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

<b>let </b><font color="Maroon">PM</font> be    <a href="metric_1.html#L1">MetrStruct</a> ;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font>;<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>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>assume </b><a NAME="E2:32"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">PM</font> is <a href="metric_1.html#V9">triangle</a>
 ;<br/>

<b>assume </b><a NAME="E3:32"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">y</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>
 ;<br/>

<a NAME="E4:32"/><b>then </b><i><font color="Green">E28</font></i>: 
not <font color="Maroon">PM</font> is <a href="struct_0.html#V3">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E5:32"/><i><font color="Green">E32</font></i>: 
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="pcomps_1.html#E2:32"><i><font color="Green">E26</font></i></a>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<b>thus </b><a NAME="E6:32"/>
 ex <font color="Olive">p</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">y</font>,<font color="Olive">p</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">p</font> = <font color="Maroon">r'</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>;<br/>
<a NAME="E1:32_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><font color="Maroon">r'</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> holds <br/><font color="Olive">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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font>;<br/>

<b>assume </b><a NAME="E1:32_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><font color="Maroon">r'</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:32_1_1"/><b>then </b>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r'</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E3:32_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E4:32_1_1"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T1" target="_self">Th1</a>, <a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>
<a NAME="E5:32_1_1"/><b>then </b>
 <a href="metric_1.html#K2">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/>
<b>hence </b><a NAME="E6:32_1_1"/>
<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="txt" href="pcomps_1.html#E3:32"><i><font color="Green">E27</font></i></a>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">r'</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
;<br/>

<b>thus </b><a NAME="E2:32_1"/>
<font color="Maroon">r'</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="pcomps_1.html#E4:32"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>

<b>thus </b><a NAME="E3:32_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><font color="Maroon">r'</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</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="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>


</div><b>end;</b></div>


</div>
