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

<b>let </b><font color="Maroon" title="c1">f</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/>

<b>assume </b><a NAME="E1:72"/><i><font color="Green" title="E63">A1</font></i>: 
(  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c1">f</font> is <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a> )
 ;<br/>

<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">f</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>   st  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Olive" title="b1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">f</font> is <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a> &amp;  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/>for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  holds <br/> ( (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Olive" title="b2">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Olive" title="b2">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> ) &amp; (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Olive" title="b2">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Olive" title="b2">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> ) &amp; (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Olive" title="b2">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> ) );<br/>
<a NAME="E2:72"/><i><font color="Green" title="E64">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<div><i><font color="Green" title="E65">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/><b>assume </b><a NAME="E1:72_1"/><i><font color="Green" title="E66">A4</font></i>: 
 <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c4">z</font> is <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>
 ;<br/><b>assume </b><a NAME="E2:72_1"/><i><font color="Green" title="E67">A5</font></i>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c3">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E3:72_1"/>
<font color="Maroon" title="c4">z</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a>  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c4">z</font>
 <b>by </b><i><a class="ref" href="hurwitz.html#T30" target="_self" title="HURWITZ:th.30">Th30</a></i>;<br/><a NAME="E4:72_1"/><b>then </b><i><font color="Green" title="E68">A6</font></i>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:72_1"><i><font color="Green" title="E66">A4</font></i></a>, <a class="ref" href="hurwitz.html#D8" target="_self" title="HURWITZ:def.8">Def8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">y'</font> = <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c6">z'</font> = <font color="Maroon" title="c4">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></i>;<br/><a NAME="E6:72_1"/><i><font color="Green" title="E69">A7</font></i>: 
(  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c5">y'</font> &amp;  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c6">z'</font> )
 <b>by </b><i><a class="ref" href="complfld.html#T4" title="COMPLFLD:th.4">COMPLFLD:4</a></i>;<br/><a NAME="E7:72_1"/>
( <font color="Maroon" title="c3">y</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c4">z</font></span>)</span>,<font color="Maroon" title="c3">y</font> &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c4">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="hurwitz.html#T29" target="_self" title="HURWITZ:th.29">Th29</a>, <a class="ref" href="hurwitz.html#T48" target="_self" title="HURWITZ:th.48">Th48</a></i>;<br/><a NAME="E8:72_1"/><b>then </b><i><font color="Green" title="E70">A8</font></i>: 
( <font color="Maroon" title="c3">y</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c4">z</font></span>)</span>,<font color="Maroon" title="c3">y</font> &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c4">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c5">y'</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">z'</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="hurwitz.html#E6:72_1"><i><font color="Green" title="E69">A7</font></i></a>, <a class="ref" href="complfld.html#T5" title="COMPLFLD:th.5">COMPLFLD:5</a></i>;<br/><a NAME="E9:72_1"/>
( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span> &amp; <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="complex1.html#T29" title="COMPLEX1:th.29">COMPLEX1:29</a></i>;<br/><a NAME="E10:72_1"/><b>then </b><i><font color="Green" title="E71">A9</font></i>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y'</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c6">z'</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y'</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">z'</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E2:72_1"><i><font color="Green" title="E67">A5</font></i></a>, <a class="txt" href="hurwitz.html#E4:72_1"><i><font color="Green" title="E68">A6</font></i></a>, <a class="txt" href="hurwitz.html#E71"><i><font color="Green" title="E59">Lm15</font></i></a></i>;<br/><a NAME="E11:72_1"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y'</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">z'</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">y'</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">z'</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>;<br/><b>hence </b><a NAME="E12:72_1"/>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c4">z</font></span>)</span>,<font color="Maroon" title="c3">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c4">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c3">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E8:72_1"><i><font color="Green" title="E70">A8</font></i></a>, <a class="txt" href="hurwitz.html#E10:72_1"><i><font color="Green" title="E71">A9</font></i></a>, <a class="ref" href="complfld.html#T5" title="COMPLFLD:th.5">COMPLFLD:5</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E66">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/><b>assume </b><a NAME="E1:72_2"/><i><font color="Green" title="E67">A11</font></i>: 
