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<div class="add">

<b>let </b><font color="Maroon" title="c1">GF</font> be   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">VectSp</a> of <font color="Maroon" title="c1">GF</font>;<br/><b>let </b><font color="Maroon" title="c3">v1</font>, <font color="Maroon" title="c4">v2</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font>;<br/>







<b>thus </b><a NAME="E1:10"/>
( <font color="Maroon" title="c3">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v2</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> implies for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">GF</font>  st <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c4">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font> holds <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:10_1"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c3">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v2</font>
 <b>and </b><br/><a NAME="E2:10_1"/><i><font color="Green" title="E7">A2</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">GF</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E3:10_1"/><i><font color="Green" title="E8">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c4">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font>
 <b>and </b><br/><a NAME="E4:10_1"/><i><font color="Green" title="E9">A4</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font> or <font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_1_1"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font> or <font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font> )
 </span><b>by </b><i><a class="txt" href="vectsp_7.html#E4:10_1"><i><font color="Green" title="E9">A4</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_1"><b>suppose </b></a><a NAME="E1:10_1_1_1_1"/><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font>
 ;<br/><div class="add"> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c4">v2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_7.html#E3:10_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="vectsp_1.html#T60" title="VECTSP_1:th.60">VECTSP_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c4">v2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">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"><font color="Maroon" title="c5">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c4">v2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">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"><font color="Maroon" title="c5">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c4">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">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" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">GF</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c4">v2</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_7.html#E1:10_1_1_1_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="vectsp_1.html#D22" title="VECTSP_1:def.22">VECTSP_1:def 22</a></i>
<br/>.= 
<font color="Maroon" title="c3">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c4">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>
;<br/><b>then </b><font color="Maroon" title="c3">v1</font> = 
 <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c4">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T63" title="VECTSP_1:th.63">VECTSP_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">GF</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c4">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T59" title="VECTSP_1:th.59">VECTSP_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">GF</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c4">v2</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>
;<br/><b>hence </b><a NAME="E4:10_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="vectsp_7.html#E1:10_1"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="vectsp_7.html#E2:10_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="vectsp_7.html#T8" target="_self" title="VECTSP_7:th.8">Th8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_2"><b>suppose </b></a><a NAME="E1:10_1_1_1_2"/><i><font color="Green" title="E10">A6</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font>
 ;<br/><div class="add"> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c4">v2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_7.html#E3:10_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="vectsp_1.html#T60" title="VECTSP_1:th.60">VECTSP_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c4">v2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">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"><font color="Maroon" title="c6">b</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c4">v2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">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"><font color="Maroon" title="c6">b</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c4">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">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"><font color="Maroon" title="c6">b</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">GF</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c4">v2</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_7.html#E1:10_1_1_1_2"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="vectsp_1.html#D22" title="VECTSP_1:def.22">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"><font color="Maroon" title="c6">b</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c4">v2</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>
;<br/><b>then </b><font color="Maroon" title="c4">v2</font> = 
 <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v1</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T63" title="VECTSP_1:th.63">VECTSP_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">GF</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v1</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T59" title="VECTSP_1:th.59">VECTSP_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">GF</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">"</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v1</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>
;<br/><b>hence </b><a NAME="E4:10_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="vectsp_7.html#E1:10_1"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="vectsp_7.html#E2:10_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="vectsp_7.html#T8" target="_self" title="VECTSP_7:th.8">Th8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:10_1"/>
contradiction
 ;<br/>


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

<b>assume </b><a NAME="E2:10"/><i><font color="Green" title="E6">A7</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">GF</font>  st <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c4">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font> holds <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font> )
 ;<br/>

<div><i><font color="Green" title="E7">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_2"/><i><font color="Green" title="E8">A9</font></i>: 
<font color="Maroon" title="c4">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font>
 ;<br/> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T60" title="VECTSP_1:th.60">VECTSP_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">GF</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c4">v2</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_7.html#E1:10_2"><i><font color="Green" title="E8">A9</font></i></a>, <a class="ref" href="vectsp_1.html#T60" title="VECTSP_1:th.60">VECTSP_1:60</a></i>
;<br/><a NAME="E3:10_2"/><b>then </b>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">GF</font>
 <b>by </b><i><a class="txt" href="vectsp_7.html#E2:10"><i><font color="Green" title="E6">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E4:10_2"/>
contradiction
 <b>by </b><i><a class="ref" href="vectsp_1.html#D21" title="VECTSP_1:def.21">VECTSP_1:def 21</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">GF</font>;<br/><b>assume </b><a NAME="E1:10_3"/>
<font color="Maroon" title="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c4">v2</font>
 ;<br/><a NAME="E2:10_3"/><b>then </b>
<font color="Maroon" title="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c4">v2</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>;<br/><b>then </b> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font> = 
<font color="Maroon" title="c3">v1</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c4">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_2.html#T22" title="VECTSP_2:th.22">VECTSP_2:22</a></i>
<br/>.= 
<font color="Maroon" title="c3">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c4">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T68" title="VECTSP_1:th.68">VECTSP_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">GF</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c3">v1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c4">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>
;<br/><a NAME="E4:10_3"/><b>then </b>
 <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">GF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">GF</font>
 <b>by </b><i><a class="txt" href="vectsp_7.html#E2:10"><i><font color="Green" title="E6">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E5:10_3"/>
contradiction
 <b>by </b><i><a class="ref" href="vectsp_1.html#D21" title="VECTSP_1:def.21">VECTSP_1:def 21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:10"/>
( <font color="Maroon" title="c3">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v2</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> )
 <b>by </b><i><a class="txt" href="vectsp_7.html#E3:10"><i><font color="Green" title="E7">A8</font></i></a>, <a class="ref" href="vectsp_7.html#T8" target="_self" title="VECTSP_7:th.8">Th8</a></i>;<br/>


</div>
