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





<a NAME="E1:14"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">M</font> is <a href="convex1.html#V1" target="_self">convex</a> implies <font color="Maroon">v</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">M</font> is <a href="convex1.html#V1" target="_self">convex</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">M</font> is <a href="convex1.html#V1" target="_self">convex</a>
 ;<br/>

<b>let </b><font color="Maroon">w1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">w2</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="convex1.html#D2" target="_self">CONVEX1:def 2</a><br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:14_1"/><i><font color="Green">E35</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>and </b><br/><a NAME="E3:14_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">M</font> &amp; <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">M</font> )
 ;<br/>

<a NAME="E4:14_1"/>
<font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is   <a href="rlvect_1.html#NM1">VECTOR</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="rusub_4.html#D9">RUSUB_4:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon">x1</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E6:14_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 ;<br/>
<a NAME="E7:14_1"/>
<font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is   <a href="rlvect_1.html#NM1">VECTOR</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="rusub_4.html#D9">RUSUB_4:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon">x2</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E9:14_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">w2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 ;<br/>
<a NAME="E10:14_1"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i>, , , , </i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w2</font></span>)</span> = 
<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="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<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="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</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="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
;<br/>
<a NAME="E12:14_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w2</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is   <a href="rlvect_1.html#NM1">VECTOR</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/>;<br/>
<b>hence </b><a NAME="E13:14_1"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w2</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="rusub_4.html#D9">RUSUB_4:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:14"/>
( <font color="Maroon">v</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">M</font> is <a href="convex1.html#V1" target="_self">convex</a> implies <font color="Maroon">M</font> is <a href="convex1.html#V1" target="_self">convex</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">v</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">M</font> is <a href="convex1.html#V1" target="_self">convex</a>
 ;<br/>

<b>let </b><font color="Maroon">w1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">w2</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="convex1.html#D2" target="_self">CONVEX1:def 2</a><br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:14_2"/><i><font color="Green">E52</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>and </b><br/><a NAME="E3:14_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 ;<br/>

<b>set </b><font color="Maroon">x1</font> = <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w1</font>;<br/>
<b>set </b><font color="Maroon">x2</font> = <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w2</font>;<br/>
<a NAME="E4:14_2"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is   <a href="rlvect_1.html#NM1">VECTOR</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/>;<br/>
<a NAME="E5:14_2"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="rusub_4.html#D9">RUSUB_4:def 9</a></i>;<br/>
<a NAME="E6:14_2"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is   <a href="rlvect_1.html#NM1">VECTOR</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/>;<br/>
<a NAME="E7:14_2"/><b>then </b>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="rusub_4.html#D9">RUSUB_4:def 9</a></i>;<br/>
<a NAME="E8:14_2"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w2</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">M</font>
 
<b>by </b><i>, , , </i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w2</font></span>)</span></span>)</span> = 
<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="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<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="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</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="Maroon">w1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
;<br/>
<a NAME="E10:14_2"/><b>then </b>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</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="Maroon">w1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is   <a href="rlvect_1.html#NM1">VECTOR</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="rusub_4.html#D9">RUSUB_4:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon">w</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E12:14_2"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</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="Maroon">w1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 ;<br/>
<b>thus </b><a NAME="E13:14_2"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w2</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#T21">RLVECT_1:21</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:14"/>
( <font color="Maroon">M</font> is <a href="convex1.html#V1" target="_self">convex</a> iff <font color="Maroon">v</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">M</font> is <a href="convex1.html#V1" target="_self">convex</a> )
 <b>by </b><i/>;<br/>


</div>
