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<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">R</font> be   <a href="vectsp_2.html#NM2">domRing</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_2.html#NM7">RightMod</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>







<b>thus </b><a NAME="E1:8"/>
( <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="rmod_5.html#K1" target="_self">Lin</a> <font color="Maroon">A</font> implies  ex <font color="Olive">l</font> being   <a href="rmod_4.html#M2">Linear_Combination</a> of <font color="Maroon">A</font> st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rmod_4.html#K5">Sum</a> <font color="Olive">l</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="rmod_5.html#K1" target="_self">Lin</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:8_1"/><b>then </b>
<font color="Maroon">x</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="rmod_5.html#K1" target="_self">Lin</a> <font color="Maroon">A</font></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"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rmod_4.html#K5">Sum</a> <font color="Olive">l</font></span>)</span> where <font color="Olive">l</font> is    <a href="rmod_4.html#M2">Linear_Combination</a> of <font color="Maroon">A</font> : verum </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:8_1"/>
 ex <font color="Olive">l</font> being   <a href="rmod_4.html#M2">Linear_Combination</a> of <font color="Maroon">A</font> st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rmod_4.html#K5">Sum</a> <font color="Olive">l</font>
 ;<br/>


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

<b>given </b><font color="Maroon">k</font> being    <a href="rmod_4.html#M2">Linear_Combination</a> of <font color="Maroon">A</font><b> such that </b><a NAME="E2:8"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rmod_4.html#K5">Sum</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E3:8"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rmod_4.html#K5">Sum</a> <font color="Olive">l</font></span>)</span> where <font color="Olive">l</font> is    <a href="rmod_4.html#M2">Linear_Combination</a> of <font color="Maroon">A</font> : verum </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:8"/><b>then </b>
<font color="Maroon">x</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="rmod_5.html#K1" target="_self">Lin</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:8"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="rmod_5.html#K1" target="_self">Lin</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


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