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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 1</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 1</span>)</span>;<br/>





<b>assume </b><a NAME="E1:73"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 1</span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<b>hence </b><a NAME="E2:73"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T66" target="_self">Th66</a></i>;<br/>

<b>assume </b><a NAME="E3:73"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="tbsp_1.html#V6">bounded</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:73"/><i><font color="Green">E68</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 1</span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) )
 <b>by </b><i><a class="ref" href="tbsp_1.html#D9">TBSP_1:def 9</a></i>;<br/>
<a NAME="E6:73"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/>
<a NAME="E7:73"/><i><font color="Green">E70</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font> <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/>
<a NAME="E8:73"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E9:73"/><b>then </b><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E7:73"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E10:73"/>
3 <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="jordan2c.html#E5:73"><i><font color="Green">E68</font></i></a>, <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/>
<a NAME="E11:73"/><b>then </b>
0 <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E9:73"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E12:73"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 3 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E6:73"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:73"/><b>then </b><i><font color="Green">E72</font></i>: 
 <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span> as    <a href="subset_1.html#M1">Element</a> of 1 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> 1 ;<br/>
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="jordan2c.html#T36" target="_self">Th36</a>, <a class="ref" href="euclid.html#D11">EUCLID:def 11</a></i>
<br/>.= 
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p4">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p0">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> 1</span>)</span></span><a href="binarith.html#K13">*&gt;</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T69">RVSUM_1:69</a></i>
;<br/>
<a NAME="E17:73"/><b>then </b><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 1</span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan2c.html#E13:73"><i><font color="Green">E72</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> 1</span>)</span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 1</span>)</span> <b>by </b><i><a class="txt" href="jordan2c.html#E1:73"><i><font color="Green">E67</font></i></a></i>;<br/>
<a NAME="E19:73"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/>
<a NAME="E20:73"/>
0 <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E5:73"><i><font color="Green">E68</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E21:73"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E19:73"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E22:73"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="jordan2c.html#T36" target="_self">Th36</a>, <a class="ref" href="euclid.html#D11">EUCLID:def 11</a></i>
<br/>.= 
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> 1</span>)</span></span><a href="binarith.html#K13">*&gt;</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T69">RVSUM_1:69</a></i>
;<br/>
<a NAME="E24:73"/><b>then </b><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E1:73"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan2c.html#E22:73"><i><font color="Green">E75</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> 1</span>)</span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 1</span>)</span> ;<br/>
<a NAME="E26:73"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E27:73"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E28:73"/><b>then </b><i><font color="Green">E78</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E5:73"><i><font color="Green">E68</font></i></a></i>;<br/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> = 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p5">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> 1</span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p0">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> 1</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="jgraph_1.html#T45">JGRAPH_1:45</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p5">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p0">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> 1</span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T54">EUCLID:54</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p0">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> 1</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="toprns_1.html#T27">TOPRNS_1:27</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p0">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p5">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p0">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> 1</span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T44">EUCLID:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> 1</span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="toprns_1.html#T8">TOPRNS_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan2c.html#T37" target="_self">Th37</a></i>
;<br/>
<a NAME="E30:73"/><b>then </b><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a>, <a class="ref" href="square_1.html#T83">SQUARE_1:83</a></i>;<br/>
<a NAME="E31:73"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E28:73"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E32:73"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E30:73"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E33:73"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E27:73"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E34:73"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:73"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan2c.html#E5:73"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan2c.html#E17:73"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan2c.html#E24:73"><i><font color="Green">E76</font></i></a></i>;<br/>


</div>
