<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="rlvect_2.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">A</font> : verum </span>}</span> ;<br/>
<a NAME="E1:83_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="rlvect_2.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">A</font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" target="_self">LC_RLSpace</a> <font color="Maroon">V</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:83_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="rlvect_2.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">A</font> : verum </span>}</span> 
 ;<br/>

<a NAME="E2:83_1_1_1"/><b>then </b>
 ex <font color="Olive">l</font> being   <a href="rlvect_2.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">A</font> st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">l</font>
 
;<br/>
<a NAME="E3:83_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="txt" href="rlvect_2.html#E4:83_1_1"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:83_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" target="_self">LC_RLSpace</a> <font color="Maroon">V</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="rlvect_2.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">A</font> : verum </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" target="_self">LC_RLSpace</a> <font color="Maroon">V</font></span>)</span> ;<br/>
<a NAME="E3:83_1_1"/>
 <a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font> is    <a href="rlvect_2.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:83_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
;<br/>
<a NAME="E5:83_1_1"/>
<font color="Maroon">X</font> is <a href="rlsub_1.html#V1">lineary-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:83_1_1_2"/>
for <font color="Olive">v</font>, <font color="Olive">u</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" target="_self">LC_RLSpace</a> <font color="Maroon">V</font></span>)</span>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</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">X</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlsub_1.html#D1">RLSUB_1:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_2_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 <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" target="_self">LC_RLSpace</a> <font color="Maroon">V</font></span>)</span>, <font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" target="_self">LC_RLSpace</a> <font color="Maroon">V</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:83_1_1_2_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E2:83_1_1_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>consider </b><font color="Maroon">l1</font> being    <a href="rlvect_2.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:83_1_1_2_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l1</font>
 <b>by </b><i><a class="txt" href="rlvect_2.html#E2:83_1_1_2_1"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">l2</font> being    <a href="rlvect_2.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E6:83_1_1_2_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l2</font>
 <b>by </b><i><a class="txt" href="rlvect_2.html#E4:83_1_1_2_1"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E7:83_1_1_2_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K1" target="_self">vector</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" target="_self">LC_RLSpace</a> <font color="Maroon">V</font></span>)</span>,<font color="Maroon">l1</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K1" target="_self">vector</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" target="_self">LC_RLSpace</a> <font color="Maroon">V</font></span>)</span>,<font color="Maroon">l2</font> )
 
<b>by </b><i><a class="txt" href="rlvect_2.html#E6:83_1_1_2_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="rlvect_2.html#E4:83_1_1_2"><i><font color="Green">E56</font></i></a>, </i>;<br/>
<a NAME="E8:83_1_1_2_1"/><b>then </b>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l1</font> <a href="rlvect_2.html#K7" target="_self">+</a> <font color="Maroon">l2</font>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E9:83_1_1_2_1"/><b>then </b>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font> is    <a href="rlvect_2.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:83_1_1_2_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">X</font>
 ;<br/>


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

<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 <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" target="_self">LC_RLSpace</a> <font color="Maroon">V</font></span>)</span>;<br/>



<b>assume </b><a NAME="E2:83_1_1_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">l</font> being    <a href="rlvect_2.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:83_1_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font>
 ;<br/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K1" target="_self">vector</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K16" target="_self">LC_RLSpace</a> <font color="Maroon">V</font></span>)</span>,<font color="Maroon">l</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">a</font> <a href="rlvect_2.html#K8" target="_self">*</a> <font color="Maroon">l</font>
<b>by </b><i/>
;<br/>
<a NAME="E6:83_1_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> is    <a href="rlvect_2.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:83_1_1_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">X</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:83_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1">Subspace</a> of  <a href="rlvect_2.html#K16" target="_self">LC_RLSpace</a> <font color="Maroon">V</font> st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="rlvect_2.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">A</font> : verum </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rlsub_1.html#T43">RLSUB_1:43</a></i>;<br/>


</div>
