<?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   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="rlvect_2.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:16"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<a NAME="E2:16"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
;<br/>
<a NAME="E3:16"/><b>then </b>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="card_4.html#T17">CARD_4:17</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:16"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</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>5</sub></font> = <font color="Maroon">c<sub>4</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#E5:16"><i><font color="Green">E10</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="rlvect_2.html#M2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span><b> such that </b><br/><a NAME="E8:16"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>from </b><i><a class="ref" href="rlvect_4.html#S2">RLVECT_4:sch 2</a>();<br/></i>
<a NAME="E9:16"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/>
<a NAME="E10:16"/><b>then </b><i><font color="Green">E12</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="convex3.html#E5:16"><i><font color="Green">E10</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E11:16"/><i><font color="Green">E13</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/>
<a NAME="E12:16"/><b>then </b>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="convex3.html#E10:16"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="rlvect_2.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/>
<a NAME="E14:16"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:16_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:16_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="convex3.html#E11:16"><i><font color="Green">E13</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:16_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="convex3.html#E8:16"><i><font color="Green">E11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 ) );<br/>
<a NAME="E15:16"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font>
 
;<br/>
<a NAME="E16:16"/><i><font color="Green">E16</font></i>: 
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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_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="E1:16_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>
 ;<br/>

<a NAME="E2:16_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
;<br/>
<b>hence </b><a NAME="E3:16_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E18:16"/><i><font color="Green">E17</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; ( 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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(<a class="txt" href="convex3.html#E15:16"><i><font color="Green">E15</font></i></a>, <a class="txt" href="convex3.html#E16:16"><i><font color="Green">E16</font></i></a>);<br/></i>
<a NAME="E19:16"/>
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>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3"><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="E1:16_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:16_3"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_3_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_1"><b>suppose </b></a><a NAME="E1:16_3_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E3:16_3_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="convex3.html#E18:16"><i><font color="Green">E17</font></i></a>, <a class="txt" href="convex3.html#E3:16_3"><i><font color="Green">E18</font></i></a>, <a class="txt" href="convex3.html#E1:16_3_1_1"><i><font color="Green">E19</font></i></a></i>;<br/><b>hence </b><a NAME="E4:16_3_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2"><b>suppose </b></a><a NAME="E1:16_3_1_2"/>
not <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:16_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="convex3.html#E18:16"><i><font color="Green">E17</font></i></a>, <a class="txt" href="convex3.html#E3:16_3"><i><font color="Green">E18</font></i></a></i>;<br/><b>hence </b><a NAME="E3:16_3_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E20:16"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E21:16"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="txt" href="convex3.html#E18:16"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E22:16"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<b>thus </b><a NAME="E2:16_4"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="convex3.html#E18:16"><i><font color="Green">E17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="convex3.html#E21:16"><i><font color="Green">E18</font></i></a>, <a class="ref" href="rlvect_2.html#D5">RLVECT_2:def 5</a></i>;<br/>
<a NAME="E24:16"/>
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>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <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="16_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:16_5"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:16_5"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="convex3.html#E1:16_5"><i><font color="Green">E19</font></i></a>, <a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>;<br/>
<a NAME="E4:16_5"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="convex3.html#E18:16"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E5:16_5"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:16_5"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E7:16_5"/>
<font color="Maroon">c<sub>10</sub></font> <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#E5:16_5"><i><font color="Green">E20</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:16"/><b>then </b>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</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>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="rlvect_2.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/>
<a NAME="E27:16"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_6"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:16_6"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="convex3.html#E8:16"><i><font color="Green">E11</font></i></a>, <a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>;<br/>
<b>hence </b><a NAME="E3:16_6"/>
contradiction
 <b>by </b><i><a class="txt" href="convex3.html#E5:16"><i><font color="Green">E10</font></i></a>, <a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:16"/><b>then </b><i><font color="Green">E19</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="convex3.html#E11:16"><i><font color="Green">E13</font></i></a>, <a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/>
<a NAME="E29:16"/>
