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

<b>let </b><font color="Maroon">d</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>set </b><font color="Maroon">c</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4;<br/>
<b>set </b><font color="Maroon">xx</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4</span>)</span>;<br/>
<b>assume </b><a NAME="E1:44"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:44"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">d</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 4
 
;<br/>
<a NAME="E3:44"/>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E4:44"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 4
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:44"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E6:44"/><b>then </b>
3 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="nat_1.html#K2">*</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E7:44"/><b>then </b>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/>
<a NAME="E8:44"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 4 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K7">/</a> 4
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<b>hence </b><a NAME="E9:44"/>
contradiction
 <b>by </b><i><a class="ref" href="scmpds_9.html#T2" target="_self">Th2</a>, </i>;<br/>


</div>
