<?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>assume </b><a NAME="E1:37"/>
<font color="Maroon">T</font> is <a href="tbsp_1.html#V1" target="_self">totally_bounded</a>
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:37"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">Y</font> is <a href="finset_1.html#V1">finite</a>
 <b>and </b><br/><a NAME="E4:37"/><i><font color="Green">E45</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E5:37"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/> ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> st <font color="Olive">P</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,1
 <b>by </b><i/>;<br/>
<a NAME="E6:37"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">B</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E8:37"/>
for <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Olive">P</font> is <a href="tbsp_1.html#V6" target="_self">bounded</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:37_1"/>
for <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Olive">P</font> is <a href="tbsp_1.html#V6" target="_self">bounded</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:37_1_1"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:37_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> st <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,1
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:37_1_1"/>
<font color="Maroon">P</font> is <a href="tbsp_1.html#V6" target="_self">bounded</a>
 <b>by </b><i/>;<br/>


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


</div><b>end;</b></div>
<a NAME="E9:37"/><b>then </b>
<font color="Maroon">B</font> is <a href="tbsp_1.html#V6" target="_self">bounded</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E10:37"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> is <a href="tbsp_1.html#V6" target="_self">bounded</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<b>hence </b><a NAME="E11:37"/>
<font color="Maroon">T</font> is <a href="tbsp_1.html#V5" target="_self">bounded</a>
 <b>by </b><i/>;<br/>


</div>
