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

<b>let </b><font color="Maroon">GF</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">GF</font>;<br/><b>let </b><font color="Maroon">v1</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v2</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/>







<b>thus </b><a NAME="E1:9"/>
( <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="domain_1.html#K7">}</a></span> is <a href="vectsp_7.html#V1" target="_self">linearly-independent</a> implies ( <font color="Maroon">v2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> &amp; ( for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">GF</font>  holds <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:9_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font>
 <b>and </b><br/><a NAME="E2:9_1"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="domain_1.html#K7">}</a></span> is <a href="vectsp_7.html#V1" target="_self">linearly-independent</a>
 ;<br/>

<b>thus </b><a NAME="E3:9_1"/>
<font color="Maroon">v2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i>, </i>;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">GF</font>;<br/>

<b>assume </b><a NAME="E4:9_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GF</font> =  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GF</font><b> such that </b><br/><a NAME="E6:9_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">GF</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>and </b><br/><a NAME="E7:9_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>  st <font color="Olive">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v1</font> &amp; <font color="Olive">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font> holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">v</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S7">FUNCT_2:sch 7</a>();<br/></i>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GF</font> <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/><b>assume </b><a NAME="E1:9_1_1"/>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/><a NAME="E2:9_1_1"/><b>then </b>
( <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:9_1_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="vectsp_6.html#D4">VECTSP_6:def 4</a></i>;<br/>
<a NAME="E11:9_1"/>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="domain_1.html#K7">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:9_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:9_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> : <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font> </span>}</span> 
 
;<br/>
<a NAME="E3:9_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
 ex <font color="Olive">u</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">u</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font> )
 
;<br/>
<b>assume </b><a NAME="E4:9_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/>

<a NAME="E5:9_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:9_1_2"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="vectsp_6.html#M2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="domain_1.html#K7">}</a></span> <b>by </b><i><a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/>
<div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_3"/>
not <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:9_1_3"/><b>then </b>
not <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> : <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font> </span>}</span> 
 ;<br/><a NAME="E3:9_1_3"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">GF</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:9_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="vectsp_6.html#T82">VECTSP_6:82</a></i>;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">w</font> = <font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font>;<br/>
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">GF</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="vectsp_6.html#T44">VECTSP_6:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T59">VECTSP_1:59</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T64">VECTSP_1:64</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T66">VECTSP_1:66</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T25">RLVECT_1:25</a></i>
;<br/>
<b>hence </b><a NAME="E15:9_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:9"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">v2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 ;<br/>

<b>assume </b><a NAME="E3:9"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">GF</font>  holds <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font>
 ;<br/>

<a NAME="E4:9"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">GF</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font> &amp; <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">GF</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> )
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>;<br/>
<b>hence </b><a NAME="E5:9"/>
<font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font>
 ;<br/>

<b>let </b><font color="Maroon">l</font> be    <a href="vectsp_6.html#M2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="domain_1.html#K7">}</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_7.html#D1" target="_self">VECTSP_7:def 1</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E6:9"/><i><font color="Green">E52</font></i>: 
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E7:9"/><i><font color="Green">E53</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l</font>;<br/>
<a NAME="E9:9"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:9"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> : <font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font> </span>}</span> 
 
;<br/>
<a NAME="E11:9"/><b>then </b><i><font color="Green">E54</font></i>: 
 ex <font color="Olive">u</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">u</font> &amp; <font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font> )
 
;<br/>
<a NAME="E12:9"/>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/>
<a NAME="E13:9"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E14:9"/><i><font color="Green">E56</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="vectsp_6.html#T44">VECTSP_6:44</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_2_1"/>
( <font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font> or ( <font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font> &amp; <font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font> ) )
 </span><b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_1"><b>suppose </b></a><a NAME="E1:9_2_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font>
 ;<br/><div class="add"> <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#T60">VECTSP_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">GF</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
<br/>.= 
<font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
;<br/><b>then </b><font color="Maroon">v1</font> = 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T63">VECTSP_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">GF</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T59">VECTSP_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">GF</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
;<br/><b>hence </b><a NAME="E4:9_2_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2"><b>suppose </b></a><a NAME="E1:9_2_1_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font> &amp; <font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font> )
 ;<br/><div class="add"> <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#T60">VECTSP_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">GF</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#T44">VECTSP_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T60">VECTSP_1:60</a></i>
<br/>.= 
<font color="Maroon">v2</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><b>hence </b><a NAME="E3:9_2_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E16:9"/>
contradiction
 ;<br/>


</div>
