<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">M</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>



<a NAME="E1:17"/><i><font color="Green">E29</font></i>: 
 <a href="rusub_4.html#K3">Up</a> <span class="p1">(<span class="default"><a href="rlsub_1.html#K2">(Omega).</a> <font color="Maroon">V</font></span>)</span> is <a href="convex1.html#V1">convex</a>
 
<b>by </b><i><a class="ref" href="convex1.html#T9">CONVEX1:9</a></i>;<br/>
<a NAME="E2:17"/>
<font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="rusub_4.html#K3">Up</a> <span class="p1">(<span class="default"><a href="rlsub_1.html#K2">(Omega).</a> <font color="Maroon">V</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><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="E1:17_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> ;<br/>
<a NAME="E3:17_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> #)
 
;<br/>
<a NAME="E4:17_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="rlsub_1.html#K2">(Omega).</a> <font color="Maroon">V</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#D4">RLSUB_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:17_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="rusub_4.html#K3">Up</a> <span class="p1">(<span class="default"><a href="rlsub_1.html#K2">(Omega).</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="rusub_4.html#D6">RUSUB_4:def 6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:17"/>
 <a href="convex1.html#K2">Convex-Family</a> <font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="convex1.html#D4">CONVEX1:def 4</a></i>;<br/>


</div>
