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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a> <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a> <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> ;<br/><b>let </b><font color="Maroon" title="c2">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c2">P</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b1">X</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> );<br/>
<a NAME="E2:31"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<a NAME="E1:31_1"/>
 <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>
 
;<br/>
<b>hence </b><a NAME="E2:31_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:31"/><i><font color="Green" title="E25">A3</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:31_2"/><i><font color="Green" title="E26">A4</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font>
 <b>and </b><br/><a NAME="E2:31_2"/>
<font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font>
 <b>and </b><br/><a NAME="E3:31_2"/><i><font color="Green" title="E27">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">B</font>]
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E5:31_2"/><i><font color="Green" title="E28">A6</font></i>: 
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font>
 <b>and </b><br/><a NAME="E6:31_2"/><i><font color="Green" title="E29">A7</font></i>: 
<font color="Maroon" title="c5">X</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>
 <b>by </b><i><a class="txt" href="tbsp_1.html#E3:31_2"><i><font color="Green" title="E27">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c3">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="tbsp_1.html#E1:31_2"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">W</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E9:31_2"/><i><font color="Green" title="E30">A8</font></i>: 
<font color="Maroon" title="c7">W</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>
 
<b>by </b><i><a class="ref" href="tbsp_1.html#T22" target="_self" title="TBSP_1:th.22">Th22</a></i>;<br/>
<a NAME="E10:31_2"/>
 ex <font color="Olive" title="b1">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Olive" title="b1">Y</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c5">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">W</font>
;<br/>

<b>thus </b><a NAME="E1:31_2_1"/>
( <font color="Maroon" title="c5">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c5">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">W</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> )
 <b>by </b><i><a class="txt" href="tbsp_1.html#E5:31_2"><i><font color="Green" title="E28">A6</font></i></a>, <a class="txt" href="tbsp_1.html#E6:31_2"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="tbsp_1.html#E9:31_2"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="tbsp_1.html#T20" target="_self" title="TBSP_1:th.20">Th20</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:31_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:31"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">P</font>]
 
<b>from </b><i><a class="ref" href="finset_1.html#S2" title="FINSET_1:sch.2">FINSET_1:sch 2</a>(<a class="txt" href="tbsp_1.html#E1:31"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="tbsp_1.html#E2:31"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="tbsp_1.html#E3:31"><i><font color="Green" title="E25">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:31"/>
<font color="Maroon" title="c2">P</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>
 ;<br/>


</div>
