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<b>let </b><font color="Maroon">L</font> be    <a href="vectsp_6.html#M1" target="_self">Linear_Combination</a> of <font color="Maroon">V</font>, <font color="Maroon">L'</font> be    <a href="vectsp_6.html#M1" target="_self">Linear_Combination</a> of <font color="Maroon">V</font>;<br/>

<b>reconsider </b><font color="Maroon">K</font> = <font color="Maroon">L</font>, <font color="Maroon">K'</font> = <font color="Maroon">L'</font> as   <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> ;<br/>
<b>assume </b><b>that </b><br/><a NAME="E2:4_1_2"/><i><font color="Green">E59</font></i>: 
 <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E3:4_1_2"/><i><font color="Green">E60</font></i>: 
 <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">L'</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 ;<br/><b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> ;<br/><div><i><font color="Green">E61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_1_2_1_1"/>
<font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</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="E2:4_1_2_1_1"/><b>then </b>
<font color="Maroon">v</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="struct_0.html#NM1">Element</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/><b>hence </b><a NAME="E3:4_1_2_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_1_2_1_2"/>
<font color="Maroon">L'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</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="E2:4_1_2_1_2"/><b>then </b>
<font color="Maroon">v</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="struct_0.html#NM1">Element</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/><b>hence </b><a NAME="E3:4_1_2_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:4_1_2_1"/>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:4_1_2"/>
<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L'</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>


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