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<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>let </b><font color="Maroon">W</font> be  <a href="vectsp_2.html#V3">strict</a>  <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">V</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:12"/><i><font color="Green">E26</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E2:12"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/><b>thus </b><a NAME="E1:12_1"/>
( <font color="Maroon">v</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 <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1_1"/>
<font color="Maroon">v</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="E2:12_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
 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">v</font> <a href="hidden.html#R1">=</a>  <a href="rmod_4.html#K5">Sum</a> <font color="Olive">l</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:12_1_1"/>
( <font color="Maroon">A</font> is <a href="rmod_2.html#V1">lineary-closed</a> &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, <a class="ref" href="rmod_2.html#T41">RMOD_2:41</a></i>;<br/>
<a NAME="E4:12_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 <font color="Maroon">W</font>
 
<b>by </b><i>, , , <a class="ref" href="rmod_4.html#T40">RMOD_4:40</a></i>;<br/>
<b>hence </b><a NAME="E5:12_1_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E2:12_1"/>
( <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> iff <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_1"/>
( <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> implies <font color="Maroon">v</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>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:12"/>
 <a href="rmod_5.html#K1" target="_self">Lin</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="rmod_2.html#T38">RMOD_2:38</a></i>;<br/>


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