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

<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">M</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>;<br/>



<a NAME="E1:9"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b2">L</font></span>)</span> where <font color="Olive" title="b2">L</font> is   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b2">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c1">V</font> </span>}</span>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:9_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b1">L</font></span>)</span> where <font color="Olive" title="b1">L</font> is   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c1">V</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">L</font> being   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E3:9_1"/><i><font color="Green" title="E6">A1</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c4">L</font> &amp; <font color="Maroon" title="c4">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c1">V</font> )
 ;<br/>
<a NAME="E4:9_1"/>
<font color="Maroon" title="c2">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="convex2.html#T8" title="CONVEX2:th.8">CONVEX2:8</a></i>;<br/>
<b>hence </b><a NAME="E5:9_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="txt" href="convex3.html#E3:9_1"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="convex2.html#T6" title="CONVEX2:th.6">CONVEX2:6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:9"/><b>then </b><i><font color="Green" title="E6">A2</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b1">L</font></span>)</span> where <font color="Olive" title="b1">L</font> is   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c1">V</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:9"/><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">m</font> = <font color="Maroon" title="c3">m</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="convex3.html#E4:9"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">LL</font> being   <a href="convex2.html#NM1" title="CONVEX2:NM.1">Convex_Combination</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E7:9"/><i><font color="Green" title="E8">A4</font></i>: 
(  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">LL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> &amp; ( for <font color="Olive" title="b1">A</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>  st <font color="Maroon" title="c4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Maroon" title="c5">LL</font> is   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Olive" title="b1">A</font> ) )
 <b>by </b><i><a class="ref" href="convex3.html#T1" target="_self" title="CONVEX3:th.1">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">LL</font> = <font color="Maroon" title="c5">LL</font> as   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Maroon" title="c2">M</font> <b>by </b><i><a class="txt" href="convex3.html#E4:9"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="convex3.html#E7:9"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/>
<a NAME="E9:9"/>
<font color="Maroon" title="c6">LL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="convex3.html#D1" target="_self" title="CONVEX3:def.1">Def1</a></i>;<br/>
<a NAME="E10:9"/><b>then </b>
<font color="Maroon" title="c4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b1">L</font></span>)</span> where <font color="Olive" title="b1">L</font> is   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c1">V</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="convex3.html#E7:9"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">N</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b1">L</font></span>)</span> where <font color="Olive" title="b1">L</font> is   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c1">V</font> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E12:9"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:9_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">N</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">L</font> being   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E3:9_2"/><i><font color="Green" title="E9">A5</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c9">L</font> &amp; <font color="Maroon" title="c9">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c1">V</font> )
 ;<br/>
<b>thus </b><a NAME="E4:9_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="convex3.html#E3:9_2"><i><font color="Green" title="E9">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">N</font> = <font color="Maroon" title="c7">N</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E14:9"/>
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b3">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b3">r</font> &amp; <font color="Olive" title="b3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 &amp; <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b3">r</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">u</font>, <font color="Maroon" title="c10">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c11">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>assume </b><a NAME="E1:9_3"/><i><font color="Green" title="E9">A6</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">r</font> &amp; <font color="Maroon" title="c11">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 &amp; <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> &amp; <font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">Lu</font> being   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E3:9_3"/><i><font color="Green" title="E10">A7</font></i>: 
( <font color="Maroon" title="c9">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c12">Lu</font> &amp; <font color="Maroon" title="c12">Lu</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c1">V</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c13">Lv</font> being   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E5:9_3"/><i><font color="Green" title="E11">A8</font></i>: 
( <font color="Maroon" title="c10">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c13">Lv</font> &amp; <font color="Maroon" title="c13">Lv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c1">V</font> )
 <b>by </b><i><a class="txt" href="convex3.html#E1:9_3"><i><font color="Green" title="E9">A6</font></i></a></i>;<br/>
<a NAME="E6:9_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c12">Lu</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c13">Lv</font></span>)</span> is   <a href="convex2.html#NM1" title="CONVEX2:NM.1">Convex_Combination</a> of <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="txt" href="convex3.html#E1:9_3"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="convex2.html#T9" title="CONVEX2:th.9">CONVEX2:9</a></i>;<br/>
<a NAME="E7:9_3"/><b>then </b><i><font color="Green" title="E12">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c12">Lu</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c13">Lv</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="convex3.html#D1" target="_self" title="CONVEX3:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">LL</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c12">Lu</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c13">Lv</font></span>)</span> as   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Maroon" title="c2">M</font> <b>by </b><i><a class="txt" href="convex3.html#E1:9_3"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="convex2.html#T10" title="CONVEX2:th.10">CONVEX2:10</a></i>;<br/>
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c14">LL</font> = 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c12">Lu</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c13">Lv</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_3.html#T1" title="RLVECT_3:th.1">RLVECT_3:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c12">Lu</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c13">Lv</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_3.html#T2" title="RLVECT_3:th.2">RLVECT_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c12">Lu</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c13">Lv</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_3.html#T2" title="RLVECT_3:th.2">RLVECT_3:2</a></i>
;<br/>
<b>hence </b><a NAME="E10:9_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c9">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c10">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font>
 <b>by </b><i><a class="txt" href="convex3.html#E3:9_3"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="convex3.html#E5:9_3"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="convex3.html#E7:9_3"><i><font color="Green" title="E12">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:9"/><b>then </b><i><font color="Green" title="E9">A10</font></i>: 
<font color="Maroon" title="c8">N</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>
 
<b>by </b><i><a class="ref" href="convex1.html#D2" title="CONVEX1:def.2">CONVEX1:def 2</a></i>;<br/>
<a NAME="E16:9"/>
for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> holds <br/><font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:9_4"/><i><font color="Green" title="E10">A11</font></i>: 
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">v</font> = <font color="Maroon" title="c9">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<b>consider </b><font color="Maroon" title="c11">LL</font> being   <a href="convex2.html#NM1" title="CONVEX2:NM.1">Convex_Combination</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E4:9_4"/><i><font color="Green" title="E11">A12</font></i>: 
(  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c11">LL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v</font> &amp; ( for <font color="Olive" title="b1">A</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>  st <font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Maroon" title="c11">LL</font> is   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Olive" title="b1">A</font> ) )
 <b>by </b><i><a class="ref" href="convex3.html#T1" target="_self" title="CONVEX3:th.1">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">LL</font> = <font color="Maroon" title="c11">LL</font> as   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Maroon" title="c2">M</font> <b>by </b><i><a class="txt" href="convex3.html#E1:9_4"><i><font color="Green" title="E10">A11</font></i></a>, <a class="txt" href="convex3.html#E4:9_4"><i><font color="Green" title="E11">A12</font></i></a></i>;<br/>
<a NAME="E6:9_4"/>
<font color="Maroon" title="c12">LL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="convex3.html#D1" target="_self" title="CONVEX3:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E7:9_4"/>
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font>
 <b>by </b><i><a class="txt" href="convex3.html#E4:9_4"><i><font color="Green" title="E11">A12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:9"/><b>then </b>
<font color="Maroon" title="c2">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">N</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E18:9"/><b>then </b>
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">N</font>
 
<b>by </b><i><a class="txt" href="convex3.html#E15:9"><i><font color="Green" title="E9">A10</font></i></a>, <a class="ref" href="convex1.html#T30" title="CONVEX1:th.30">CONVEX1:30</a></i>;<br/>
<b>hence </b><a NAME="E19:9"/>
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b1">L</font></span>)</span> where <font color="Olive" title="b1">L</font> is   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c1">V</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="convex3.html#E2:9"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
