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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="rltopsp1.html#L1" target="_self">RLTopStruct</a> ;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<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/>





<b>thus </b><a NAME="E1:93"/>
<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#K10">.:</a> <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">V</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="93_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:93_1"/>
<font color="Maroon">y</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#K10">.:</a> <font color="Maroon">V</font>
 ;<br/>

<b>then consider </b><font color="Maroon">c</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:93_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E4:93_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">y</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="funct_2.html#K3">.</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E5:93_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#T5" target="_self">Th5</a>, </i>;<br/>
<b>hence </b><a NAME="E6:93_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font>
 <b>by </b><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="E2:93"/>
<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">V</font>
 ;<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="E4:93"/><i><font color="Green">E38</font></i>: 
<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">u</font>
 <b>and </b><br/><a NAME="E5:93"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/>
<a NAME="E6:93"/>
<font color="Maroon">x</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="funct_2.html#K3">.</a> <font color="Maroon">u</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:93"/>
<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#K10">.:</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>


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