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

<b>let </b><font color="Maroon" title="c1">q</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E9">A1</font></i>: 
( <font color="Maroon" title="c1">q</font> is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> &amp; <font color="Maroon" title="c1">q</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>let </b><font color="Maroon" title="c2">r</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><b>that </b><br/><a NAME="E2:9"/><i><font color="Green" title="E10">A2</font></i>: 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>and </b><br/><a NAME="E3:9"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>
 ;<br/>

<b>set </b><font color="Maroon" title="c3">a</font> =  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">r</font>;<br/>
<b>set </b><font color="Maroon" title="c4">b</font> =  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c2">r</font>;<br/>
<a NAME="E4:9"/><i><font color="Green" title="E12">A4</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</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="c2">r</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="uniroots.html#E2:9"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="comptrig.html#T7" title="COMPTRIG:th.7">COMPTRIG:7</a>, <a class="ref" href="square_1.html#T59" title="SQUARE_1:th.59">SQUARE_1:59</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1"/><i><font color="Green" title="E13">A5</font></i>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:9_1"/><b>then </b>
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="uniroots.html#E4:9"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/><b>hence </b><a NAME="E3:9_1"/>
contradiction
 <b>by </b><i><a class="txt" href="uniroots.html#E3:9"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="uniroots.html#E1:9_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="complex1.html#T29" title="COMPLEX1:th.29">COMPLEX1:29</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:9"/><b>then </b>
(  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp;  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="uniroots.html#E2:9"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="complex1.html#T140" title="COMPLEX1:th.140">COMPLEX1:140</a></i>;<br/>
<a NAME="E7:9"/><b>then </b><i><font color="Green" title="E13">A6</font></i>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 
<b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E8:9"/><i><font color="Green" title="E14">A7</font></i>: 
(  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">r</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c2">r</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">q</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">r</font></span>)</span> &amp;  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">r</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c2">r</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c2">r</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="complex1.html#T28" title="COMPLEX1:th.28">COMPLEX1:28</a></i>;<br/>
<a NAME="E9:9"/>
2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">q</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> 
 
<b>by </b><i><a class="txt" href="uniroots.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="real_2.html#T122" title="REAL_2:th.122">REAL_2:122</a></i>;<br/>
<a NAME="E10:9"/><b>then </b>
2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">q</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E7:9"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="real_2.html#T145" title="REAL_2:th.145">REAL_2:145</a></i>;<br/>
<a NAME="E11:9"/><b>then </b>
 <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">q</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">r</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<a NAME="E12:9"/><b>then </b>
<span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">q</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">q</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">r</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E13:9"/><b>then </b><i><font color="Green" title="E15">A8</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">q</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">r</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">q</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">qc</font> = <span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c1">q</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> as   <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/>
<a NAME="E15:9"/><i><font color="Green" title="E16">A9</font></i>: 
<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="c5">qc</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">r</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</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"><font color="Maroon" title="c5">qc</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">r</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</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"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c1">q</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">r</font></span>)</span>,<span class="p4">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c2">r</font></span>)</span></span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
<b>by </b><i><a class="ref" href="complex1.html#T29" title="COMPLEX1:th.29">COMPLEX1:29</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p4">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">r</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p4">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c2">r</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
<b>by </b><i><a class="ref" href="polynom5.html#T6" title="POLYNOM5:th.6">POLYNOM5:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">r</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</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="c2">r</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E8:9"><i><font color="Green" title="E14">A7</font></i></a>, <a class="ref" href="comptrig.html#T7" title="COMPTRIG:th.7">COMPTRIG:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">q</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">r</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1
<b>by </b><i><a class="txt" href="uniroots.html#E4:9"><i><font color="Green" title="E12">A4</font></i></a></i>
;<br/>
<a NAME="E17:9"/><b>then </b>
<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="c5">qc</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">r</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="uniroots.html#E13:9"><i><font color="Green" title="E15">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E18:9"/>
<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"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c1">q</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">r</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> <font color="Maroon" title="c1">q</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 <b>by </b><i><a class="txt" href="uniroots.html#E15:9"><i><font color="Green" title="E16">A9</font></i></a>, <a class="ref" href="square_1.html#T118" title="SQUARE_1:th.118">SQUARE_1:118</a></i>;<br/>


</div>
