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

<b>set </b><font color="Maroon" title="c3">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c2">A</font> : verum </span>}</span> ;<br/>
<a NAME="E1:83_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c2">A</font> : verum </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">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" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:83_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c2">A</font> : verum </span>}</span> 
 ;<br/>

<a NAME="E2:83_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">l</font> being   <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c2">A</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">l</font>
 
;<br/>
<a NAME="E3:83_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K11" title="RLVECT_2:func.11">LinComb</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#D16" target="_self" title="RLVECT_2:def.16">Def16</a></i>;<br/>
<b>hence </b><a NAME="E4:83_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c2">A</font> : verum </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span> ;<br/>
<a NAME="E3:83_1_1"/>
 <a href="rlvect_2.html#K4" title="RLVECT_2:func.4">ZeroLC</a> <font color="Maroon" title="c1">V</font> is    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#T34" target="_self" title="RLVECT_2:th.34">Th34</a></i>;<br/>
<a NAME="E4:83_1_1"/><b>then </b><i><font color="Green" title="E52">A1</font></i>: 
 <a href="rlvect_2.html#K4" title="RLVECT_2:func.4">ZeroLC</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 
;<br/>
<a NAME="E5:83_1_1"/>
<font color="Maroon" title="c4">X</font> is <a href="rlsub_1.html#V1" title="RLSUB_1:attr.1">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" title="b1">v</font>, <font color="Olive" title="b2">u</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> holds <br/><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlsub_1.html#D1" title="RLSUB_1:def.1">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" title="c5">v</font>, <font color="Maroon" title="c6">u</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:83_1_1_2_1"/><i><font color="Green" title="E53">A2</font></i>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>and </b><br/><a NAME="E2:83_1_1_2_1"/><i><font color="Green" title="E54">A3</font></i>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">l1</font> being    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E4:83_1_1_2_1"/><i><font color="Green" title="E55">A4</font></i>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">l1</font>
 <b>by </b><i><a class="txt" href="rlvect_2.html#E1:83_1_1_2_1"><i><font color="Green" title="E53">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">l2</font> being    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E6:83_1_1_2_1"/><i><font color="Green" title="E56">A5</font></i>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">l2</font>
 <b>by </b><i><a class="txt" href="rlvect_2.html#E2:83_1_1_2_1"><i><font color="Green" title="E54">A3</font></i></a></i>;<br/>
<a NAME="E7:83_1_1_2_1"/>
( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K1" title="RLVECT_2:func.1">vector</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c7">l1</font> &amp; <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K1" title="RLVECT_2:func.1">vector</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c8">l2</font> )
 
<b>by </b><i><a class="txt" href="rlvect_2.html#E4:83_1_1_2_1"><i><font color="Green" title="E55">A4</font></i></a>, <a class="txt" href="rlvect_2.html#E6:83_1_1_2_1"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="rlvect_2.html#T3" target="_self" title="RLVECT_2:th.3">Th3</a></i>;<br/>
<a NAME="E8:83_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">l1</font> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <font color="Maroon" title="c8">l2</font>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#T96" target="_self" title="RLVECT_2:th.96">Th96</a></i>;<br/>
<a NAME="E9:83_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c6">u</font> is    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#T59" target="_self" title="RLVECT_2:th.59">Th59</a></i>;<br/>
<b>hence </b><a NAME="E10:83_1_1_2_1"/>
<font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ;<br/>


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

<b>let </b><font color="Maroon" title="c5">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c6">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>;<br/>



<b>assume </b><a NAME="E2:83_1_1_2"/>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">l</font> being    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E4:83_1_1_2"/><i><font color="Green" title="E53">A6</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">l</font>
 ;<br/>
<font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c6">v</font> = 
<font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K1" title="RLVECT_2:func.1">vector</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>,<font color="Maroon" title="c7">l</font></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_2.html#E4:83_1_1_2"><i><font color="Green" title="E53">A6</font></i></a>, <a class="ref" href="rlvect_2.html#T3" target="_self" title="RLVECT_2:th.3">Th3</a></i>
<br/>.= 
<font color="Maroon" title="c5">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c7">l</font>
<b>by </b><i><a class="ref" href="rlvect_2.html#T97" target="_self" title="RLVECT_2:th.97">Th97</a></i>
;<br/>
<a NAME="E6:83_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c6">v</font> is    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#T67" target="_self" title="RLVECT_2:th.67">Th67</a></i>;<br/>
<b>hence </b><a NAME="E7:83_1_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">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" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of  <a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font> st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">l</font> where <font color="Olive" title="b2">l</font> is    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c2">A</font> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="rlvect_2.html#E4:83_1_1"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="rlsub_1.html#T43" title="RLSUB_1:th.43">RLSUB_1:43</a></i>;<br/>


</div>