(  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font> is <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  )
 ;<br/><b>assume </b><a NAME="E2:72_2"/><i><font color="Green" title="E68">A12</font></i>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>reconsider </b><font color="Maroon" title="c6">a'</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c7">e'</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span>,<font color="Maroon" title="c4">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></i>;<br/><a NAME="E4:72_2"/><i><font color="Green" title="E69">A13</font></i>: 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E3:72"><i><font color="Green" title="E65">A3</font></i></a>, <a class="txt" href="hurwitz.html#E1:72_2"><i><font color="Green" title="E67">A11</font></i></a>, <a class="txt" href="hurwitz.html#E2:72_2"><i><font color="Green" title="E68">A12</font></i></a></i>;<br/><a NAME="E5:72_2"/><i><font color="Green" title="E70">A14</font></i>: 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:72_2"><i><font color="Green" title="E67">A11</font></i></a>, <a class="ref" href="complex1.html#T133" title="COMPLEX1:th.133">COMPLEX1:133</a>, <a class="ref" href="complfld.html#T9" title="COMPLFLD:th.9">COMPLFLD:9</a></i>;<br/><i><font color="Green" title="E71">A15</font></i>: <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> = 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p4">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#T31" title="POLYNOM5:th.31">POLYNOM5:31</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c8">r'</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c4">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></i>;<br/><span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> = 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> </span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T139" title="COMPLEX1:th.139">COMPLEX1:139</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> </span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span></span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#T31" title="POLYNOM5:th.31">POLYNOM5:31</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="hurwitz.html#T44" target="_self" title="HURWITZ:th.44">Th44</a></i>
;<br/><b>hence </b><a NAME="E9:72_2"/>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E4:72_2"><i><font color="Green" title="E69">A13</font></i></a>, <a class="txt" href="hurwitz.html#E5:72_2"><i><font color="Green" title="E70">A14</font></i></a>, <a class="txt" href="hurwitz.html#E6:72_2"><i><font color="Green" title="E71">A15</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E67">A16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/><b>assume </b><a NAME="E1:72_3"/><i><font color="Green" title="E68">A17</font></i>: 
 <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c4">z</font> is <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>
 ;<br/><b>assume </b><a NAME="E2:72_3"/><i><font color="Green" title="E69">A18</font></i>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c3">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E3:72_3"/>
<font color="Maroon" title="c4">z</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a>  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c4">z</font>
 <b>by </b><i><a class="ref" href="hurwitz.html#T30" target="_self" title="HURWITZ:th.30">Th30</a></i>;<br/><a NAME="E4:72_3"/><b>then </b><i><font color="Green" title="E70">A19</font></i>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:72_3"><i><font color="Green" title="E68">A17</font></i></a>, <a class="ref" href="hurwitz.html#D8" target="_self" title="HURWITZ:def.8">Def8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">y'</font> = <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c6">z'</font> = <font color="Maroon" title="c4">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></i>;<br/><a NAME="E6:72_3"/><i><font color="Green" title="E71">A20</font></i>: 
(  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c5">y'</font> &amp;  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c6">z'</font> )
 <b>by </b><i><a class="ref" href="complfld.html#T4" title="COMPLFLD:th.4">COMPLFLD:4</a></i>;<br/><a NAME="E7:72_3"/>
( <font color="Maroon" title="c3">y</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c4">z</font></span>)</span>,<font color="Maroon" title="c3">y</font> &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c4">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="hurwitz.html#T29" target="_self" title="HURWITZ:th.29">Th29</a>, <a class="ref" href="hurwitz.html#T48" target="_self" title="HURWITZ:th.48">Th48</a></i>;<br/><a NAME="E8:72_3"/><b>then </b><i><font color="Green" title="E72">A21</font></i>: 
( <font color="Maroon" title="c3">y</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c4">z</font></span>)</span>,<font color="Maroon" title="c3">y</font> &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c4">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c5">y'</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">z'</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="hurwitz.html#E6:72_3"><i><font color="Green" title="E71">A20</font></i></a>, <a class="ref" href="complfld.html#T5" title="COMPLFLD:th.5">COMPLFLD:5</a></i>;<br/><a NAME="E9:72_3"/>
( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span> &amp; <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="complex1.html#T29" title="COMPLEX1:th.29">COMPLEX1:29</a></i>;<br/><a NAME="E10:72_3"/><b>then </b><i><font color="Green" title="E73">A22</font></i>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y'</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c6">z'</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y'</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">z'</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E2:72_3"><i><font color="Green" title="E69">A18</font></i></a>, <a class="txt" href="hurwitz.html#E4:72_3"><i><font color="Green" title="E70">A19</font></i></a>, <a class="txt" href="hurwitz.html#E71"><i><font color="Green" title="E59">Lm15</font></i></a></i>;<br/><a NAME="E11:72_3"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y'</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">z'</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">y'</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">z'</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>;<br/><b>hence </b><a NAME="E12:72_3"/>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c4">z</font></span>)</span>,<font color="Maroon" title="c3">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c4">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c3">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E8:72_3"><i><font color="Green" title="E72">A21</font></i></a>, <a class="txt" href="hurwitz.html#E10:72_3"><i><font color="Green" title="E73">A22</font></i></a>, <a class="ref" href="complfld.html#T5" title="COMPLFLD:th.5">COMPLFLD:5</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E68">A23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/><b>assume </b><a NAME="E1:72_4"/><i><font color="Green" title="E69">A24</font></i>: 
(  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font> is <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  )
 ;<br/><b>assume </b><a NAME="E2:72_4"/><i><font color="Green" title="E70">A25</font></i>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>reconsider </b><font color="Maroon" title="c6">a'</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c7">e'</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span>,<font color="Maroon" title="c4">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></i>;<br/><a NAME="E4:72_4"/><i><font color="Green" title="E71">A26</font></i>: 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E5:72"><i><font color="Green" title="E67">A16</font></i></a>, <a class="txt" href="hurwitz.html#E1:72_4"><i><font color="Green" title="E69">A24</font></i></a>, <a class="txt" href="hurwitz.html#E2:72_4"><i><font color="Green" title="E70">A25</font></i></a></i>;<br/><a NAME="E5:72_4"/><i><font color="Green" title="E72">A27</font></i>: 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:72_4"><i><font color="Green" title="E69">A24</font></i></a>, <a class="ref" href="complex1.html#T133" title="COMPLEX1:th.133">COMPLEX1:133</a>, <a class="ref" href="complfld.html#T9" title="COMPLFLD:th.9">COMPLFLD:9</a></i>;<br/><i><font color="Green" title="E73">A28</font></i>: <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> = 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p4">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#T31" title="POLYNOM5:th.31">POLYNOM5:31</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c8">r'</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c4">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></i>;<br/><span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> = 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> </span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T139" title="COMPLEX1:th.139">COMPLEX1:139</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> </span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span></span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#T31" title="POLYNOM5:th.31">POLYNOM5:31</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="hurwitz.html#T44" target="_self" title="HURWITZ:th.44">Th44</a></i>
;<br/><b>hence </b><a NAME="E9:72_4"/>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E4:72_4"><i><font color="Green" title="E71">A26</font></i></a>, <a class="txt" href="hurwitz.html#E5:72_4"><i><font color="Green" title="E72">A27</font></i></a>, <a class="txt" href="hurwitz.html#E6:72_4"><i><font color="Green" title="E73">A28</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E69">A29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/><b>assume </b><a NAME="E1:72_5"/><i><font color="Green" title="E70">A30</font></i>: 
 <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c4">z</font> is <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>
 ;<br/><b>assume </b><a NAME="E2:72_5"/><i><font color="Green" title="E71">A31</font></i>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E3:72_5"/>
<font color="Maroon" title="c4">z</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a>  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c4">z</font>
 <b>by </b><i><a class="ref" href="hurwitz.html#T30" target="_self" title="HURWITZ:th.30">Th30</a></i>;<br/><a NAME="E4:72_5"/><b>then </b><i><font color="Green" title="E72">A32</font></i>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:72_5"><i><font color="Green" title="E70">A30</font></i></a>, <a class="ref" href="hurwitz.html#D8" target="_self" title="HURWITZ:def.8">Def8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">y'</font> = <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c6">z'</font> = <font color="Maroon" title="c4">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></i>;<br/><a NAME="E6:72_5"/><i><font color="Green" title="E73">A33</font></i>: 
