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

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="rltopsp1.html#NM1" title="RLTOPSP1:NM.1">LinearTopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">r</font> be  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<a NAME="E1:97"/><i><font color="Green" title="E60">A1</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="c3">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><a NAME="E1:97_1"/><i><font color="Green" title="E61">A2</font></i>: 
<font color="Maroon" title="c3">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>] <b>means </b>( $1 is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c2">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> $1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font> );<br/><b>consider </b><font color="Maroon" title="c4">F</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:97_1"/><i><font color="Green" title="E62">A3</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3" title="SUBSET_1:sch.3">SUBSET_1:sch 3</a>();<br/></i><b>reconsider </b><font color="Maroon" title="c5">F</font> = <font color="Maroon" title="c4">F</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E5:97_1"/><i><font color="Green" title="E63">A4</font></i>: 
<font color="Maroon" title="c5">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">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" title="c6">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:97_1_1"/>
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font>
 ;<br/>

<b>hence </b><a NAME="E2:97_1_1"/>
<font color="Maroon" title="c6">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E3:97_1"><i><font color="Green" title="E62">A3</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:97_1"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K4" title="RLTOPSP1:func.4">mlt</a> <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">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" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K4" title="RLTOPSP1:func.4">mlt</a> <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">P</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">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" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:97_1_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:97_1_2_1"/><i><font color="Green" title="E64">A5</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 <b>and </b><br/><a NAME="E4:97_1_2_1"/><i><font color="Green" title="E65">A6</font></i>: 
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">A</font> = <font color="Maroon" title="c7">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="rltopsp1.html#E4:97_1_2_1"><i><font color="Green" title="E65">A6</font></i></a></i>;<br/>
<a NAME="E6:97_1_2_1"/><i><font color="Green" title="E66">A7</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c8">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E3:97_1"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="rltopsp1.html#E4:97_1_2_1"><i><font color="Green" title="E65">A6</font></i></a></i>;<br/>
<a NAME="E7:97_1_2_1"/>
<font color="Maroon" title="c8">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c6">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="rltopsp1.html#E3:97_1_2_1"><i><font color="Green" title="E64">A5</font></i></a></i>;<br/>
<a NAME="E9:97_1_2_1"/><i><font color="Green" title="E67">A8</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c8">A</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E3:97_1_2_1"><i><font color="Green" title="E64">A5</font></i></a></i>;<br/>
<a NAME="E10:97_1_2_1"/>
<span class="p1">(<span class="default"><a href="rltopsp1.html#K4" title="RLTOPSP1:func.4">mlt</a> <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c9">x</font>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#D13" target="_self" title="RLTOPSP1:def.13">Def13</a></i>;<br/>
<b>hence </b><a NAME="E11:97_1_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K4" title="RLTOPSP1:func.4">mlt</a> <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E6:97_1_2_1"><i><font color="Green" title="E66">A7</font></i></a>, <a class="txt" href="rltopsp1.html#E9:97_1_2_1"><i><font color="Green" title="E67">A8</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:97_1_2"/><i><font color="Green" title="E64">A9</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K4" title="RLTOPSP1:func.4">mlt</a> <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">P</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E4:97_1_2"/>
<span class="p1">(<span class="default"><a href="rltopsp1.html#K4" title="RLTOPSP1:func.4">mlt</a> <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E2:97_1_2"><i><font color="Green" title="E64">A9</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/>
<a NAME="E5:97_1_2"/><b>then </b>
<font color="Maroon" title="c2">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#D13" target="_self" title="RLTOPSP1:def.13">Def13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">e</font> being  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a> <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>, <font color="Maroon" title="c9">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E7:97_1_2"/><i><font color="Green" title="E65">A10</font></i>: 
<font color="Maroon" title="c9">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E8:97_1_2"/><i><font color="Green" title="E66">A11</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">W</font>
 <b>and </b><br/><a NAME="E9:97_1_2"/><i><font color="Green" title="E67">A12</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">e</font> holds <br/><font color="Olive" title="b1">s</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c9">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:97_1"><i><font color="Green" title="E61">A2</font></i></a>, <a class="ref" href="rltopsp1.html#D9" target="_self" title="RLTOPSP1:def.9">Def9</a></i>;<br/>
<a NAME="E10:97_1_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">e</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>;<br/>
<a NAME="E11:97_1_2"/><b>then </b>
<font color="Maroon" title="c2">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c9">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E9:97_1_2"><i><font color="Green" title="E67">A12</font></i></a></i>;<br/>
<a NAME="E12:97_1_2"/><b>then </b>
<font color="Maroon" title="c9">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E3:97_1"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="rltopsp1.html#E7:97_1_2"><i><font color="Green" title="E65">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:97_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E8:97_1_2"><i><font color="Green" title="E66">A11</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">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" title="RLTOPSP1:func.4">mlt</a> <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E5:97_1"><i><font color="Green" title="E63">A4</font></i></a>, <a class="ref" href="tops_2.html#T26" title="TOPS_2:th.26">TOPS_2:26</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:97"/>
 <a href="rltopsp1.html#K4" title="RLTOPSP1:func.4">mlt</a> <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c1">X</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:97"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>


</div>
