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

<b>let </b><font color="Maroon">X</font> be   <a href="rltopsp1.html#NM1">LinearTopSpace</a>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:89"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">F</font> is <a href="finset_1.html#V1">finite</a>
 <b>and </b><br/><a NAME="E2:89"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">P</font> where <font color="Olive">P</font> is  <a href="rltopsp1.html#V9" target="_self">bounded</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : verum </span>}</span> 
 ;<br/>

<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><b>now </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">X</font>;<br/><b>assume </b><a NAME="E1:89_1"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E2:89_1"/><b>then </b>
 ex <font color="Olive">A</font> being <a href="rltopsp1.html#V9" target="_self">bounded</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:89_1"/>
<font color="Maroon">P</font> is <a href="rltopsp1.html#V9" target="_self">bounded</a>
 ;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">A</font> is <a href="rltopsp1.html#V9" target="_self">bounded</a> );<br/>
<a NAME="E4:89"/>
(  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font> is <a href="rltopsp1.html#V9" target="_self">bounded</a> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/>
<a NAME="E5:89"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
;<br/>
<a NAME="E6:89"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:89_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E2:89_2"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E3:89_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font>]
 ;<br/>

<b>consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:89_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E6:89_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">A</font> is <a href="rltopsp1.html#V9" target="_self">bounded</a>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="rltopsp1.html#T6" target="_self">Th6</a></i>;<br/>
<i><font color="Green">E60</font></i>:  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">B</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T96">ZFMISC_1:96</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">B</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>
;<br/>
<a NAME="E9:89_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> is <a href="rltopsp1.html#V9" target="_self">bounded</a>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E6:89_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="rltopsp1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E10:89_2"/>
 ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp; <font color="Olive">Y</font> is <a href="rltopsp1.html#V9" target="_self">bounded</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">x</font>
;<br/>

<b>thus </b><a NAME="E1:89_2_2"/>
( <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp; <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">x</font> is <a href="rltopsp1.html#V9" target="_self">bounded</a> )
 <b>by </b><i><a class="txt" href="rltopsp1.html#E8:89_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="rltopsp1.html#T7" target="_self">Th7</a>, <a class="txt" href="rltopsp1.html#E9:89_2"><i><font color="Green">E62</font></i></a>, , <a class="txt" href="rltopsp1.html#E1:89_2"><i><font color="Green">E52</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:89_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:89"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F</font>]
 
<b>from </b><i><a class="ref" href="finset_1.html#S2">FINSET_1:sch 2</a>(, , <a class="ref" href="rltopsp1.html#T5" target="_self">Th5</a>);<br/></i>
<b>hence </b><a NAME="E8:89"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> is <a href="rltopsp1.html#V9" target="_self">bounded</a>
 ;<br/>


</div>
