<?xml version="1.0"?>
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<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> 2</span>)</span>;<br/><b>let </b><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> 2</span>)</span>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">r1</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">s1</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">r2</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">s2</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</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> 2</span>)</span>;<br/>

















<b>assume </b><b>that </b><br/><a NAME="E1:23"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font>
 <b>and </b><br/><a NAME="E2:23"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">s1</font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E3:23"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r2</font>,<font color="Maroon">s2</font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E4:23"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r2</font>,<font color="Maroon">s1</font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E5:23"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">r</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">r1</font>, <font color="Maroon">s1</font> = <font color="Maroon">s1</font>, <font color="Maroon">r2</font> = <font color="Maroon">r2</font>, <font color="Maroon">s2</font> = <font color="Maroon">s2</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/>
<a NAME="E7:23"/>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u6</font> where <font color="Olive">u6</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">u</font>,<font color="Olive">u6</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/>
<a NAME="E8:23"/><b>then </b><i><font color="Green">E65</font></i>: 
 ex <font color="Olive">u5</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> st <br/>( <font color="Olive">u5</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="Olive">u5</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> )
 
;<br/>
<b>reconsider </b><font color="Maroon">p1'</font> = <font color="Maroon">p1</font>, <font color="Maroon">p2'</font> = <font color="Maroon">p2</font>, <font color="Maroon">p'</font> = <font color="Maroon">p</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> 2 <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E10:23"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> 2</span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">p1'</font>,<font color="Maroon">p2'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1'</font> <a href="euclid.html#K8">-</a> <font color="Maroon">p2'</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#D6">EUCLID:def 6</a></i>;<br/>
<a NAME="E11:23"/><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> 2</span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">p1'</font>,<font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1'</font> <a href="euclid.html#K8">-</a> <font color="Maroon">p'</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#D6">EUCLID:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">up</font> = <font color="Maroon">p</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/>;<br/>
<a NAME="E13:23"/><i><font color="Green">E72</font></i>: 
 <a href="euclid.html#K14">Euclid</a> 2 <a href="hidden.html#R1">=</a>  <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> 2</span>)</span> #)
 
<b>by </b><i><a class="ref" href="euclid.html#D7">EUCLID:def 7</a></i>;<br/>
<a NAME="E14:23"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">u</font>,<font color="Maroon">up</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1'</font> <a href="euclid.html#K8">-</a> <font color="Maroon">p'</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i>, , <a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E15:23"/><i><font color="Green">E76</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1'</font> <a href="euclid.html#K8">-</a> <font color="Maroon">p2'</font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , , , <a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E16:23"/><i><font color="Green">E77</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1'</font> <a href="euclid.html#K8">-</a> <font color="Maroon">p'</font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">p1'</font> <a href="euclid.html#K8">-</a> <font color="Maroon">p'</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#D5">EUCLID:def 5</a></i>;<br/>
<a NAME="E17:23"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">p1'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">s1</font></span><a href="finseq_4.html#K2">*&gt;</a></span> &amp; <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">r2</font>,<font color="Maroon">s1</font></span><a href="finseq_4.html#K2">*&gt;</a></span> &amp; <font color="Maroon">p2'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">r2</font>,<font color="Maroon">s2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> )
 
<b>by </b><i>, , , <a class="ref" href="euclid.html#D16">EUCLID:def 16</a></i>;<br/>
<a NAME="E18:23"/>
( <font color="Maroon">p1'</font> <a href="euclid.html#K8">-</a> <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p1'</font> <a href="euclid.html#K8">-</a> <font color="Maroon">p2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font> )
 
<b>by </b><i><a class="ref" href="euclid.html#D13">EUCLID:def 13</a></i>;<br/>
<a NAME="E19:23"/><b>then </b><i><font color="Green">E79</font></i>: 
( <font color="Maroon">p1'</font> <a href="euclid.html#K8">-</a> <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> &amp; <font color="Maroon">p1'</font> <a href="euclid.html#K8">-</a> <font color="Maroon">p2'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> )
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E80</font></i>:  <a href="euclid.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">p1'</font> <a href="euclid.html#K8">-</a> <font color="Maroon">p'</font></span>)</span> = 
<a href="rvsum_1.html#K1">sqrreal</a>  <a href="finseqop.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1'</font> <a href="euclid.html#K8">-</a> <font color="Maroon">p'</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#D8">RVSUM_1:def 8</a></i>
<br/>.= 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="finseq_2.html#T40">FINSEQ_2:40</a></i>
;<br/>
<i><font color="Green">E81</font></i>:  <a href="euclid.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">p1'</font> <a href="euclid.html#K8">-</a> <font color="Maroon">p2'</font></span>)</span> = 
<a href="rvsum_1.html#K1">sqrreal</a>  <a href="finseqop.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1'</font> <a href="euclid.html#K8">-</a> <font color="Maroon">p2'</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#D8">RVSUM_1:def 8</a></i>
<br/>.= 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="finseq_2.html#T40">FINSEQ_2:40</a></i>
;<br/>
<i><font color="Green">E82</font></i>:  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">p1'</font> <a href="euclid.html#K8">-</a> <font color="Maroon">p'</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> 0</span>)</span>
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T107">RVSUM_1:107</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="ref" href="rvsum_1.html#D2">RVSUM_1:def 2</a>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>
<br/>.= 
<a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span>

;<br/>
<a NAME="E23:23"/>
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E24:23"/><b>then </b>
<a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="rvsum_1.html#D2">RVSUM_1:def 2</a></i>;<br/>
<a NAME="E25:23"/><b>then </b><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E26:23"/>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E27:23"/><b>then </b>
<a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="rvsum_1.html#D2">RVSUM_1:def 2</a></i>;<br/>
<a NAME="E28:23"/><b>then </b><i><font color="Green">E100</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1'</font> <a href="euclid.html#K8">-</a> <font color="Maroon">p'</font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="square_1.html#T94">SQUARE_1:94</a></i>;<br/>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1'</font> <a href="euclid.html#K8">-</a> <font color="Maroon">p2'</font></span>)</span></span><a href="euclid.html#K12">.|</a></span> = 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="euclid.html#D5">EUCLID:def 5</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T107">RVSUM_1:107</a></i>
;<br/>
<a NAME="E30:23"/><b>then </b>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1'</font> <a href="euclid.html#K8">-</a> <font color="Maroon">p'</font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&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="E31:23"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u6</font> where <font color="Olive">u6</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">u</font>,<font color="Olive">u6</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E32:23"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/>


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