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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>assume </b><a NAME="E1:17"/><i><font color="Green">E43</font></i>: 
(  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> is <a href="vectsp_4.html#V1">lineary-closed</a> )
 ;<br/>

<b>thus </b><a NAME="E2:17"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="mod_3.html#K1">Lin</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:17"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="mod_3.html#K1">Lin</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>

<a NAME="E4:17"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="mod_3.html#K1">Lin</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:17"/><b>then </b>
 ex <font color="Olive">l</font> being   <a href="vectsp_6.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="vectsp_6.html#K4">Sum</a> <font color="Olive">l</font>
 
<b>by </b><i><a class="ref" href="mod_3.html#T11">MOD_3:11</a></i>;<br/>
<b>hence </b><a NAME="E6:17"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/>


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