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<div class="add">

<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">v1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v2</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>thus </b><a NAME="E1:85"/>
(  ex <font color="Olive">C</font> being   <a href="vectsp_4.html#M2" target="_self">Coset</a> of <font color="Maroon">W</font> st <br/>( <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) implies <font color="Maroon">v1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">C</font> being    <a href="vectsp_4.html#M2" target="_self">Coset</a> of <font color="Maroon">W</font><b> such that </b><a NAME="E1:85_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 ;<br/>

<a NAME="E2:85_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="E3:85_1"/>
<font color="Maroon">v1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="vectsp_4.html#D2" target="_self">Def2</a></i>;<br/>


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

<b>assume </b><a NAME="E2:85"/>
<font color="Maroon">v1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W</font>
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:85"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="vectsp_4.html#K3" target="_self">+</a> <font color="Maroon">W</font> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="vectsp_4.html#K3" target="_self">+</a> <font color="Maroon">W</font> )
 <b>by </b><i><a class="ref" href="vectsp_4.html#D2" target="_self">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">v</font> <a href="vectsp_4.html#K3" target="_self">+</a> <font color="Maroon">W</font> as    <a href="vectsp_4.html#M2" target="_self">Coset</a> of <font color="Maroon">W</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">C</font>
;<br/>

<b>thus </b><a NAME="E6:85"/>
( <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i/>;<br/>


</div>
