<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></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">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="vectsp_4.html#M1" target="_self">Subspace</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="vectsp_4.html#M2" target="_self">Coset</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>







<b>thus </b><a NAME="E1:74"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="vectsp_4.html#V1" target="_self">lineary-closed</a> implies <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:74_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="vectsp_4.html#V1" target="_self">lineary-closed</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:74_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="vectsp_4.html#K3" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E4:74_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="vectsp_4.html#E3:74_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="vectsp_4.html#T59" target="_self">Th59</a></i>;<br/>
<a NAME="E5:74_1"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="vectsp_4.html#K3" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_4.html#E1:74_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="vectsp_4.html#E3:74_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="vectsp_4.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E6:74_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_4.html#E3:74_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="vectsp_4.html#T63" target="_self">Th63</a></i>;<br/>


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

<b>thus </b><a NAME="E2:74"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>4</sub></font> is <a href="vectsp_4.html#V1" target="_self">lineary-closed</a> )
 <b>by </b><i><a class="txt" href="vectsp_4.html#E28"><i><font color="Green">Lemma14</font></i></a></i>;<br/>


</div>
