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

<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">B</font> be  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><a NAME="E1:52"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 ;<br/>

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E2:52"/><i><font color="Green">E40</font></i>: 
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E3:52"/><i><font color="Green">E42</font></i>: 
 <a href="jordan1k.html#K5" target="_self">dist</a> <font color="Maroon">p</font>,<font color="Maroon">B</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan1k.html#K5" target="_self">dist</a> <font color="Maroon">p</font>,<font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="jordan1k.html#E2:52"><i><font color="Green">E40</font></i></a>, <a class="ref" href="jordan1k.html#T3" target="_self">Th3</a></i>;<br/>
<b>assume </b><a NAME="E4:52"/>
 <a href="jordan1k.html#K5" target="_self">dist</a> <font color="Maroon">p</font>,<font color="Maroon">B</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="jordan1k.html#K5" target="_self">dist</a> <font color="Maroon">p</font>,<font color="Maroon">A</font>
 ;<br/>

<a NAME="E5:52"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="jordan1k.html#K5" target="_self">dist</a> <font color="Maroon">p</font>,<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="jordan1k.html#K5" target="_self">dist</a> <font color="Maroon">p</font>,<font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="jordan1k.html#E3:52"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>consider </b><font color="Maroon">q</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E7:52"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E8:52"/><i><font color="Green">E48</font></i>: 
 <a href="jordan1k.html#K5" target="_self">dist</a> <font color="Maroon">p</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="jordan1k.html#E1:52"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">B</font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">q</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E11:52"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan1k.html#E2:52"><i><font color="Green">E40</font></i></a>, <a class="ref" href="jordan1k.html#T4" target="_self">Th4</a></i>;<br/>
<b>consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E13:52"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E14:52"/><i><font color="Green">E95</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">a</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan1k.html#E2:52"><i><font color="Green">E40</font></i></a>, <a class="ref" href="jordan1k.html#T4" target="_self">Th4</a>, <a class="ref" href="topmetr.html#T22">TOPMETR:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">s</font> = <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">q'</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span>;<br/>
<a NAME="E16:52"/><i><font color="Green">E98</font></i>: 
 <a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="txt" href="jordan1k.html#E7:52"><i><font color="Green">E46</font></i></a>, </i>;<br/>
<a NAME="E17:52"/><b>then </b><i><font color="Green">E99</font></i>: 
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/>
<a NAME="E18:52"/><b>then </b><i><font color="Green">E102</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<b>then </b> <a href="topreal6.html#K1">dist</a> <font color="Maroon">q</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="real_1.html#K6">/</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T85">XCMPLX_1:85</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
<b>by </b><i><a class="txt" href="jordan1k.html#E11:52"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
;<br/>
<a NAME="E20:52"/><b>then </b><i><font color="Green">E103</font></i>: 
 <a href="topreal6.html#K1">dist</a> <font color="Maroon">q</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 1
 
<b>by </b><i>, <a class="ref" href="real_2.html#T200">REAL_2:200</a></i>;<br/>
<a NAME="E21:52"/>
 ex <font color="Olive">p1</font>, <font color="Olive">q1</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <font color="Olive">q1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> &amp;  <a href="topreal6.html#K1">dist</a> <font color="Maroon">q</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Olive">p1</font>,<font color="Olive">q1</font> )
 
<b>by </b><i><a class="ref" href="topreal6.html#D1">TOPREAL6:def 1</a></i>;<br/>
<a NAME="E22:52"/><b>then </b><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">a</font>,<font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 ;<br/><a NAME="E2:52_2"/>
 ex <font color="Olive">p1</font>, <font color="Olive">q1</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Olive">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp;  <a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Olive">p1</font>,<font color="Olive">q1</font> )
 <b>by </b><i><a class="ref" href="topreal6.html#D1">TOPREAL6:def 1</a></i>;<br/><a NAME="E3:52_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">a</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan1k.html#T7" target="_self">Th7</a>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><b>hence </b><a NAME="E4:52_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan1k.html#E8:52"><i><font color="Green">E48</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E24:52"/><b>then </b>
1 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="real_2.html#T199">REAL_2:199</a></i>;<br/>
<a NAME="E25:52"/><b>then </b>
<font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1
 
<b>by </b><i><a class="ref" href="jordan1k.html#T5" target="_self">Th5</a>, <a class="ref" href="real_2.html#T118">REAL_2:118</a></i>;<br/>
<a NAME="E26:52"/><b>then </b><i><font color="Green">E106</font></i>: 
 <a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E27:52"/>
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K5">-</a> 0
 
<b>by </b><i><a class="ref" href="jordan1k.html#T6" target="_self">Th6</a>, <a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/>
<a NAME="E28:52"/><b>then </b>
 <a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 
<b>by </b><i><a class="txt" href="jordan1k.html#E11:52"><i><font color="Green">E49</font></i></a>, <a class="ref" href="jordan1k.html#T7" target="_self">Th7</a>, <a class="ref" href="real_2.html#T145">REAL_2:145</a></i>;<br/>
<b>hence </b><a NAME="E29:52"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1k.html#E5:52"><i><font color="Green">E43</font></i></a>, , <a class="txt" href="jordan1k.html#E8:52"><i><font color="Green">E48</font></i></a>, , </i>;<br/>


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