<?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">U</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>, <font color="Maroon">V</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>;<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>let </b><font color="Maroon">r</font> be  <a href="xxreal_0.html#V2">positive</a> <a href="real_1.html#NM1">Real</a>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:125"/><i><font color="Green">E36</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> <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> holds <br/><font color="Olive">s</font> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font>
 <b>and </b><br/><a NAME="E2:125"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font></span>)</span> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> :  <a href="complex1.html#K18">abs</a> <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> </span>}</span> 
 ;<br/>

<b>set </b><font color="Maroon">W</font> =  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>;<br/>
<b>thus </b><a NAME="E3:125"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">F'</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="convex1.html#K1">*</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span></span>)</span> where <font color="Olive">a</font> is  non <a href="xboole_0.html#V1">zero</a> <a href="real_1.html#NM1">Real</a> :  <a href="complex1.html#K18">abs</a> <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">a</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E2:125_1"/><i><font color="Green">E38</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E3:125_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T7">XREAL_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E5:125_1"/><i><font color="Green">E49</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#T13" target="_self">Th13</a>, <a class="ref" href="rltopsp1.html#T15" target="_self">Th15</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E6:125_1"/><i><font color="Green">E52</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E7:125_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">a</font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="convex1.html#K1">*</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span></span>)</span> where <font color="Olive">a</font> is  non <a href="xboole_0.html#V1">zero</a> <a href="real_1.html#NM1">Real</a> :  <a href="complex1.html#K18">abs</a> <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="rltopsp1.html#T13" target="_self">Th13</a>, </i>;<br/>
<a NAME="E8:125_1"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E9:125_1"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>;<br/>
<a NAME="E10:125_1"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="convex1.html#K1">*</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span></span>)</span> where <font color="Olive">a</font> is  non <a href="xboole_0.html#V1">zero</a> <a href="real_1.html#NM1">Real</a> :  <a href="complex1.html#K18">abs</a> <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E11:125_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="convex1.html#K1">*</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span></span>)</span> where <font color="Olive">a</font> is  non <a href="xboole_0.html#V1">zero</a> <a href="real_1.html#NM1">Real</a> :  <a href="complex1.html#K18">abs</a> <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</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="E1:125_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="convex1.html#K1">*</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span></span>)</span> where <font color="Olive">a</font> is  non <a href="xboole_0.html#V1">zero</a> <a href="real_1.html#NM1">Real</a> :  <a href="complex1.html#K18">abs</a> <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> </span>}</span> 
 ;<br/>

<a NAME="E2:125_1_1"/><b>then </b>
 ex <font color="Olive">a</font> being non <a href="xboole_0.html#V1">zero</a> <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 
;<br/>
<b>hence </b><a NAME="E3:125_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F'</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="convex1.html#K1">*</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span></span>)</span> where <font color="Olive">a</font> is  non <a href="xboole_0.html#V1">zero</a> <a href="real_1.html#NM1">Real</a> :  <a href="complex1.html#K18">abs</a> <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E13:125_1"/>
<font color="Maroon">F'</font> is <a href="tops_2.html#V1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_2"><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">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="E1:125_1_2"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F'</font>
 ;<br/>

<a NAME="E2:125_1_2"/><b>then </b>
 ex <font color="Olive">a</font> being non <a href="xboole_0.html#V1">zero</a> <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 
;<br/>
<b>hence </b><a NAME="E3:125_1_2"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E14:125_1"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F'</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/>
<a NAME="E15:125_1"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F'</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/>
<a NAME="E16:125_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F'</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_3"><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="E1:125_1_3"/>
<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">P</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:125_1_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E4:125_1_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">P</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>consider </b><font color="Maroon">a</font> being  non <a href="xboole_0.html#V1">zero</a> <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:125_1_3"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span>
 <b>and </b><br/><a NAME="E7:125_1_3"/><i><font color="Green">E65</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T19" target="_self">Th19</a></i>;<br/>
<b>consider </b><font color="Maroon">v</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E9:125_1_3"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E10:125_1_3"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E21"><i><font color="Green">Lemma87</font></i></a>, <a class="ref" href="rltopsp1.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E11:125_1_3"/><i><font color="Green">E68</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E12:125_1_3"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">a</font> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#T12" target="_self">Th12</a>, </i>;<br/>
<a NAME="E13:125_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="rltopsp1.html#T21" target="_self">Th21</a>, </i>;<br/>
<b>hence </b><a NAME="E14:125_1_3"/>
<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="rltopsp1.html#D3" target="_self">Def3</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:125_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F'</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T48">TOPS_1:48</a></i>;<br/>
<b>hence </b><a NAME="E18:125_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="rltopsp1.html#D1" target="_self">Def1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a><br/>


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

<b>thus </b><a NAME="E4:125"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> is <a href="rltopsp1.html#V3" target="_self">circled</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be   <a href="real_1.html#NM1">Real</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="rltopsp1.html#D6" target="_self">RLTOPSP1:def 6</a><br/>

<b>assume </b><a NAME="E1:125_2"/><i><font color="Green">E169</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/>

<b>let </b><font color="Maroon">z</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="E2:125_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">s</font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:125_2"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font>
 <b>and </b><br/><a NAME="E5:125_2"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 ;<br/>
<b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:125_2"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E8:125_2"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">Y</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>consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E10:125_2"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E11:125_2"/><i><font color="Green">E179</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T12" target="_self">Th12</a>, </i>;<br/>
<b>consider </b><font color="Maroon">v</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E13:125_2"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E14:125_2"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E15:125_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#T15" target="_self">Th15</a>, <a class="txt" href="rltopsp1.html#E21"><i><font color="Green">Lemma87</font></i></a>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/>
<a NAME="E16:125_2"/><b>then </b><i><font color="Green">E182</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E17:125_2"/><i><font color="Green">E184</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">s</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#T13" target="_self">Th13</a>, <a class="ref" href="real_2.html#T147">REAL_2:147</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:125_2_1"/>
( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">s</font> or  <a href="complex1.html#K18">abs</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_2_1_1"><b>suppose </b></a><a NAME="E1:125_2_1_1"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:125_2_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/><a NAME="E3:125_2_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="complex1.html#T133">COMPLEX1:133</a></i>;<br/><a NAME="E4:125_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">s</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">s</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#D1" target="_self">Def1</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E5:125_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">s</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:125_2_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>;<br/><a NAME="E7:125_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T12" target="_self">Th12</a></i>;<br/><b>hence </b><a NAME="E8:125_2_1_1"/>
<font color="Maroon">z</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="rltopsp1.html#T20" target="_self">Th20</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_2_1_2"><b>suppose </b></a><a NAME="E1:125_2_1_2"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:125_2_1_2"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/><a NAME="E3:125_2_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E4:125_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T12" target="_self">Th12</a></i>;<br/><b>hence </b><a NAME="E5:125_2_1_2"/>
<font color="Maroon">z</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="rltopsp1.html#T20" target="_self">Th20</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>let </b><font color="Maroon">z</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="E5:125"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> ;<br/>
<b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:125"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E9:125"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T13" target="_self">Th13</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E11:125"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E12:125"/><i><font color="Green">E189</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T12" target="_self">Th12</a>, </i>;<br/>
<a NAME="E13:125"/>
<font color="Maroon">a</font> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E14:125"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T15" target="_self">Th15</a>, </i>;<br/>


</div>
