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<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">R</font>;<br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>;<br/><b>let </b><font color="Maroon" title="c4">W</font> be  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:10"/><i><font color="Green" title="E10">A1</font></i>: 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font>
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">v</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font>;<br/><b>thus </b><a NAME="E1:10_1"/>
( <font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c3">A</font> implies <font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">W</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1_1"/>
<font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c3">A</font>
 ;<br/>

<a NAME="E2:10_1_1"/><b>then </b><i><font color="Green" title="E12">A3</font></i>: 
 ex <font color="Olive" title="b1">l</font> being   <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">A</font> st <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b1">l</font>
 
<b>by </b><i><a class="ref" href="mod_3.html#T11" target="_self" title="MOD_3:th.11">Th11</a></i>;<br/>
<a NAME="E3:10_1_1"/>
( <font color="Maroon" title="c3">A</font> is <a href="vectsp_4.html#V1" title="VECTSP_4:attr.1">lineary-closed</a> &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="mod_3.html#E2:10"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="vectsp_4.html#T41" title="VECTSP_4:th.41">VECTSP_4:41</a></i>;<br/>
<a NAME="E4:10_1_1"/><b>then </b>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W</font>
 
<b>by </b><i><a class="txt" href="mod_3.html#E1:10"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="mod_3.html#E2:10"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="mod_3.html#E2:10_1_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="vectsp_6.html#T40" title="VECTSP_6:th.40">VECTSP_6:40</a></i>;<br/>
<b>hence </b><a NAME="E5:10_1_1"/>
<font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">W</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E2:10_1"/>
( <font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">W</font> iff <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W</font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:10_1"/>
( <font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">W</font> implies <font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c3">A</font> )
 <b>by </b><i><a class="txt" href="mod_3.html#E2:10"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="mod_3.html#T12" target="_self" title="MOD_3:th.12">Th12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:10"/>
 <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T38" title="VECTSP_4:th.38">VECTSP_4:38</a></i>;<br/>


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