<?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="rlvect_1.html#NM1">VECTOR</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="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:21"/><i><font color="Green">E19</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>3</sub></font> <a href="rlsub_2.html#K1">+</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/>



<b>assume </b><a NAME="E2:21"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E3:21"/><b>then </b>
 <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>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</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>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="hahnban.html#T23" target="_self">Th23</a></i>;<br/>
<b>hence </b><a NAME="E4:21"/>
 <a href="rlvect_1.html#G2">RLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>,the <a href="struct_0.html#U2">Zero</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>,the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>,the <a href="rlvect_1.html#U2">Mult</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> #) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="rlsub_2.html#K1">+</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="hahnban.html#E2:21"><i><font color="Green">E20</font></i></a>, <a class="ref" href="hahnban.html#T22" target="_self">Th22</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlsub_2.html#D4">RLSUB_2:def 4</a><br/>

<b>assume </b><a NAME="E5:21"/>
<font color="Maroon">c<sub>5</sub></font> <a href="rlsub_2.html#K2">/\</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlsub_1.html#K1">(0).</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E7:21"/><i><font color="Green">E21</font></i>: 
( ( <font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="rlsub_2.html#K2">/\</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp; not <font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rlsub_1.html#K1">(0).</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> ) or ( <font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rlsub_1.html#K1">(0).</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> &amp; not <font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="rlsub_2.html#K2">/\</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="rlsub_1.html#T39">RLSUB_1:39</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1"/>
( ( <font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="rlsub_2.html#K2">/\</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp; not <font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rlsub_1.html#K1">(0).</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> ) or ( not <font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="rlsub_2.html#K2">/\</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rlsub_1.html#K1">(0).</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="hahnban.html#E7:21"><i><font color="Green">E21</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:21_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="rlsub_2.html#K2">/\</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E2:21_1_1"/><i><font color="Green">E23</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rlsub_1.html#K1">(0).</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E3:21_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="hahnban.html#E1:21_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="rlsub_2.html#T7">RLSUB_2:7</a></i>;<br/><a NAME="E4:21_1_1"/>
<font color="Maroon">c<sub>6</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>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="hahnban.html#E1:21_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="rlsub_2.html#T7">RLSUB_2:7</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:21_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_4.html#T11">RLVECT_4:11</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:21_1_1_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:21_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="hahnban.html#E6:21_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>;<br/><b>hence </b><a NAME="E3:21_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="hahnban.html#E2:21_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="rlsub_1.html#T25">RLSUB_1:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:21_1_1_1_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><font color="Maroon">c<sub>4</sub></font> = 
1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="hahnban.html#E1:21_1_1_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/><b>hence </b><a NAME="E3:21_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="hahnban.html#E1:21"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hahnban.html#E2:21"><i><font color="Green">E20</font></i></a>, <a class="txt" href="hahnban.html#E3:21_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="hahnban.html#E6:21_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="rlsub_1.html#T29">RLSUB_1:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:21_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:21_1_2"/><i><font color="Green">E22</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="rlsub_2.html#K2">/\</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E2:21_1_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="rlsub_1.html#K1">(0).</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E3:21_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="hahnban.html#E2:21_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="rlvect_3.html#T36">RLVECT_3:36</a></i>;<br/><b>hence </b><a NAME="E4:21_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="hahnban.html#E1:21_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="rlsub_1.html#T25">RLSUB_1:25</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
