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

<b>let </b><font color="Maroon" title="c1">g</font>, <font color="Maroon" title="c2">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<b>thus </b><a NAME="E1:1"/>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font> &amp; <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">g</font> implies  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">g</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:1_1"/><i><font color="Green" title="E2">A1</font></i>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font>
 <b>and </b><br/><a NAME="E2:1_1"/><i><font color="Green" title="E3">A2</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">g</font>
 ;<br/>

<a NAME="E3:1_1"/><i><font color="Green" title="E4">A3</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">g</font>
 
<b>by </b><i><a class="txt" href="seq_2.html#E1:1_1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="seq_2.html#E2:1_1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="absvalue.html#T12" title="ABSVALUE:th.12">ABSVALUE:12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1"/><i><font color="Green" title="E5">A4</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">g</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1"/>
<font color="Maroon" title="c2">r</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="E2:1_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">r</font>
 <b>by </b><i><a class="txt" href="seq_2.html#E1:1_1_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="seq_2.html#E1:1_1"><i><font color="Green" title="E2">A1</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="seq_2.html#E2:1_1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="seq_2.html#E1:1_1_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">g</font>
 <b>by </b><i><a class="txt" href="seq_2.html#E3:1_1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:1"/><i><font color="Green" title="E2">A5</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">g</font>
 ;<br/>

<a NAME="E3:1"/><b>then </b><i><font color="Green" title="E3">A6</font></i>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#T12" title="ABSVALUE:th.12">ABSVALUE:12</a></i>;<br/>
<a NAME="E4:1"/><i><font color="Green" title="E4">A7</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>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">r</font>
 
<b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/>
<a NAME="E5:1"/><i><font color="Green" title="E5">A8</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">g</font>
 
<b>by </b><i><a class="txt" href="seq_2.html#E2:1"><i><font color="Green" title="E2">A5</font></i></a>, <a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/>
<a NAME="E6:1"/><i><font color="Green" title="E6">A9</font></i>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="seq_2.html#E2:1"><i><font color="Green" title="E2">A5</font></i></a>, <a class="txt" href="seq_2.html#E4:1"><i><font color="Green" title="E4">A7</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_2"/>
<font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">g</font>
 ;<br/><b>then </b> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">r</font> = 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">g</font></span>)</span>
<b>by </b><i><a class="txt" href="seq_2.html#E6:1"><i><font color="Green" title="E6">A9</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c1">g</font>

;<br/><b>hence </b><a NAME="E3:1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="seq_2.html#E2:1"><i><font color="Green" title="E2">A5</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:1"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font>
 <b>by </b><i><a class="txt" href="seq_2.html#E3:1"><i><font color="Green" title="E3">A6</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>

<b>thus </b><a NAME="E9:1"/>
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">g</font>
 <b>by </b><i><a class="txt" href="seq_2.html#E2:1"><i><font color="Green" title="E2">A5</font></i></a>, <a class="txt" href="seq_2.html#E5:1"><i><font color="Green" title="E5">A8</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>


</div>