(  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c5">y'</font> &amp;  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c6">z'</font> )
 <b>by </b><i><a class="ref" href="complfld.html#T4" title="COMPLFLD:th.4">COMPLFLD:4</a></i>;<br/><a NAME="E7:72_5"/>
( <font color="Maroon" title="c3">y</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c4">z</font></span>)</span>,<font color="Maroon" title="c3">y</font> &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c4">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="hurwitz.html#T29" target="_self" title="HURWITZ:th.29">Th29</a>, <a class="ref" href="hurwitz.html#T48" target="_self" title="HURWITZ:th.48">Th48</a></i>;<br/><a NAME="E8:72_5"/><b>then </b><i><font color="Green" title="E74">A34</font></i>: 
( <font color="Maroon" title="c3">y</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c4">z</font></span>)</span>,<font color="Maroon" title="c3">y</font> &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c4">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c5">y'</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">z'</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="hurwitz.html#E6:72_5"><i><font color="Green" title="E73">A33</font></i></a>, <a class="ref" href="complfld.html#T5" title="COMPLFLD:th.5">COMPLFLD:5</a></i>;<br/><a NAME="E9:72_5"/>
( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span> &amp; <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="complex1.html#T29" title="COMPLEX1:th.29">COMPLEX1:29</a></i>;<br/><a NAME="E10:72_5"/><b>then </b><i><font color="Green" title="E75">A35</font></i>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y'</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c6">z'</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y'</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">z'</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E2:72_5"><i><font color="Green" title="E71">A31</font></i></a>, <a class="txt" href="hurwitz.html#E4:72_5"><i><font color="Green" title="E72">A32</font></i></a>, <a class="txt" href="hurwitz.html#E71"><i><font color="Green" title="E59">Lm15</font></i></a></i>;<br/><a NAME="E11:72_5"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y'</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">z'</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">y'</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">z'</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>;<br/><b>hence </b><a NAME="E12:72_5"/>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c4">z</font></span>)</span>,<font color="Maroon" title="c3">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c4">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c3">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E8:72_5"><i><font color="Green" title="E74">A34</font></i></a>, <a class="txt" href="hurwitz.html#E10:72_5"><i><font color="Green" title="E75">A35</font></i></a>, <a class="ref" href="complfld.html#T5" title="COMPLFLD:th.5">COMPLFLD:5</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E70">A36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/><b>assume </b><a NAME="E1:72_6"/><i><font color="Green" title="E71">A37</font></i>: 
(  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font> is <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  )
 ;<br/><b>assume </b><a NAME="E2:72_6"/><i><font color="Green" title="E72">A38</font></i>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>reconsider </b><font color="Maroon" title="c6">a'</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c7">e'</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span>,<font color="Maroon" title="c4">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></i>;<br/><i><font color="Green" title="E73">A39</font></i>: <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> = 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p4">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#T31" title="POLYNOM5:th.31">POLYNOM5:31</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c8">r'</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c4">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></i>;<br/><span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> = 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> </span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T139" title="COMPLEX1:th.139">COMPLEX1:139</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> </span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span></span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#T31" title="POLYNOM5:th.31">POLYNOM5:31</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="hurwitz.html#T44" target="_self" title="HURWITZ:th.44">Th44</a></i>
;<br/><b>hence </b><a NAME="E7:72_6"/>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p4">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p5">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c5">z</font></span>)</span></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c4">y</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E7:72"><i><font color="Green" title="E69">A29</font></i></a>, <a class="txt" href="hurwitz.html#E1:72_6"><i><font color="Green" title="E71">A37</font></i></a>, <a class="txt" href="hurwitz.html#E2:72_6"><i><font color="Green" title="E72">A38</font></i></a>, <a class="txt" href="hurwitz.html#E4:72_6"><i><font color="Green" title="E73">A39</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E71">A40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:72_7"/><i><font color="Green" title="E72">A41</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/><b>assume </b><a NAME="E1:72_7_1"/><i><font color="Green" title="E73">A42</font></i>: 
