<?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>assume </b><a NAME="E1:18"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="rlvect_3.html#V1">linearly-independent</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</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="E2:18"/><i><font color="Green">E20</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>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E3:18"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <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>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>assume </b><a NAME="E4:18"/>
<font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rlvect_3.html#K1">Lin</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="rlvect_2.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E6:18"/><i><font color="Green">E22</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>5</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_3.html#T17">RLVECT_3:17</a></i>;<br/>
<a NAME="E7:18"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:18"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rlvect_3.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="rlvect_3.html#T18">RLVECT_3:18</a></i>;<br/>
<b>then 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>3</sub></font></span><a href="domain_1.html#K6">}</a></span><b> such that </b><br/><a NAME="E10:18"/><i><font color="Green">E23</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>6</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_3.html#T17">RLVECT_3:17</a></i>;<br/>
<i><font color="Green">E24</font></i>:  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_5.html#E6:18"><i><font color="Green">E22</font></i></a>, <a class="txt" href="rlvect_5.html#E10:18"><i><font color="Green">E23</font></i></a>, <a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>5</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> <span class="p2">(<span class="default"><a href="rlvect_2.html#K9">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_3.html#T3">RLVECT_3:3</a></i>
<br/>.= 
 <a href="rlvect_2.html#K6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_2.html#K7">+</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K9">-</a> <font color="Maroon">c<sub>6</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 href="rlvect_2.html#K6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_2.html#D15">RLVECT_2:def 15</a></i>
;<br/>
<a NAME="E12:18"/><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="rlvect_5.html#E2:18"><i><font color="Green">E20</font></i></a>, <a class="ref" href="subset_1.html#T63">SUBSET_1:63</a></i>;<br/>
<a NAME="E13:18"/><b>then </b><i><font color="Green">E26</font></i>: 
( <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</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> &amp; <font color="Maroon">c<sub>4</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="rlvect_5.html#E3:18"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E14:18"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> is <a href="rlvect_3.html#V1">linearly-independent</a>
 
<b>by </b><i><a class="txt" href="rlvect_5.html#E1:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="rlvect_5.html#E12:18"><i><font color="Green">E25</font></i></a>, <a class="ref" href="rlvect_3.html#T6">RLVECT_3:6</a></i>;<br/>
<a NAME="E15:18"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_3.html#T9">RLVECT_3:9</a></i>;<br/>
<a NAME="E16:18"/><b>then </b>
not  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="rlvect_5.html#E6:18"><i><font color="Green">E22</font></i></a>, <a class="ref" href="rlvect_2.html#T52">RLVECT_2:52</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E18:18"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E19:18"/><i><font color="Green">E28</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <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>5</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#T80">RLVECT_2:80</a></i>;<br/>
<a NAME="E20:18"/><i><font color="Green">E29</font></i>: 
(  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <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>3</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="E21:18"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E22:18"/>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</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> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="rlvect_5.html#E13:18"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E23:18"/><b>then </b>
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="rlvect_5.html#E21:18"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E24:18"/><b>then </b>
 <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="rlvect_5.html#E19:18"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E25:18"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font> is    <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="E26:18"/><i><font color="Green">E32</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>5</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><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="E1:18_2"/>
<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>5</sub></font>
 ;<br/>

<a NAME="E2:18_2"/><b>then </b>
not <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>6</sub></font>
 
<b>by </b><i><a class="txt" href="rlvect_5.html#E3:18"><i><font color="Green">E21</font></i></a>, <a class="txt" href="rlvect_5.html#E20:18"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:18_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</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/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3"><b>now </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><b>that </b><br/><a NAME="E1:18_3"/><i><font color="Green">E33</font></i>: 
<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>5</sub></font>
 <b>and </b><br/><a NAME="E2:18_3"/><i><font color="Green">E34</font></i>: 
not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</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="rlvect_5.html#E1:18_3"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E4:18_3"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="rlvect_5.html#E1:18_3"><i><font color="Green">E33</font></i></a>, <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>5</sub></font> <a href="rlvect_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_2.html#T79">RLVECT_2:79</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 0
<b>by </b><i><a class="txt" href="rlvect_5.html#E26:18"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rlvect_5.html#E1:18_3"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>

;<br/><b>hence </b><a NAME="E6:18_3"/>
contradiction
 <b>by </b><i><a class="txt" href="rlvect_5.html#E2:18_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rlvect_5.html#E4:18_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>;<br/></div><b>end;</b></div>
<a NAME="E28:18"/><b>then </b>
not  <a href="rlvect_2.html#K3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="rlvect_5.html#E18:18"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>hence </b><a NAME="E29:18"/>
contradiction
 <b>by </b><i><a class="txt" href="rlvect_5.html#E1:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="rlvect_5.html#E11:18"><i><font color="Green">E24</font></i></a>, <a class="txt" href="rlvect_5.html#E25:18"><i><font color="Green">E31</font></i></a>, <a class="ref" href="rlvect_3.html#D1">RLVECT_3:def 1</a></i>;<br/>


</div>
