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<b>let </b><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font>, <font color="Maroon" title="c3">W</font> be   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">F</font>;<br/><b>let </b><font color="Maroon" title="c4">T</font> be   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>;<br/><b>let </b><font color="Maroon" title="c5">l</font> be    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font>;<br/><b>let </b><font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">W</font>;<br/>











<b>assume </b><a NAME="E1:50"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon" title="c6">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K8" title="RANKNULL:func.8">@</a> <font color="Maroon" title="c5">l</font></span>)</span>
 ;<br/>

<a NAME="E2:50"/><i><font color="Green" title="E37">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K8" title="RANKNULL:func.8">@</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</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">F</font>
 
<b>by </b><i><a class="txt" href="ranknull.html#E1:50"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="vectsp_6.html#T20" title="VECTSP_6:th.20">VECTSP_6:20</a></i>;<br/>
<b>assume </b><a NAME="E3:50"/><i><font color="Green" title="E38">A4</font></i>: 
<font color="Maroon" title="c4">T</font> <a href="ranknull.html#K7" title="RANKNULL:func.7">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">w</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">l</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1"/>
( <font color="Maroon" title="c4">T</font> <a href="ranknull.html#K7" title="RANKNULL:func.7">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">w</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c4">T</font> <a href="ranknull.html#K7" title="RANKNULL:func.7">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">w</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1"><b>suppose </b></a><a NAME="E1:50_1_1"/>
<font color="Maroon" title="c4">T</font> <a href="ranknull.html#K7" title="RANKNULL:func.7">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">w</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>then </b> <a href="rlvect_2.html#K2" title="RLVECT_2:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K7" title="RANKNULL:func.7">"</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">w</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> = 
 <a href="rlvect_2.html#K2" title="RLVECT_2:func.2">Sum</a> <span class="p1">(<span class="default"><a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">F</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T149" title="RELAT_1:th.149">RELAT_1:149</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font>
<b>by </b><i><a class="ref" href="rlvect_2.html#T14" title="RLVECT_2:th.14">RLVECT_2:14</a></i>
;<br/><b>hence </b><a NAME="E3:50_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ranknull.html#E2:50"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="ranknull.html#D5" target="_self" title="RANKNULL:def.5">Def5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2"><b>suppose </b></a><a NAME="E1:50_1_2"/>
<font color="Maroon" title="c4">T</font> <a href="ranknull.html#K7" title="RANKNULL:func.7">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">w</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:50_1_2"/><b>then </b>
<font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K7" title="RANKNULL:func.7">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">w</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="ranknull.html#T35" target="_self" title="RANKNULL:th.35">Th33</a>, <a class="txt" href="ranknull.html#E3:50"><i><font color="Green" title="E38">A4</font></i></a></i>;<br/><a NAME="E3:50_1_2"/><b>then </b>
 <a href="rlvect_2.html#K2" title="RLVECT_2:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="ranknull.html#K5" title="RANKNULL:func.5">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K7" title="RANKNULL:func.7">"</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">w</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font>
 <b>by </b><i><a class="ref" href="rlvect_2.html#T15" title="RLVECT_2:th.15">RLVECT_2:15</a></i>;<br/><b>hence </b><a NAME="E4:50_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="ranknull.html#E2:50"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="ranknull.html#D5" target="_self" title="RANKNULL:def.5">Def5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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