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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">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">V</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font>;<br/>







<a NAME="E1:63"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</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/>
<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">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:63"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E4:63"/><i><font color="Green">E37</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="E5:63"/><i><font color="Green">E38</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">a</font></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">W</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E3:63"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">r</font>
;<br/>

<a NAME="E6:63"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#T3" target="_self">Th3</a>, <a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">W</font> = <font color="Maroon">W</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font> <b>by </b><i>, <a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">W</font>
;<br/>

<b>let </b><font color="Maroon">s</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E8:63"/>
 <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> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E9:63"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">s</font> <a href="convex1.html#K1">*</a> <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:63"/>
 <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/>
<b>hence </b><a NAME="E11:63"/>
<font color="Maroon">s</font> <a href="convex1.html#K1">*</a> <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


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