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

<b>set </b><font color="Maroon" title="c3">A1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b1">l</font></span>)</span> 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="E1:16_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b1">l</font></span>)</span> 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 <font color="Maroon" title="c1">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_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="E1:16_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"> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b1">l</font></span>)</span> 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="E2:16_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>  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b1">l</font>
 
;<br/>
<b>hence </b><a NAME="E3:16_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 <font color="Maroon" title="c1">V</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">A1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b1">l</font></span>)</span> 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 <font color="Maroon" title="c1">V</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c5">l</font> =  <a href="rlvect_2.html#K4" title="RLVECT_2:func.4">ZeroLC</a> <font color="Maroon" title="c1">V</font> as    <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" title="RLVECT_2:th.34">RLVECT_2:34</a></i>;<br/>
<a NAME="E4:16_1_1"/>
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#T48" title="RLVECT_2:th.48">RLVECT_2:48</a></i>;<br/>
<a NAME="E5:16_1_1"/><b>then </b><i><font color="Green" title="E12">A1</font></i>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font>
 
;<br/>
<a NAME="E6:16_1_1"/>
<font color="Maroon" title="c4">A1</font> is <a href="rlsub_1.html#V1" title="RLSUB_1:attr.1">linearly-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:16_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 <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</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">A1</font> holds <br/><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</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="16_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">v</font>, <font color="Maroon" title="c7">u</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:16_1_1_2_1"/><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font>
 <b>and </b><br/><a NAME="E2:16_1_1_2_1"/><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">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="E4:16_1_1_2_1"/><i><font color="Green" title="E15">A4</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c8">l1</font>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E1:16_1_1_2_1"><i><font color="Green" title="E13">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">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="E6:16_1_1_2_1"/><i><font color="Green" title="E16">A5</font></i>: 
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c9">l2</font>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E2:16_1_1_2_1"><i><font color="Green" title="E14">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c8">l1</font> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <font color="Maroon" title="c9">l2</font> as    <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" title="RLVECT_2:th.59">RLVECT_2:59</a></i>;<br/>
<a NAME="E8:16_1_1_2_1"/>
<font color="Maroon" title="c6">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c10">f</font>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E4:16_1_1_2_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="rlvect_3.html#E6:16_1_1_2_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="rlvect_3.html#T1" target="_self" title="RLVECT_3:th.1">Th1</a></i>;<br/>
<b>hence </b><a NAME="E9:16_1_1_2_1"/>
<font color="Maroon" title="c6">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font>
 ;<br/>


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

<b>let </b><font color="Maroon" title="c6">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c7">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/>



<b>assume </b><a NAME="E2:16_1_1_2"/>
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">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="E4:16_1_1_2"/><i><font color="Green" title="E13">A6</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c8">l</font>
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c6">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c8">l</font> as    <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" title="RLVECT_2:th.67">RLVECT_2:67</a></i>;<br/>
<a NAME="E6:16_1_1_2"/>
<font color="Maroon" title="c6">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c9">f</font>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E4:16_1_1_2"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="rlvect_3.html#T2" target="_self" title="RLVECT_3:th.2">Th2</a></i>;<br/>
<b>hence </b><a NAME="E7:16_1_1_2"/>
<font color="Maroon" title="c6">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:16_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 <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"> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b2">l</font></span>)</span> 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_3.html#E5:16_1_1"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="rlsub_1.html#T43" title="RLSUB_1:th.43">RLSUB_1:43</a></i>;<br/>


</div>