(  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c4">f</font> is <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a> &amp;  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/><b>let </b><font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:72_7_1_1"/>
( <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="hurwitz.html#E1:72_7_1"><i><font color="Green" title="E73">A42</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_7_1_1_1"><b>suppose </b></a><a NAME="E1:72_7_1_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">z1</font>, <font color="Maroon" title="c7">z2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> <b> such that </b><br/><a NAME="E3:72_7_1_1_1"/><i><font color="Green" title="E74">A43</font></i>: 
( <font color="Maroon" title="c6">z1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  &amp; <font color="Maroon" title="c4">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">z1</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c7">z2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:72_7_1"><i><font color="Green" title="E73">A42</font></i></a>, <a class="ref" href="hurwitz.html#T28" target="_self" title="HURWITZ:th.28">Th28</a></i>;<br/><a NAME="E4:72_7_1_1_1"/>
 <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c7">z2</font> is <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:72_7_1"><i><font color="Green" title="E73">A42</font></i></a>, <a class="txt" href="hurwitz.html#E3:72_7_1_1_1"><i><font color="Green" title="E74">A43</font></i></a>, <a class="ref" href="hurwitz.html#T40" target="_self" title="HURWITZ:th.40">Th40</a></i>;<br/><b>hence </b><a NAME="E5:72_7_1_1_1"/>
( (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> ) &amp; (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> ) &amp; (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> ) )
 <b>by </b><i><a class="txt" href="hurwitz.html#E4:72"><i><font color="Green" title="E66">A10</font></i></a>, <a class="txt" href="hurwitz.html#E6:72"><i><font color="Green" title="E68">A23</font></i></a>, <a class="txt" href="hurwitz.html#E8:72"><i><font color="Green" title="E70">A36</font></i></a>, <a class="txt" href="hurwitz.html#E3:72_7_1_1_1"><i><font color="Green" title="E74">A43</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_7_1_1_2"><b>suppose </b></a><a NAME="E1:72_7_1_1_2"/><i><font color="Green" title="E74">A44</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:72_7_1_1_2"/><i><font color="Green" title="E75">A45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:72_7_1_1_2"/><b>then </b>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:72_7_1"><i><font color="Green" title="E73">A42</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:72_7_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">f</font> is <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a>
 <b>by </b><i><a class="ref" href="polynom5.html#T75" title="POLYNOM5:th.75">POLYNOM5:75</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> <b> such that </b><br/><a NAME="E6:72_7_1_1_2"/><i><font color="Green" title="E76">A46</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="polynom5.html#D7" title="POLYNOM5:def.7">POLYNOM5:def 7</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">f1</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> <b> such that </b><br/><a NAME="E8:72_7_1_1_2"/><i><font color="Green" title="E77">A47</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c7">f1</font>
 <b>by </b><i><a class="txt" href="hurwitz.html#E6:72_7_1_1_2"><i><font color="Green" title="E76">A46</font></i></a>, <a class="ref" href="hurwitz.html#T33" target="_self" title="HURWITZ:th.33">Th33</a></i>;<br/><a NAME="E9:72_7_1_1_2"/><i><font color="Green" title="E78">A48</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:72_7_1"><i><font color="Green" title="E73">A42</font></i></a>, <a class="txt" href="hurwitz.html#E2:72_7_1_1_2"><i><font color="Green" title="E75">A45</font></i></a>, <a class="ref" href="polynom4.html#T6" title="POLYNOM4:th.6">POLYNOM4:6</a></i>;<br/><a NAME="E10:72_7_1_1_2"/><b>then </b><i><font color="Green" title="E79">A49</font></i>: 
 <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 <b>by </b><i><a class="txt" href="hurwitz.html#E8:72_7_1_1_2"><i><font color="Green" title="E77">A47</font></i></a>, <a class="ref" href="polynom3.html#T35" title="POLYNOM3:th.35">POLYNOM3:35</a></i>;<br/><a NAME="E11:72_7_1_1_2"/>
<font color="Maroon" title="c7">f1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 <b>by </b><i><a class="txt" href="hurwitz.html#E8:72_7_1_1_2"><i><font color="Green" title="E77">A47</font></i></a>, <a class="txt" href="hurwitz.html#E9:72_7_1_1_2"><i><font color="Green" title="E78">A48</font></i></a>, <a class="ref" href="polynom3.html#T35" title="POLYNOM3:th.35">POLYNOM3:35</a></i>;<br/><b>then </b><i><font color="Green" title="E80">A50</font></i>:  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c4">f</font> = 
<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c7">f1</font></span>)</span>
<b>by </b><i><a class="txt" href="hurwitz.html#E8:72_7_1_1_2"><i><font color="Green" title="E77">A47</font></i></a>, <a class="txt" href="hurwitz.html#E10:72_7_1_1_2"><i><font color="Green" title="E79">A49</font></i></a>, <a class="ref" href="hurwitz.html#T23" target="_self" title="HURWITZ:th.23">Th23</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c7">f1</font></span>)</span>
