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<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="vectsp_2.html#NM1" title="VECTSP_2:NM.1">comRing</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_2.html#NM7" title="VECTSP_2:NM.7">RightMod</a> of <font color="Maroon" title="c1">R</font>;<br/><b>let </b><font color="Maroon" title="c3">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>thus </b><a NAME="E1:50"/>
<font color="Maroon" title="c3">L</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K3" title="RMOD_4:func.3">ZeroLC</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="rmod_4.html#R1" title="RMOD_4:pred.1">=</a> <font color="Maroon" title="c3">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">v</font> be   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="rmod_4.html#D10" target="_self" title="RMOD_4:def.10">RMOD_4:def 10</a><br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <span class="p2">(<span class="default"><a href="rmod_4.html#K3" title="RMOD_4:func.3">ZeroLC</a> <font color="Maroon" title="c2">V</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v</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="rmod_4.html#K3" title="RMOD_4:func.3">ZeroLC</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v</font></span>)</span>
<b>by </b><i><a class="ref" href="rmod_4.html#D11" target="_self" title="RMOD_4:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">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="c1">R</font></span>)</span>
<b>by </b><i><a class="ref" href="rmod_4.html#T22" target="_self" title="RMOD_4:th.22">Th22</a></i>
<br/>.= 
<font color="Maroon" title="c3">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/>


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

<b>hence </b><a NAME="E2:50"/>
<span class="p1">(<span class="default"><a href="rmod_4.html#K3" title="RMOD_4:func.3">ZeroLC</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="rmod_4.html#K6" title="RMOD_4:func.6">+</a> <font color="Maroon" title="c3">L</font> <a href="rmod_4.html#R1" title="RMOD_4:pred.1">=</a> <font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="ref" href="rmod_4.html#T53" target="_self" title="RMOD_4:th.53">Th53</a></i>;<br/>


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