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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="vectsp_2.html#NM7">RightMod</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="rmod_2.html#M1" target="_self">Submodule</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="rmod_2.html#M2" target="_self">Coset</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>







<b>thus </b><a NAME="E1:70"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="rmod_2.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="70_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:70_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="rmod_2.html#V1" target="_self">lineary-closed</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:70_1"/><i><font color="Green">E49</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="rmod_2.html#K3" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="rmod_2.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E4:70_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="rmod_2.html#E3:70_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="rmod_2.html#T59" target="_self">Th59</a></i>;<br/>
<a NAME="E5:70_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="rmod_2.html#K3" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rmod_2.html#E1:70_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rmod_2.html#E3:70_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="rmod_2.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E6:70_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="rmod_2.html#E3:70_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="rmod_2.html#T63" target="_self">Th63</a></i>;<br/>


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

<b>thus </b><a NAME="E2:70"/>
( <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="rmod_2.html#V1" target="_self">lineary-closed</a> )
 <b>by </b><i><a class="txt" href="rmod_2.html#E27"><i><font color="Green">Lemma13</font></i></a></i>;<br/>


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