<b>by </b><i><a class="ref" href="hurwitz.html#T27" target="_self" title="HURWITZ:th.27">Th27</a></i>
;<br/><a NAME="E13:72_7_1_1_2"/><i><font color="Green" title="E81">A51</font></i>: 
( <font color="Maroon" title="c7">f1</font> is <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a> &amp;  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font> is <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a> )
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:72_7_1"><i><font color="Green" title="E73">A42</font></i></a>, <a class="txt" href="hurwitz.html#E8:72_7_1_1_2"><i><font color="Green" title="E77">A47</font></i></a>, <a class="ref" href="hurwitz.html#T41" target="_self" title="HURWITZ:th.41">Th41</a></i>;<br/><a NAME="E14:72_7_1_1_2"/><i><font color="Green" title="E82">A52</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:72_7_1_1_2"><i><font color="Green" title="E74">A44</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><i><font color="Green" title="E83">A53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_7_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_7_1_1_2_2"/><i><font color="Green" title="E84">A54</font></i>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>reconsider </b><font color="Maroon" title="c8">r'</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span>,<font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c9">e'</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c7">f1</font>,<font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">r1'</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c11">e1'</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></i>;<br/><a NAME="E4:72_7_1_1_2_2"/><i><font color="Green" title="E85">A55</font></i>: 
( <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c9">e'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c11">e1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> &amp; <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c8">r'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c10">r1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> )
 <b>by </b><i><a class="txt" href="hurwitz.html#E3:72"><i><font color="Green" title="E65">A3</font></i></a>, <a class="txt" href="hurwitz.html#E1:72_7"><i><font color="Green" title="E72">A41</font></i></a>, <a class="txt" href="hurwitz.html#E1:72_7_1"><i><font color="Green" title="E73">A42</font></i></a>, <a class="txt" href="hurwitz.html#E12:72_7_1_1_2"><i><font color="Green" title="E80">A50</font></i></a>, <a class="txt" href="hurwitz.html#E13:72_7_1_1_2"><i><font color="Green" title="E81">A51</font></i></a>, <a class="txt" href="hurwitz.html#E14:72_7_1_1_2"><i><font color="Green" title="E82">A52</font></i></a>, <a class="txt" href="hurwitz.html#E1:72_7_1_1_2_2"><i><font color="Green" title="E84">A54</font></i></a></i>;<br/><a NAME="E5:72_7_1_1_2_2"/><i><font color="Green" title="E86">A56</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c9">e'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c8">r'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> )
 <b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/><a NAME="E6:72_7_1_1_2_2"/><b>then </b><i><font color="Green" title="E87">A57</font></i>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c10">r1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c9">e'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c10">r1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c11">e1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E4:72_7_1_1_2_2"><i><font color="Green" title="E85">A55</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E7:72_7_1_1_2_2"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c8">r'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c9">e'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c10">r1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c9">e'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E4:72_7_1_1_2_2"><i><font color="Green" title="E85">A55</font></i></a>, <a class="txt" href="hurwitz.html#E5:72_7_1_1_2_2"><i><font color="Green" title="E86">A56</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E8:72_7_1_1_2_2"/><b>then </b>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c8">r'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c9">e'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c10">r1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c11">e1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E6:72_7_1_1_2_2"><i><font color="Green" title="E87">A57</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:72_7_1_1_2_2"/><b>then </b>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r'</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <font color="Maroon" title="c9">e'</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c10">r1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c11">e1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>;<br/><a NAME="E10:72_7_1_1_2_2"/><b>then </b><i><font color="Green" title="E88">A58</font></i>: 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p4">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c7">f1</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>;<br/><a NAME="E11:72_7_1_1_2_2"/>
( <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c7">f1</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c7">f1</font></span>)</span>,<font color="Maroon" title="c5">x</font> &amp; <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">x</font> )
