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<b>let </b><font color="Maroon">V</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="rusub_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">V1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>, <font color="Maroon">V2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>





<b>assume </b><a NAME="E1:14"/><i><font color="Green">E19</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V1</font>
 ;<br/>

<b>set </b><font color="Maroon">VW</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>;<br/>
<b>reconsider </b><font color="Maroon">WW</font> = <font color="Maroon">W</font> as   <a href="bhsp_1.html#NM1">RealUnitarySpace</a> ;<br/>
<a NAME="E3:14"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">v</font>, <font color="Olive">u</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> holds <br/><font color="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>

<b>assume </b><a NAME="E1:14_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">vv</font> = <font color="Maroon">v</font>, <font color="Maroon">uu</font> = <font color="Maroon">u</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">WW</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">vw</font> = <font color="Maroon">vv</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">uu</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> ;<br/>
<a NAME="E4:14_1"/>
<font color="Maroon">vw</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:14_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:14"/>
for <font color="Olive">a</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> holds <br/><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>



<b>assume </b><a NAME="E1:14_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">vv</font> = <font color="Maroon">v</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">WW</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">vw</font> = <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">vv</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> ;<br/>
<a NAME="E4:14_2"/>
<font color="Maroon">vw</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:14_2"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:14"/>
<font color="Maroon">V1</font> is <a href="rlsub_1.html#V1">lineary-closed</a>
 <b>by </b><i>, <a class="ref" href="rlsub_1.html#D1">RLSUB_1:def 1</a></i>;<br/>


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