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<b>let </b><font color="Maroon" title="c1">K</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">K</font>;<br/>



<b>assume </b><a NAME="E1:15"/><i><font color="Green" title="E5">A1</font></i>: 
not <font color="Maroon" title="c2">V</font> is <a href="realset2.html#V3" title="REALSET2:attr.3">trivial</a>
 ;<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>assume </b><b>that </b><br/><a NAME="E2:15"/><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon" title="c3">A</font> is <a href="mod_3.html#V1" title="MOD_3:attr.1">base</a>
 <b>and </b><br/><a NAME="E3:15"/><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E4:15"/><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="txt" href="lmod_7.html#E3:15"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/>
 <a href="vectsp_1.html#G4" title="VECTSP_1:aggr.4">VectSpStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c2">V</font>,the <a href="vectsp_1.html#U2" title="VECTSP_1:sel.2">lmult</a> of <font color="Maroon" title="c2">V</font> #) = 
 <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="lmod_7.html#E2:15"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="mod_3.html#D2" title="MOD_3:def.2">MOD_3:def 2</a></i>
<br/>.= 
 <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
<b>by </b><i><a class="txt" href="lmod_7.html#E4:15"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="mod_3.html#T13" title="MOD_3:th.13">MOD_3:13</a></i>
;<br/>
<b>hence </b><a NAME="E6:15"/>
contradiction
 <b>by </b><i><a class="txt" href="lmod_7.html#E1:15"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="lmod_6.html#T7" title="LMOD_6:th.7">LMOD_6:7</a></i>;<br/>


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