 <b>by </b><i><a class="ref" href="polynom4.html#T27" title="POLYNOM4:th.27">POLYNOM4:27</a></i>;<br/><b>hence </b><a NAME="E12:72_7_1_1_2_2"/>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E8:72_7_1_1_2"><i><font color="Green" title="E77">A47</font></i></a>, <a class="txt" href="hurwitz.html#E10:72_7_1_1_2_2"><i><font color="Green" title="E88">A58</font></i></a>, <a class="ref" href="hurwitz.html#T47" target="_self" title="HURWITZ:th.47">Th47</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E84">A59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_7_1_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_7_1_1_2_3"/><i><font color="Green" title="E85">A60</font></i>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>reconsider </b><font color="Maroon" title="c8">r'</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span>,<font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c9">e'</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c7">f1</font>,<font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">r1'</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c11">e1'</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></i>;<br/><a NAME="E4:72_7_1_1_2_3"/><i><font color="Green" title="E86">A61</font></i>: 
( <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c9">e'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c11">e1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> &amp; <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c8">r'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c10">r1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> )
 <b>by </b><i><a class="txt" href="hurwitz.html#E5:72"><i><font color="Green" title="E67">A16</font></i></a>, <a class="txt" href="hurwitz.html#E1:72_7"><i><font color="Green" title="E72">A41</font></i></a>, <a class="txt" href="hurwitz.html#E1:72_7_1"><i><font color="Green" title="E73">A42</font></i></a>, <a class="txt" href="hurwitz.html#E12:72_7_1_1_2"><i><font color="Green" title="E80">A50</font></i></a>, <a class="txt" href="hurwitz.html#E13:72_7_1_1_2"><i><font color="Green" title="E81">A51</font></i></a>, <a class="txt" href="hurwitz.html#E14:72_7_1_1_2"><i><font color="Green" title="E82">A52</font></i></a>, <a class="txt" href="hurwitz.html#E1:72_7_1_1_2_3"><i><font color="Green" title="E85">A60</font></i></a></i>;<br/><a NAME="E5:72_7_1_1_2_3"/><i><font color="Green" title="E87">A62</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c11">e1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c10">r1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> )
 <b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/><a NAME="E6:72_7_1_1_2_3"/><b>then </b><i><font color="Green" title="E88">A63</font></i>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c8">r'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c9">e'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c10">r1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c9">e'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E4:72_7_1_1_2_3"><i><font color="Green" title="E86">A61</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E7:72_7_1_1_2_3"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c10">r1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c9">e'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c10">r1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c11">e1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E4:72_7_1_1_2_3"><i><font color="Green" title="E86">A61</font></i></a>, <a class="txt" href="hurwitz.html#E5:72_7_1_1_2_3"><i><font color="Green" title="E87">A62</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E8:72_7_1_1_2_3"/><b>then </b>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c8">r'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c9">e'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c10">r1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c11">e1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E6:72_7_1_1_2_3"><i><font color="Green" title="E88">A63</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:72_7_1_1_2_3"/><b>then </b>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r'</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <font color="Maroon" title="c9">e'</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c10">r1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c11">e1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>;<br/><a NAME="E10:72_7_1_1_2_3"/><b>then </b><i><font color="Green" title="E89">A64</font></i>: 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p4">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c7">f1</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>;<br/><a NAME="E11:72_7_1_1_2_3"/>
( <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c7">f1</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c7">f1</font></span>)</span>,<font color="Maroon" title="c5">x</font> &amp; <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">x</font> )
