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

<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>, <font color="Maroon">q</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="E1:5"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E2:5"/>
for <font color="Olive">G</font> being  <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>  st <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">G</font> holds <br/><span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="tarski.html#K2">}</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</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> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:5_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E2:5_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">G</font> as   <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">x</font> = <font color="Maroon">p</font>, <font color="Maroon">y</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/>
<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="E6:5_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E7:5_1"/><i><font color="Green">E69</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <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="real_1.html#NM1">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">t</font> =  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>;<br/>
<b>set </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>;<br/>
<a NAME="E9:5_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E10:5_1"/><i><font color="Green">E72</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="sub_metr.html#T2">SUB_METR:2</a></i>;<br/>
<a NAME="E11:5_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E12:5_1"/><b>then </b><i><font color="Green">E73</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T21">XXREAL_0:21</a></i>;<br/>
<a NAME="E13:5_1"/><b>then </b><i><font color="Green">E74</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E14:5_1"/><i><font color="Green">E75</font></i>: 
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<a NAME="E15:5_1"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <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="E16:5_1"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E17:5_1"/><b>then </b><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1
 
<b>by </b><i>, <a class="ref" href="real_2.html#T142">REAL_2:142</a></i>;<br/>
<b>set </b><font color="Maroon">pp</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"><span class="p4">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">z</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"><span class="p4">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></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> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E19:5_1"/><i><font color="Green">E80</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"><span class="p4">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 
<b>by </b><i>, , <a class="ref" href="sppol_1.html#T22">SPPOL_1:22</a></i>;<br/>
<i><font color="Green">E81</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"><span class="p4">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <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="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></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/>
<div><i><font color="Green">E82</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_1_2"/>
<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"><span class="p4">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 ;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</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"><span class="p5">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</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"><span class="p4">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, <a class="ref" href="euclid.html#T52">EUCLID:52</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="euclid.html#T52">EUCLID:52</a></i>
;<br/><b>hence </b><a NAME="E3:5_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="euclid.html#T38">EUCLID:38</a></i>;<br/></div><b>end;</b></div>
<i><font color="Green">E83</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"><span class="p4">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></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"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span> = 
<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"><span class="p4">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="euclid.html#T37">EUCLID:37</a></i>
<br/>.= 
<font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
;<br/>
<a NAME="E23:5_1"/><i><font color="Green">E84</font></i>: 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, , <a class="ref" href="real_2.html#T142">REAL_2:142</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_1_4"/>
<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"><span class="p4">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/><b>then </b><span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</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"><span class="p5">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i>, <a class="ref" href="euclid.html#T52">EUCLID:52</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T52">EUCLID:52</a></i>
;<br/><b>hence </b><a NAME="E3:5_1_4"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="euclid.html#T38">EUCLID:38</a></i>;<br/></div><b>end;</b></div>
<a NAME="E25:5_1"/><b>then </b>
not <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"><span class="p4">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E26:5_1"/><b>then </b><i><font color="Green">E85</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"><span class="p4">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> 2 ;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x'</font>, <font color="Maroon">b</font> = <font color="Maroon">y'</font> as    <a href="subset_1.html#M1">Element</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">a</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">b</font>, <font color="Maroon">v</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">b</font>, <font color="Maroon">w</font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">a</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> 2 ;<br/>
<a NAME="E30:5_1"/><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font> <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"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="euclid.html#D11">EUCLID:def 11</a></i>;<br/>
<a NAME="E31:5_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">y'</font>
 
<b>by </b><i><a class="ref" href="euclid.html#D11">EUCLID:def 11</a></i>;<br/>
<a NAME="E32:5_1"/><b>then </b><i><font color="Green">E95</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"><span class="p4">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="euclid.html#K7">+</a> <font color="Maroon">v</font>
 
<b>by </b><i>, <a class="ref" href="euclid.html#D10">EUCLID:def 10</a></i>;<br/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> = 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="euclid.html#K8">-</a> <font color="Maroon">z'</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="sppol_1.html#T20">SPPOL_1:20</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K8">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="rvsum_1.html#K8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T60">RVSUM_1:60</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</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="rvsum_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span> <a href="rvsum_1.html#K8">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="rvsum_1.html#K8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T74">RVSUM_1:74</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</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="rvsum_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K6">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T52">RVSUM_1:52</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</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="rvsum_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rvsum_1.html#K10">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T76">RVSUM_1:76</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</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="rvsum_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="rvsum_1.html#K8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T71">RVSUM_1:71</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</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="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span> <a href="rvsum_1.html#K8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T72">RVSUM_1:72</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</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"><a href="square_1.html#K1">min</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T52">RVSUM_1:52</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</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"><a href="square_1.html#K1">min</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T76">RVSUM_1:76</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</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"><a href="square_1.html#K1">min</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T71">RVSUM_1:71</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</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"><a href="square_1.html#K1">min</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T71">RVSUM_1:71</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K4">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T73">RVSUM_1:73</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K4">+</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K6">-</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T76">RVSUM_1:76</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">b</font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T52">RVSUM_1:52</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"><a href="square_1.html#K1">min</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">u</font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T14">EUCLID:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">b</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="sppol_1.html#T20">SPPOL_1:20</a></i>
<br/>.= 
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
;<br/>
<a NAME="E34:5_1"/><b>then </b><i><font color="Green">E96</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<a NAME="E35:5_1"/>
<font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2 <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="E36:5_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E37:5_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E38:5_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="tarski.html#K2">}</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:5"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="tarski.html#K2">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="tops_1.html#T39">TOPS_1:39</a></i>;<br/>


</div>
