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<b>let </b><font color="Maroon">K</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">K</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">l</font> be    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">A</font>;<br/>







<b>assume </b><a NAME="E1:16"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> is <a href="vectsp_4.html#V1">lineary-closed</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1"/>
(  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font> or  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_1"/>
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="vectsp_6.html#T40">VECTSP_6:40</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_1_2"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font>
 ;<br/><div class="add"><a NAME="E2:16_1_1_2"/><b>then </b>
<font color="Maroon">K</font> is <a href="realset2.html#V3">trivial</a>
 <b>by </b><i/>;<br/><a NAME="E3:16_1_1_2"/><b>then </b>
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:16_1_1_2"/>
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="vectsp_4.html#T4">VECTSP_4:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:16"/>
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>


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