 <b>by </b><i><a class="ref" href="polynom4.html#T27" title="POLYNOM4:th.27">POLYNOM4:27</a></i>;<br/><b>hence </b><a NAME="E12:72_7_1_1_2_3"/>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E8:72_7_1_1_2"><i><font color="Green" title="E77">A47</font></i></a>, <a class="txt" href="hurwitz.html#E10:72_7_1_1_2_3"><i><font color="Green" title="E89">A64</font></i></a>, <a class="ref" href="hurwitz.html#T47" target="_self" title="HURWITZ:th.47">Th47</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_7_1_1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_7_1_1_2_4"/><i><font color="Green" title="E85">A65</font></i>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:72_7_1_1_2_4"/><b>then </b><i><font color="Green" title="E86">A66</font></i>: 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c7">f1</font>,<font color="Maroon" title="c5">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:72_7"><i><font color="Green" title="E72">A41</font></i></a>, <a class="txt" href="hurwitz.html#E1:72_7_1"><i><font color="Green" title="E73">A42</font></i></a>, <a class="txt" href="hurwitz.html#E12:72_7_1_1_2"><i><font color="Green" title="E80">A50</font></i></a>, <a class="txt" href="hurwitz.html#E13:72_7_1_1_2"><i><font color="Green" title="E81">A51</font></i></a>, <a class="txt" href="hurwitz.html#E14:72_7_1_1_2"><i><font color="Green" title="E82">A52</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">r'</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span>,<font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c9">e'</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c7">f1</font>,<font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">r1'</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c11">e1'</font> =  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></i>;<br/><a NAME="E5:72_7_1_1_2_4"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c8">r'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c9">e'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c10">r1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c11">e1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E7:72"><i><font color="Green" title="E69">A29</font></i></a>, <a class="txt" href="hurwitz.html#E13:72_7_1_1_2"><i><font color="Green" title="E81">A51</font></i></a>, <a class="txt" href="hurwitz.html#E1:72_7_1_1_2_4"><i><font color="Green" title="E85">A65</font></i></a>, <a class="txt" href="hurwitz.html#E2:72_7_1_1_2_4"><i><font color="Green" title="E86">A66</font></i></a></i>;<br/><a NAME="E6:72_7_1_1_2_4"/><b>then </b>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r'</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <font color="Maroon" title="c9">e'</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c10">r1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c11">e1'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>;<br/><a NAME="E7:72_7_1_1_2_4"/><b>then </b><i><font color="Green" title="E87">A67</font></i>: 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p4">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c7">f1</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>;<br/><a NAME="E8:72_7_1_1_2_4"/>
( <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c7">f1</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c7">f1</font></span>)</span>,<font color="Maroon" title="c5">x</font> &amp; <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> 1,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">x</font> )
 <b>by </b><i><a class="ref" href="polynom4.html#T27" title="POLYNOM4:th.27">POLYNOM4:27</a></i>;<br/><b>hence </b><a NAME="E9:72_7_1_1_2_4"/>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <b>by </b><i><a class="txt" href="hurwitz.html#E8:72_7_1_1_2"><i><font color="Green" title="E77">A47</font></i></a>, <a class="txt" href="hurwitz.html#E7:72_7_1_1_2_4"><i><font color="Green" title="E87">A67</font></i></a>, <a class="ref" href="hurwitz.html#T47" target="_self" title="HURWITZ:th.47">Th47</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E18:72_7_1_1_2"/>
( (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> ) &amp; (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> ) &amp; (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> ) )
 <b>by </b><i><a class="txt" href="hurwitz.html#E15:72_7_1_1_2"><i><font color="Green" title="E83">A53</font></i></a>, <a class="txt" href="hurwitz.html#E16:72_7_1_1_2"><i><font color="Green" title="E84">A59</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="E3:72_7"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E10:72"/><i><font color="Green" title="E72">A68</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="hurwitz.html#E2:72"><i><font color="Green" title="E64">A2</font></i></a>, <a class="txt" href="hurwitz.html#E9:72"><i><font color="Green" title="E71">A40</font></i></a>);<br/></i>
<a NAME="E11:72"/>
<font color="Maroon" title="c1">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 
<b>by </b><i><a class="txt" href="hurwitz.html#E1:72"><i><font color="Green" title="E63">A1</font></i></a>, <a class="ref" href="hurwitz.html#T20" target="_self" title="HURWITZ:th.20">Th20</a></i>;<br/>
<a NAME="E12:72"/><b>then </b>
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="txt" href="hurwitz.html#E31"><i><font color="Green" title="E29">Lm9</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E14:72"/><i><font color="Green" title="E73">A69</font></i>: 
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font>
 ;<br/>
<b>thus </b><a NAME="E15:72"/>
( (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c2">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> ) &amp; (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c2">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> ) &amp; (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="hurwitz.html#K7" title="HURWITZ:func.7">*'</a> </span>)</span>,<font color="Maroon" title="c2">x</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> ) )
 <b>by </b><i><a class="txt" href="hurwitz.html#E1:72"><i><font color="Green" title="E63">A1</font></i></a>, <a class="txt" href="hurwitz.html#E10:72"><i><font color="Green" title="E72">A68</font></i></a>, <a class="txt" href="hurwitz.html#E14:72"><i><font color="Green" title="E73">A69</font></i></a></i>;<br/>


</div>
