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<b>let </b><font color="Maroon">R</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">R</font>;<br/>



<b>set </b><font color="Maroon">A</font> =  <a href="mod_3.html#K1" target="_self">Lin</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/><b>thus </b><a NAME="E1:8_1"/>
( <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="mod_3.html#K1" target="_self">Lin</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span>)</span> implies <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="mod_3.html#K1" target="_self">Lin</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span>)</span>
 ;<br/>

<a NAME="E2:8_1_1"/><b>then </b>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="mod_3.html#K1" target="_self">Lin</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span>)</span></span>)</span> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="mod_3.html#K1" target="_self">Lin</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Olive">l0</font></span>)</span> where <font color="Olive">l0</font> is    <a href="vectsp_6.html#M2">Linear_Combination</a> of  <a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> : verum </span>}</span>  )
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E3:8_1_1"/><b>then </b>
 ex <font color="Olive">l0</font> being   <a href="vectsp_6.html#M2">Linear_Combination</a> of  <a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> st <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Olive">l0</font>
 
;<br/>
<a NAME="E4:8_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#T42">VECTSP_6:42</a></i>;<br/>
<b>hence </b><a NAME="E5:8_1_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T46">VECTSP_4:46</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:8_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
 ;<br/><a NAME="E3:8_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T46">VECTSP_4:46</a></i>;<br/><b>hence </b><a NAME="E4:8_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="mod_3.html#K1" target="_self">Lin</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T25">VECTSP_4:25</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:8"/>
 <a href="mod_3.html#K1" target="_self">Lin</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T38">VECTSP_4:38</a></i>;<br/>


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