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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>assume </b><a NAME="E1:15"/><i><font color="Green">E32</font></i>: 
not <font color="Maroon">V</font> is <a href="realset2.html#V3">trivial</a>
 ;<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><b>that </b><br/><a NAME="E2:15"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">A</font> is <a href="mod_3.html#V1">base</a>
 <b>and </b><br/><a NAME="E3:15"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E4:15"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 
<b>by </b><i/>;<br/>
 <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font> #) = 
 <a href="mod_3.html#K1">Lin</a> <font color="Maroon">A</font>
<b>by </b><i>, <a class="ref" href="mod_3.html#D2">MOD_3:def 2</a></i>
<br/>.= 
 <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
<b>by </b><i>, <a class="ref" href="mod_3.html#T13">MOD_3:13</a></i>
;<br/>
<b>hence </b><a NAME="E6:15"/>
contradiction
 <b>by </b><i>, <a class="ref" href="lmod_6.html#T7">LMOD_6:7</a></i>;<br/>


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