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<b>let </b><font color="Maroon">GF</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V4">associative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">V</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="vectsp_1.html#V12">VectSp-like</a>  <a href="vectsp_1.html#L4">VectSpStr</a> of <font color="Maroon">GF</font>;<br/><b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="vectsp_4.html#M1" target="_self">Subspace</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">C</font> be    <a href="vectsp_4.html#M2" target="_self">Coset</a> of <font color="Maroon">W</font>;<br/>









<b>thus </b><a NAME="E1:82"/>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> implies <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="vectsp_4.html#K3" target="_self">+</a> <font color="Maroon">W</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:82_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E2:82_1"/>
 ex <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> st <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> <a href="vectsp_4.html#K3" target="_self">+</a> <font color="Maroon">W</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:82_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="vectsp_4.html#K3" target="_self">+</a> <font color="Maroon">W</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:82"/>
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="vectsp_4.html#K3" target="_self">+</a> <font color="Maroon">W</font> implies <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i/>;<br/>


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