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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">u</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">P</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> <font color="Maroon">n</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:83"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:83"/>
for <font color="Olive">p1</font>, <font color="Olive">p2</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> <font color="Maroon">n</font></span>)</span>  st <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/> <a href="topreal1.html#K3">LSeg</a> <font color="Olive">p1</font>,<font color="Olive">p2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p1</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> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">p2</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> <font color="Maroon">n</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:83_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 ;<br/>

<a NAME="E2:83_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">u</font>,<font color="Olive">q</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/>
<b>then consider </b><font color="Maroon">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> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E4:83_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">u</font>,<font color="Maroon">q1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">u</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E6:83_1"/><i><font color="Green">E59</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="jgraph_1.html#T45">JGRAPH_1:45</a></i>;<br/>
<b>consider </b><font color="Maroon">q2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E8:83_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">u</font>,<font color="Maroon">q2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E9:83_1"/><i><font color="Green">E61</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T11" target="_self">Th11</a>, <a class="ref" href="jgraph_1.html#T45">JGRAPH_1:45</a></i>;<br/>
<a NAME="E10:83_1"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">p3</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> <font color="Maroon">n</font></span>)</span>  st <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Olive">p3</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> holds <br/><font color="Olive">p3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p3</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> <font color="Maroon">n</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:83_1_1"/><i><font color="Green">E64</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p3</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">u3</font> = <font color="Maroon">p3</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E3:83_1_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">u</font>,<font color="Maroon">u3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T13" target="_self">Th13</a>, <a class="ref" href="jgraph_1.html#T45">JGRAPH_1:45</a></i>;<br/>
<b>hence </b><a NAME="E4:83_1_1"/>
<font color="Maroon">p3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E11:83_1"/>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:83_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/>

<a NAME="E2:83_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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">1 <a href="binop_2.html#K10">-</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="topreal1.html#D3">TOPREAL1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:83_1_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="jordan2c.html#T13" target="_self">Th13</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T37">EUCLID:37</a></i>
<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
;<br/>
<b>then </b><span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> = 
<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"><span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T50">EUCLID:50</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"><span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID: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"><span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID: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"><span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p5">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T30">EUCLID:30</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"><span class="p5">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T30">EUCLID:30</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"><span class="p5">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p5">(<span class="default"><a href="euclid.html#K19">-</a> <font color="Maroon">p1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span></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"><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">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K17">+</a> <span class="p5">(<span class="default"><a href="euclid.html#K19">-</a> <font color="Maroon">p1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T36">EUCLID:36</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">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID: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">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <span class="p5">(<span class="default"><a href="euclid.html#K19">-</a> <font color="Maroon">p2</font></span>)</span></span>)</span></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"><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">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K17">+</a> <span class="p5">(<span class="default"><a href="euclid.html#K19">-</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T36">EUCLID:36</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">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>
;<br/>
<a NAME="E8:83_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <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">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K9">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="ref" href="toprns_1.html#T30">TOPRNS_1:30</a></i>;<br/>
<a NAME="E9:83_1_2"/><i><font color="Green">E130</font></i>: 
<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">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></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"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font></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/>
<a NAME="E10:83_1_2"/><i><font color="Green">E131</font></i>: 
1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E11:83_1_2"/><b>then </b><i><font color="Green">E152</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_2_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_2_3"/>
( 1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or 1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_2_3_1"><b>suppose </b></a><a NAME="E1:83_1_2_3_1"/>
1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:83_1_2_3_1"/><b>then </b><i><font color="Green">E154</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></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"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T9" target="_self">Th9</a>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:83_1_2_3_1"/><i><font color="Green">E157</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></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"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></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/><a NAME="E4:83_1_2_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E5:83_1_2_3_1"/><b>then </b><i><font color="Green">E158</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></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"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T12" target="_self">Th12</a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E6:83_1_2_3_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E7:83_1_2_3_1"/><b>then </b>
<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">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K9">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan2c.html#D2" target="_self">Def2</a>, <a class="ref" href="jordan2c.html#T16" target="_self">Th16</a>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E8:83_1_2_3_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T15" target="_self">Th15</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E9:83_1_2_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_2_3_2"><b>suppose </b></a><a NAME="E1:83_1_2_3_2"/>
1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:83_1_2_3_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0 <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E3:83_1_2_3_2"/><b>then </b><i><font color="Green">E159</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E4:83_1_2_3_2"/><i><font color="Green">E160</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></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"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T9" target="_self">Th9</a>, , , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E5:83_1_2_3_2"/><i><font color="Green">E161</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></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"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></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/><a NAME="E6:83_1_2_3_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#D2" target="_self">Def2</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E7:83_1_2_3_2"/><b>then </b><i><font color="Green">E162</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></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"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T12" target="_self">Th12</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E8:83_1_2_3_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E9:83_1_2_3_2"/><b>then </b>
<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">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K9">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan2c.html#T16" target="_self">Th16</a>, , , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E10:83_1_2_3_2"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T15" target="_self">Th15</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E11:83_1_2_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E12:83_1"/>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:83"/>
<font color="Maroon">P</font> is <a href="jordan1.html#V1">convex</a>
 <b>by </b><i><a class="ref" href="jordan1.html#D1">JORDAN1:def 1</a></i>;<br/>


</div>
