<?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">r</font> be  non <a href="xboole_0.html#V1">zero</a> <a href="real_1.html#NM1">Real</a>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_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="E1:97_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> );<br/><b>consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:97_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> holds <br/> ( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3">SUBSET_1:sch 3</a>();<br/></i><b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> ;<br/><a NAME="E5:97_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">F</font> is <a href="tops_2.html#V1">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:97_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>hence </b><a NAME="E2:97_1_1"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T7" target="_self">Th7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:97_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K4" target="_self">mlt</a> <font color="Maroon">r</font>,<font color="Maroon">X</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:97_1_2"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K4" target="_self">mlt</a> <font color="Maroon">r</font>,<font color="Maroon">X</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:97_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:97_1_2_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:97_1_2_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E6:97_1_2_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#T7" target="_self">Th7</a>, </i>;<br/>
<a NAME="E7:97_1_2_1"/>
<font color="Maroon">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E9:97_1_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:97_1_2_1"/>
<span class="p1">(<span class="default"><a href="rltopsp1.html#K4" target="_self">mlt</a> <font color="Maroon">r</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E2:97_1_2"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:97_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K4" target="_self">mlt</a> <font color="Maroon">r</font>,<font color="Maroon">X</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>


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

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:97_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K4" target="_self">mlt</a> <font color="Maroon">r</font>,<font color="Maroon">X</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E4:97_1_2"/>
<span class="p1">(<span class="default"><a href="rltopsp1.html#K4" target="_self">mlt</a> <font color="Maroon">r</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<a NAME="E5:97_1_2"/><b>then </b>
<font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E2:97_1_2"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">e</font> being  <a href="xxreal_0.html#V2">positive</a> <a href="real_1.html#NM1">Real</a>, <font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E7:97_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E8:97_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E9:97_1_2"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a>  st  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> holds <br/><font color="Olive">s</font> <a href="convex1.html#K1">*</a> <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E8:97_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="rltopsp1.html#E1:97_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E10:97_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/>
<a NAME="E11:97_1_2"/><b>then </b>
<font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:97_1_2"/><b>then </b>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#T7" target="_self">Th7</a>, </i>;<br/>
<b>hence </b><a NAME="E13:97_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:97_1"/>
<span class="p1">(<span class="default"><a href="rltopsp1.html#K4" target="_self">mlt</a> <font color="Maroon">r</font>,<font color="Maroon">X</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T26">TOPS_2:26</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:97"/>
 <a href="rltopsp1.html#K4" target="_self">mlt</a> <font color="Maroon">r</font>,<font color="Maroon">X</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>


</div>
