<?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">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">Q</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:44"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">P</font> is <a href="tbsp_1.html#V6" target="_self">bounded</a>
 <b>and </b><br/><a NAME="E2:44"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">Q</font> is <a href="tbsp_1.html#V6" target="_self">bounded</a>
 <b>and </b><br/><a NAME="E3:44"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<a NAME="E4:44"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Q</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="E5:44"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font> is <a href="tbsp_1.html#V6" target="_self">bounded</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:44"/>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E7:44"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<b>set </b><font color="Maroon">a</font> =  <a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">P</font>;<br/>
<a NAME="E8:44"/><i><font color="Green">E60</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">P</font>
 
<b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">b</font> =  <a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">Q</font>;<br/>
<a NAME="E9:44"/><i><font color="Green">E62</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">Q</font>
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">g</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>;<br/>
<a NAME="E11:44"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">Q</font></span>)</span>;<br/>
<a NAME="E13:44"/><i><font color="Green">E64</font></i>: 
 <a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">P</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">Q</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="real_2.html#T173">REAL_2:173</a></i>;<br/>
<a NAME="E14:44"/><i><font color="Green">E65</font></i>: 
 <a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">Q</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">Q</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="real_2.html#T173">REAL_2:173</a></i>;<br/>
<a NAME="E15:44"/>
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> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</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> <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">Q</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_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="E1:44_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font> )
 ;<br/>

<a NAME="E2:44_1"/><i><font color="Green">E76</font></i>: 
 <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">g</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">g</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1"><b>suppose </b></a><a NAME="E1:44_1_1_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_1_1_1_1_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:44_1_1_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><a NAME="E2:44_1_1_1_1_1_1_1"/><b>then </b>
 <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>  <a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">P</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:44_1_1_1_1_1_1_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="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">Q</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:44_1_1_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/><div class="add"><a NAME="E2:44_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">g</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">Q</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E3:44_1_1_1_1_1_1_2"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">P</font>
 <b>by </b><i>, , , </i>;<br/><a NAME="E4:44_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">g</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">g</font>,<font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">Q</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E5:44_1_1_1_1_1_1_2"/>
 <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="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">Q</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:44_1_1_1_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="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">Q</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_2"><b>suppose </b></a><a NAME="E1:44_1_1_1_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_1_1_1_2_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:44_1_1_1_2_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><a NAME="E2:44_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">g</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">P</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E3:44_1_1_1_2_1_1_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">Q</font>
 <b>by </b><i>, , , </i>;<br/><a NAME="E4:44_1_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">g</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">g</font>,<font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E5:44_1_1_1_2_1_1_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="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">Q</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:44_1_1_1_2_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/><div class="add"><a NAME="E2:44_1_1_1_2_1_1_2"/><b>then </b>
 <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>  <a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">Q</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:44_1_1_1_2_1_1_2"/>
 <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="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">Q</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:44_1_1_1_2"/>
 <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="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">Q</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:44_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="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">Q</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:44"/>
 <a href="tbsp_1.html#K2" target="_self">diameter</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">Q</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div>
