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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:77"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">A</font> is <a href="compts_1.html#V6">compact</a>
 ;<br/>

<a NAME="E2:77"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">A2</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:77_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_1"><b>suppose </b></a><a NAME="E1:77_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:77_1_1"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E3:77_1_1"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i>, <a class="ref" href="compts_1.html#T11">COMPTS_1:11</a></i>;<br/><a NAME="E4:77_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font> is <a href="compts_1.html#V2">compact</a>
 <b>by </b><i><a class="ref" href="compts_1.html#T10">COMPTS_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">A</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i>, <a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/><a NAME="E6:77_1_1"/>
 <a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <a href="topmetr.html#K1">|</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="topmetr.html#T20">TOPMETR:20</a></i>;<br/><a NAME="E7:77_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <a href="topmetr.html#K1">|</a> <font color="Maroon">A1</font> is <a href="tbsp_1.html#V1">totally_bounded</a>
 <b>by </b><i>, <a class="ref" href="tbsp_1.html#T12">TBSP_1:12</a></i>;<br/><a NAME="E8:77_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <a href="topmetr.html#K1">|</a> <font color="Maroon">A1</font> is <a href="tbsp_1.html#V5">bounded</a>
 <b>by </b><i><a class="ref" href="tbsp_1.html#T26">TBSP_1:26</a></i>;<br/><a NAME="E9:77_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <a href="topmetr.html#K1">|</a> <font color="Maroon">A1</font></span>)</span> is <a href="tbsp_1.html#V6">bounded</a>
 <b>by </b><i><a class="ref" href="tbsp_1.html#T25">TBSP_1:25</a></i>;<br/><a NAME="E10:77_1_1"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <a href="topmetr.html#K1">|</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="topmetr.html#D2">TOPMETR:def 2</a></i>;<br/><a NAME="E11:77_1_1"/><b>then </b>
<font color="Maroon">A1</font> is <a href="tbsp_1.html#V6">bounded</a>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E12:77_1_1"/>
<font color="Maroon">A</font> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2"><b>suppose </b></a><a NAME="E1:77_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:77_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/><b>thus </b><a NAME="E4:77_1_2"/>
<font color="Maroon">A</font> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
