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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">K</font>  st <font color="Maroon" title="c1">K</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> holds <br/><font color="Olive" title="b1">V</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">K</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> implies <font color="Maroon" title="c2">V</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> )</span><br/>



<b>assume </b><a NAME="E1:8"/><i><font color="Green" title="E5">A1</font></i>: 
( <font color="Maroon" title="c1">K</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> &amp; not <font color="Maroon" title="c2">V</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:8"/><b>then </b>
 ex <font color="Olive" title="b1">a</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font> st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="ref" href="anproj_1.html#D8" title="ANPROJ_1:def.8">ANPROJ_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E3:8"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="lmod_6.html#T5" target="_self" title="LMOD_6:th.5">Th5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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