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





<b>assume </b><a NAME="E1:5"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">M</font> is <a href="rltopsp1.html#V3">circled</a>
 ;<br/>

<a NAME="E2:5"/>
for <font color="Olive">u</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><br/> for <font color="Olive">p</font> being  <a href="real_1.html#NM1">Real</a>  st  <a href="complex1.html#K18">abs</a> <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M</font> holds <br/><font color="Olive">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:5_1"/><i><font color="Green">E37</font></i>: 
(  <a href="complex1.html#K18">abs</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M</font> )
 ;<br/>

<a NAME="E2:5_1"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : <font color="Olive">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="convex1.html#D1">CONVEX1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">u'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:5_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u'</font> &amp; <font color="Maroon">u'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 ;<br/>
<i><font color="Green">E40</font></i>: <font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font> = 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u'</font>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u'</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/>
<a NAME="E6:5_1"/>
<font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E7:5_1"/>
<font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="rltopsp1.html#T1">RLTOPSP1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:5"/>
<font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M</font> is <a href="rltopsp1.html#V3">circled</a>
 <b>by </b><i/>;<br/>


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