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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>, <font color="Maroon">c<sub>3</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>4</sub></font> be    <a href="rlsub_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>thus </b><a NAME="E1:85"/>
( ex <font color="Olive">b<sub>1</sub></font> being   <a href="rlsub_1.html#M2" target="_self">Coset</a> of <font color="Maroon">c<sub>4</sub></font> st <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) implies <font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></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<sub>5</sub></font> being    <a href="rlsub_1.html#M2" target="_self">Coset</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><a NAME="E1:85_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>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<a NAME="E2:85_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>5</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>4</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="E3:85_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_1.html#E1:85_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="rlsub_1.html#T81" target="_self">Th81</a></i>;<br/>


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

<b>assume </b><a NAME="E2:85"/>
<font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:85"/><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>5</sub></font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="rlsub_1.html#T81" target="_self">Th81</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="rlsub_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font> as    <a href="rlsub_1.html#M2" target="_self">Coset</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="rlsub_1.html#D6" target="_self">Def6</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E6:85"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="rlsub_1.html#E4:85"><i><font color="Green">E55</font></i></a></i>;<br/>


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