class="txt" href="hurwitz.html#E28"><i><font color="Green">Lemma111</font></i></a>, , <a class="ref" href="vectsp_1.html#T66">VECTSP_1:66</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2_24_1_2_3_2"><b>suppose </b></a><a NAME="E1:74_1_2_24_1_2_3_2"/>
 <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:74_1_2_24_1_2_3_2"/><b>then </b><i><font color="Green">E189</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"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">z</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"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E4:74_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="hurwitz.html#T17" target="_self">Th17</a>, <a class="ref" href="hurwitz.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E3:74_1_2_24_1_2_3_2"/><b>then </b>
<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> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">z</font></span>)</span></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#T132">COMPLEX1:132</a></i>;<br/><a NAME="E4:74_1_2_24_1_2_3_2"/><b>then </b><i><font color="Green">E190</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"><font color="Maroon">a</font> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></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"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">z</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="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></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"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E5:74_1_2_24_1_2_3_2"/>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></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"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">zz</font></span>)</span></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"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span>,<font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="ref" href="hurwitz.html#T20" target="_self">Th20</a>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E6:74_1_2_24_1_2_3_2"/>
contradiction
 <b>by </b><i><a class="ref" href="hurwitz.html#T19" target="_self">Th19</a>, <a class="txt" href="hurwitz.html#E28"><i><font color="Green">Lemma111</font></i></a>, , <a class="ref" href="vectsp_1.html#T66">VECTSP_1:66</a>, <a class="txt" href="hurwitz.html#E30"><i><font color="Green">Lemma115</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:74_1_2_24_1"/>
 <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:74_1_2_24"/>
<font color="Maroon">f</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a>
 <b>by </b><i><a class="ref" href="hurwitz.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E80:74_1_2"/>
( <font color="Maroon">f</font> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a> iff <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">f</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hurwitz.html#K7" target="_self">*'</a> </span>)</span></span>)</span> is <a href="hurwitz.html#V1" target="_self">Hurwitz</a> )
 <b>by </b><i><a class="ref" href="hurwitz.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
