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<b>let </b><font color="Maroon" title="c3">L</font>, <font color="Maroon" title="c4">L'</font> be    <a href="rmod_4.html#M1" title="RMOD_4:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:21_1_2"/><i><font color="Green" title="E11">A2</font></i>: 
 <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E2:21_1_2"/><i><font color="Green" title="E12">A3</font></i>: 
 <a href="rmod_4.html#K2" title="RMOD_4:func.2">Carrier</a> <font color="Maroon" title="c4">L'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:21_1_2_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">v</font> = <font color="Maroon" title="c5">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> ;<br/><div><i><font color="Green" title="E13">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_2_1_1"/>
<font color="Maroon" title="c3">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 ;<br/><a NAME="E2:21_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> : <font color="Maroon" title="c3">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E3:21_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rmod_4.html#E1:21_1_2"><i><font color="Green" title="E11">A2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_2_1_2"/>
<font color="Maroon" title="c4">L'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 ;<br/><a NAME="E2:21_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> : <font color="Maroon" title="c4">L'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E3:21_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rmod_4.html#E2:21_1_2"><i><font color="Green" title="E12">A3</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:21_1_2_1"/>
<font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="rmod_4.html#E3:21_1_2_1"><i><font color="Green" title="E13">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:21_1_2"/>
<font color="Maroon" title="c3">L</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">L'</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>;<br/>


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