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

<span class="kw">set </span><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="E1:113_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="113_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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>  or <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:113_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<a NAME="E2:113_1_1_1"/><span class="kw">then </span>
 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/>
<span class="kw">hence </span><a NAME="E3:113_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D16" target="_self" title="RLVECT_2:def.16">Def16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><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="E3:113_1_1"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
<font color="Maroon" title="c4">X</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="113_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:113_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#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">X</font>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="rlsub_1.html#D1" title="RLSUB_1:def.1">RLSUB_1:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> or <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; <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> implies <font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:113_1_1_2_1"/><span class="lab"><font color="Green" title="E48">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:113_1_1_2_1"/><span class="lab"><font color="Green" title="E49">A3</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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></span><br/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:113_1_1_2_1"/><span class="lab"><font color="Green" title="E50">A4</font></span>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">l1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:113_1_1_2_1"><span class="lab"><font color="Green" title="E48">A2</font></span></a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E6:113_1_1_2_1"/><span class="lab"><font color="Green" title="E51">A5</font></span>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">l2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:113_1_1_2_1"><span class="lab"><font color="Green" title="E49">A3</font></span></a></span>;<br/>
<a NAME="E7:113_1_1_2_1"/><span class="lab"><font color="Green" title="E52">A6</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:113_1_1_2_1"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="ref" href="rlvect_2.html#D1" target="_self" title="RLVECT_2:def.1">Def1</a>, <a class="ref" href="rlvect_1.html#T3" title="RLVECT_1:th.3">RLVECT_1:3</a></span>;<br/>
<a NAME="E8:113_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:113_1_1_2_1"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="ref" href="rlvect_2.html#D1" target="_self" title="RLVECT_2:def.1">Def1</a>, <a class="ref" href="rlvect_1.html#T3" title="RLVECT_1:th.3">RLVECT_1:3</a></span>;<br/>
<a NAME="E9:113_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:113_1_1_2_1"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="ref" href="rlvect_2.html#T96" target="_self" title="RLVECT_2:th.96">Th96</a></span>;<br/>
<a NAME="E10:113_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#T59" target="_self" title="RLVECT_2:th.59">Th59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:113_1_1_2_1"/>
<font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> or <font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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> or <font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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> )</span><br/>



<span class="kw">assume </span><a NAME="E2:113_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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></span><br/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:113_1_1_2"/><span class="lab"><font color="Green" title="E48">A7</font></span>: 
<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#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v</font> = 
<font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:113_1_1_2"><span class="lab"><font color="Green" title="E48">A7</font></span></a>, <a class="ref" href="rlvect_2.html#D1" target="_self" title="RLVECT_2:def.1">Def1</a>, <a class="ref" href="rlvect_1.html#T3" title="RLVECT_1:th.3">RLVECT_1:3</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#T97" target="_self" title="RLVECT_2:th.97">Th97</a></span>
;<br/>
<a NAME="E6:113_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#T67" target="_self" title="RLVECT_2:th.67">Th67</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:113_1_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:113_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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#T34" target="_self" title="RLVECT_2:th.34">Th34</a></span>;<br/>
<a NAME="E5:113_1_1"/><span class="kw">then </span>
 <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/>
<span class="kw">hence </span><a NAME="E6:113_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#V1" title="RLVECT_1:attr.1">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:113_1_1"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="ref" href="rlsub_1.html#T43" title="RLSUB_1:th.43">RLSUB_1:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
