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

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:31"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="tbsp_1.html#V6" target="_self">bounded</a> );<br/>
<a NAME="E2:31"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</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="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> is <a href="tbsp_1.html#V6" target="_self">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">{}</a> ]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:31"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K1">}</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">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:31_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:31_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E3:31_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:31_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E6:31_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="tbsp_1.html#V6" target="_self">bounded</a>
 <b>by </b><i><a class="txt" href="tbsp_1.html#E3:31_2"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="tbsp_1.html#E1:31_2"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E9:31_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="tbsp_1.html#V6" target="_self">bounded</a>
 
<b>by </b><i><a class="ref" href="tbsp_1.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E10:31_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="tbsp_1.html#V6" target="_self">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">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E1:31_2_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font> is <a href="tbsp_1.html#V6" target="_self">bounded</a> )
 <b>by </b><i><a class="txt" href="tbsp_1.html#E5:31_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="tbsp_1.html#E6:31_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="tbsp_1.html#E9:31_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="tbsp_1.html#T20" target="_self">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">c<sub>4</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span>]
 ;<br/>


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


</div>
