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<b>let </b><font color="Maroon">V</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="rlvect_1.html#L2">RLSStruct</a> ;<br/><b>let </b><font color="Maroon">M</font> be  <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<a NAME="E1:21"/>
for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>

<b>assume </b><a NAME="E1:21_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font>
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:21_1"/><i><font color="Green">E30</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">v</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>
<b>thus </b><a NAME="E4:21_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:21"/><b>then </b>
<font color="Maroon">r</font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>
<b>hence </b><a NAME="E3:21"/>
<font color="Maroon">r</font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>


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