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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="rusub_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>3</sub></font> be    <a href="rusub_1.html#M2" target="_self">Coset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>thus </b><a NAME="E1:67"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="rlsub_1.html#V1">lineary-closed</a> implies <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="rlsub_1.html#V1">lineary-closed</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</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="E3:67_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="rusub_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="rusub_1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E4:67_1"/>
<font color="Maroon">c<sub>3</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="rusub_1.html#E3:67_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="rusub_1.html#T37" target="_self">Th37</a></i>;<br/>
<a NAME="E5:67_1"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="rusub_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_1.html#E1:67_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="rusub_1.html#E3:67_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="rlsub_1.html#T4">RLSUB_1:4</a></i>;<br/>
<b>hence </b><a NAME="E6:67_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="rusub_1.html#E3:67_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="rusub_1.html#T41" target="_self">Th41</a></i>;<br/>


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

<b>thus </b><a NAME="E2:67"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>3</sub></font> is <a href="rlsub_1.html#V1">lineary-closed</a> )
 <b>by </b><i><a class="txt" href="rusub_1.html#E15"><i><font color="Green">Lemma10</font></i></a></i>;<br/>


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