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<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>;<br/>



<a NAME="E1:126"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> 0 <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>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being  <a href="xxreal_0.html#V2">positive</a> <a href="real_1.html#NM1">Real</a>, <font color="Maroon">V</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:126"/><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> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="real_1.html#K5">-</a> 0</span>)</span> <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>by </b><i/>;<br/>
<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> <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/>
<a NAME="E4:126"/>
<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>  <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="126_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="E1:126_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> <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/>

<a NAME="E2:126_1"/><b>then </b>
 ex <font color="Olive">a</font> being  <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> <font color="Maroon">V</font> &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="E3:126_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> <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>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<b>take </b><font color="Maroon">W</font> =  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="126_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:126_2"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E2:126_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#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/><b>hence </b><a NAME="E3:126_2"/>
<font color="Maroon">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>by </b><i><a class="ref" href="rltopsp1.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:126"/>
( <font color="Maroon">W</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> &amp; <font color="Maroon">W</font> is <a href="rltopsp1.html#V3" target="_self">circled</a> &amp; <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> )
 <b>by </b><i/>;<br/>


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