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>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>10</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:16_7"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:16_7"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="convex3.html#E1:16_7"><i><font color="Green">E20</font></i></a>, <a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>;<br/>
<b>hence </b><a NAME="E4:16_7"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="convex3.html#E18:16"><i><font color="Green">E17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E30:16"/><b>then </b><i><font color="Green">E20</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E31:16"/>
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"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>10</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:16_8"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:16_8"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="convex3.html#E18:16"><i><font color="Green">E17</font></i></a>, <a class="txt" href="convex3.html#E1:16_8"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E4:16_8"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="convex3.html#E1:16_8"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E5:16_8"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>;<br/>
<b>hence </b><a NAME="E6:16_8"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="convex3.html#E3:16_8"><i><font color="Green">E22</font></i></a>, <a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:16"/><b>then </b>
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E33:16"/><b>then </b><i><font color="Green">E21</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="convex3.html#E30:16"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E34:16"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_9_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font> or  not <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_9_1_1"><b>suppose </b></a><a NAME="E1:16_9_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_9_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_9_1_1_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_9_1_1_1_1"><b>suppose </b></a><a NAME="E1:16_9_1_1_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:16_9_1_1_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
not <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="convex3.html#E30:16"><i><font color="Green">E20</font></i></a>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_2.html#D12">RLVECT_2:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="txt" href="convex3.html#E8:16"><i><font color="Green">E11</font></i></a>, <a class="txt" href="convex3.html#E1:16_9_1_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="convex3.html#E2:16_9_1_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>
;<br/><b>hence </b><a NAME="E4:16_9_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_9_1_1_1_2"><b>suppose </b></a><a NAME="E1:16_9_1_1_1_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:16_9_1_1_1_2"/><b>then </b>
not <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="convex3.html#E11:16"><i><font color="Green">E13</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:16_9_1_1_1_2"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>;<br/><a NAME="E4:16_9_1_1_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="convex3.html#E1:16_9_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="convex3.html#E1:16_9_1_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_2.html#D12">RLVECT_2:def 12</a></i>
<br/>.= 
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="convex3.html#E18:16"><i><font color="Green">E17</font></i></a>, <a class="txt" href="convex3.html#E3:16_9_1_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="convex3.html#E4:16_9_1_1_1_2"><i><font color="Green">E25</font></i></a></i>
;<br/><b>hence </b><a NAME="E6:16_9_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_9_1_2"><b>suppose </b></a><a NAME="E1:16_9_1_2"/><i><font color="Green">E22</font></i>: 
not <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:16_9_1_2"/><b>then </b><i><font color="Green">E23</font></i>: 
not <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="convex3.html#E5:16"><i><font color="Green">E10</font></i></a>, <a class="txt" href="convex3.html#E11:16"><i><font color="Green">E13</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:16_9_1_2"/>
not <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="convex3.html#E30:16"><i><font color="Green">E20</font></i></a>, <a class="txt" href="convex3.html#E1:16_9_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/><a NAME="E4:16_9_1_2"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_2.html#D12">RLVECT_2:def 12</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="convex3.html#E2:16_9_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="convex3.html#E4:16_9_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>
;<br/><b>hence </b><a NAME="E6:16_9_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="convex3.html#E1:16_9_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E35:16"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="rlvect_2.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="rlvect_2.html#K7">+</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#D11">RLVECT_2:def 11</a></i>;<br/>
<a NAME="E36:16"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="convex3.html#E5:16"><i><font color="Green">E10</font></i></a>, <a class="txt" href="convex3.html#E28:16"><i><font color="Green">E19</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>then </b><i><font color="Green">E23</font></i>:  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="convex3.html#E28:16"><i><font color="Green">E19</font></i></a>, <a class="txt" href="convex3.html#E33:16"><i><font color="Green">E21</font></i></a>, <a class="ref" href="card_2.html#T63">CARD_2:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="txt" href="convex3.html#E28:16"><i><font color="Green">E19</font></i></a>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<a NAME="E38:16"/>
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<b>hence </b><a NAME="E39:16"/>
(  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 &amp;  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>10</sub></font> holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="convex3.html#E5:16"><i><font color="Green">E10</font></i></a>, <a class="txt" href="convex3.html#E14:16"><i><font color="Green">E14</font></i></a>, <a class="txt" href="convex3.html#E18:16"><i><font color="Green">E17</font></i></a>, <a class="txt" href="convex3.html#E28:16"><i><font color="Green">E19</font></i></a>, <a class="txt" href="convex3.html#E30:16"><i><font color="Green">E20</font></i></a>, <a class="txt" href="convex3.html#E35:16"><i><font color="Green">E22</font></i></a>, <a class="txt" href="convex3.html#E37:16"><i><font color="Green">E23</font></i></a>, <a class="ref" href="card_1.html#T79">CARD_1:79</a>, <a class="ref" href="rlvect_3.html#T1">RLVECT_3:1</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>


</div>
