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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</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="rlsub_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="rlsub_1.html#M2" target="_self">Coset</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>







<b>thus </b><a NAME="E1:82"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>3</sub></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">E55</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:82_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#D6" target="_self">Def6</a></i>;<br/>
<b>hence </b><a NAME="E3:82_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_1.html#E1:82_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="rlsub_1.html#T70" target="_self">Th70</a></i>;<br/>


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

<b>thus </b><a NAME="E2:82"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="rlsub_1.html#T59" target="_self">Th59</a></i>;<br/>


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