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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#V6">Reflexive</a> <a href="metric_1.html#V8">symmetric</a> <a href="metric_1.html#V9">triangle</a>  <a href="metric_1.html#L1">MetrStruct</a> ;<br/><b>let </b><font color="Maroon">t1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><a NAME="E1:42"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:42"/><b>then </b><i><font color="Green">E44</font></i>: 
( <font color="Maroon">t1</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">t1</font>,<font color="Maroon">r</font> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">t1</font>,<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E3:42"/><i><font color="Green">E45</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">t1</font>,<font color="Maroon">r</font> is <a href="tbsp_1.html#V6" target="_self">bounded</a>
 
<b>by </b><i/>;<br/>
<a NAME="E4:42"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">t1</font>,<font color="Maroon">r</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">t1</font>,<font color="Maroon">r</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>, <font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E1:42_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">t1</font>,<font color="Maroon">r</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">t1</font>,<font color="Maroon">r</font> )
 ;<br/>

<a NAME="E2:42_1"/><b>then </b>
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">t1</font>,<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">t1</font>,<font color="Maroon">y</font> <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#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E3:42_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">t1</font>,<font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">t1</font>,<font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E4:42_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <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">t1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">t1</font>,<font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>
<b>hence </b><a NAME="E5:42_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:42"/>
 <a href="tbsp_1.html#K2" target="_self">diameter</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">t1</font>,<font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font>
 <b>by </b><i>, , </i>;<br/>


</div>
