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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:9"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="convex2.html#NM2">Convex_Combination</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="convex3.html#K1" target="_self">ConvexComb</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="convex1.html#K3">conv</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:9_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="convex2.html#NM2">Convex_Combination</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="convex3.html#K1" target="_self">ConvexComb</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="convex2.html#NM2">Convex_Combination</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:9_1"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="convex3.html#K1" target="_self">ConvexComb</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>
<a NAME="E4:9_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="convex1.html#K3">conv</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="convex2.html#T8">CONVEX2:8</a></i>;<br/>
<b>hence </b><a NAME="E5:9_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="convex1.html#K3">conv</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="convex3.html#E3:9_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="convex2.html#T6">CONVEX2:6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:9"/><b>then </b><i><font color="Green">E6</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="convex2.html#NM2">Convex_Combination</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="convex3.html#K1" target="_self">ConvexComb</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="convex1.html#K3">conv</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:9"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="convex3.html#E4:9"><i><font color="Green">E7</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="convex2.html#NM1">Convex_Combination</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:9"/><i><font color="Green">E8</font></i>: 
(  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> is   <a href="convex2.html#NM2">Convex_Combination</a> of <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="convex3.html#T1" target="_self">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="convex2.html#NM2">Convex_Combination</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="convex3.html#E4:9"><i><font color="Green">E7</font></i></a>, <a class="txt" href="convex3.html#E7:9"><i><font color="Green">E8</font></i></a></i>;<br/>
<a NAME="E9:9"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="convex3.html#K1" target="_self">ConvexComb</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="convex3.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E10:9"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="convex2.html#NM2">Convex_Combination</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="convex3.html#K1" target="_self">ConvexComb</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="convex3.html#E7:9"><i><font color="Green">E8</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="convex2.html#NM2">Convex_Combination</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="convex3.html#K1" target="_self">ConvexComb</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E12:9"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:9_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="convex2.html#NM2">Convex_Combination</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:9_2"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="convex3.html#K1" target="_self">ConvexComb</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>
<b>thus </b><a NAME="E4:9_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="convex3.html#E3:9_2"><i><font color="Green">E9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E14:9"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="real_1.html#NM1">Real</a> st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></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="Olive">b<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></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">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:9_3"/><i><font color="Green">E9</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="convex2.html#NM2">Convex_Combination</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:9_3"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="convex3.html#K1" target="_self">ConvexComb</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="convex2.html#NM2">Convex_Combination</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:9_3"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="convex3.html#K1" target="_self">ConvexComb</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="convex3.html#E1:9_3"><i><font color="Green">E9</font></i></a></i>;<br/>
<a NAME="E6:9_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> is   <a href="convex2.html#NM1">Convex_Combination</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="convex3.html#E1:9_3"><i><font color="Green">E9</font></i></a>, <a class="ref" href="convex2.html#T9">CONVEX2:9</a></i>;<br/>
<a NAME="E7:9_3"/><b>then </b><i><font color="Green">E12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="convex3.html#K1" target="_self">ConvexComb</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="convex3.html#D1" target="_self">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="rlvect_2.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> as   <a href="convex2.html#NM2">Convex_Combination</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="convex3.html#E1:9_3"><i><font color="Green">E9</font></i></a>, <a class="ref" href="convex2.html#T10">CONVEX2:10</a></i>;<br/>
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_3.html#T1">RLVECT_3:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_3.html#T2">RLVECT_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_3.html#T2">RLVECT_3:2</a></i>
;<br/>
<b>hence </b><a NAME="E10:9_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></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">c<sub>11</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="convex3.html#E3:9_3"><i><font color="Green">E10</font></i></a>, <a class="txt" href="convex3.html#E5:9_3"><i><font color="Green">E11</font></i></a>, <a class="txt" href="convex3.html#E7:9_3"><i><font color="Green">E12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:9"/><b>then </b><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="convex1.html#V1">convex</a>
 
<b>by </b><i><a class="ref" href="convex1.html#D2">CONVEX1:def 2</a></i>;<br/>
<a NAME="E16:9"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></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">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:9_4"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="convex2.html#NM1">Convex_Combination</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:9_4"/><i><font color="Green">E11</font></i>: 
(  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>11</sub></font> is   <a href="convex2.html#NM2">Convex_Combination</a> of <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="convex3.html#T1" target="_self">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="convex2.html#NM2">Convex_Combination</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="convex3.html#E1:9_4"><i><font color="Green">E10</font></i></a>, <a class="txt" href="convex3.html#E4:9_4"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E6:9_4"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="convex3.html#K1" target="_self">ConvexComb</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="convex3.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E7:9_4"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="convex3.html#E4:9_4"><i><font color="Green">E11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:9"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E18:9"/><b>then </b>
 <a href="convex1.html#K3">conv</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="convex3.html#E15:9"><i><font color="Green">E9</font></i></a>, <a class="ref" href="convex1.html#T30">CONVEX1:30</a></i>;<br/>
<b>hence </b><a NAME="E19:9"/>
 <a href="convex1.html#K3">conv</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="convex2.html#NM2">Convex_Combination</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="convex3.html#K1" target="_self">ConvexComb</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="convex3.html#E2:9"><i><font color="Green">E6</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
