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<b>let </b><font color="Maroon">R</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_2.html#NM7">RightMod</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">u</font> be   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="rmod_2.html#M1" target="_self">Submodule</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">C</font> be    <a href="rmod_2.html#M2" target="_self">Coset</a> of <font color="Maroon">W</font>;<br/>









<b>thus </b><a NAME="E1:78"/>
( <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="rmod_2.html#K3" target="_self">+</a> <font color="Maroon">W</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:78_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E2:78_1"/>
 ex <font color="Olive">v</font> being  <a href="vectsp_2.html#NM5">Vector</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="rmod_2.html#K3" target="_self">+</a> <font color="Maroon">W</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:78_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="rmod_2.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="E2:78"/>
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="rmod_2.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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