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

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> , <font color="Maroon">xv</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>let </b><font color="Maroon">u</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">v</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">uv</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">vv</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:69"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> &amp; <font color="Maroon">xv</font> <a href="hidden.html#R1">=</a> <font color="Maroon">uv</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">vv</font> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> &amp; <font color="Maroon">uv</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_1"/>
<font color="Maroon">u</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><a NAME="E2:69_1"/><b>then </b>
<font color="Maroon">u</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">uv</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">uv</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/><a NAME="E3:69_1"/><b>then </b>
4 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">uv</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 4 <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="E4:69_1"/><b>then </b>
<span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">uv</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E5:69_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/><a NAME="E6:69_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p5">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:69_1"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="square_1.html#D3">SQUARE_1:def 3</a></i>;<br/><a NAME="E8:69_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="square_1.html#D3">SQUARE_1:def 3</a></i>;<br/><a NAME="E9:69_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> )
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><b>hence </b><a NAME="E10:69_1"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T68">XREAL_1:68</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:69"/>
( <font color="Maroon">u</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 implies <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_2"/>
<font color="Maroon">u</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E2:69_2"/><b>then </b>
<font color="Maroon">u</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">uv</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">uv</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/><a NAME="E3:69_2"/><b>then </b>
4 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">uv</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 <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="E4:69_2"/><b>then </b>
<span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">uv</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><a NAME="E5:69_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i>, </i>;<br/><a NAME="E6:69_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p5">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:69_2"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="square_1.html#D3">SQUARE_1:def 3</a></i>;<br/><a NAME="E8:69_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="square_1.html#D3">SQUARE_1:def 3</a></i>;<br/><a NAME="E9:69_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> )
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><b>hence </b><a NAME="E10:69_2"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T68">XREAL_1:68</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:69"/>
( <font color="Maroon">u</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:69_3"/><b>then </b>
<span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">uv</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E3:69_3"/><b>then </b><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_3_1"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 ;<br/><a NAME="E2:69_3_1"/><i><font color="Green">E37</font></i>: 
( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_3_1_1"/>
( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> or <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_3_1_1_1"><b>suppose </b></a><a NAME="E1:69_3_1_1_1"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:69_3_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="square_1.html#T78">SQUARE_1:78</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_3_1_1_2"><b>suppose </b></a><a NAME="E1:69_3_1_1_2"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:69_3_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="square_1.html#T78">SQUARE_1:78</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:69_3"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:69"/>
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 0 implies <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K11">-</a> <font color="Maroon">xv</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">xv</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> )
 ;<br/>


</div>
