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<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/>



<b>assume </b><a NAME="E1:6"/><i><font color="Green">E4</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> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:6"/>
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>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</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>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</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>5</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>set </b><font color="Maroon">c<sub>6</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> ;<br/>
<b>assume </b><a NAME="E1:6_1"/><i><font color="Green">E5</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>7</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="E3:6_1"/><i><font color="Green">E6</font></i>: 
(  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</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>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>7</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>8</sub></font> = <font color="Maroon">c<sub>7</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#E1:6_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="convex3.html#E3:6_1"><i><font color="Green">E6</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</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="E6:6_1"/><i><font color="Green">E7</font></i>: 
(  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>9</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>9</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>10</sub></font> = <font color="Maroon">c<sub>9</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#E1:6_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="convex3.html#E6:6_1"><i><font color="Green">E7</font></i></a></i>;<br/>
<i><font color="Green">E8</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</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>5</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<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>5</sub></font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">c<sub>8</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>5</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>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="convex3.html#E3:6_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="convex3.html#E6:6_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="rlvect_3.html#T2">RLVECT_3:2</a></i>
<br/>.= 
<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>5</sub></font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">c<sub>8</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>5</sub></font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_3.html#T2">RLVECT_3:2</a></i>
<br/>.= 
 <a href="rlvect_2.html#K6">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_2.html#K7">+</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>5</sub></font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_3.html#T1">RLVECT_3:1</a></i>
;<br/>
<a NAME="E9:6_1"/><i><font color="Green">E9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">c<sub>8</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>5</sub></font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> is   <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#E1:6_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="convex2.html#T10">CONVEX2:10</a></i>;<br/>
<a NAME="E10:6_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">c<sub>8</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>5</sub></font></span>)</span> <a href="rlvect_2.html#K8">*</a> <font color="Maroon">c<sub>10</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/>
<a NAME="E11:6_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</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>5</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <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#E8:6_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="convex3.html#E9:6_1"><i><font color="Green">E9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:6_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</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>5</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="convex3.html#E1:6"><i><font color="Green">E4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6"/>
<font color="Maroon">c<sub>2</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/>


</div>
