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

<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">L</font> be    <a href="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>





<b>assume </b><a NAME="E1:21"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font>
 ;<br/>

<b>consider </b><font color="Maroon">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:21"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 ) );<br/>
<a NAME="E4:21"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
;<br/>
<a NAME="E5:21"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:21_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1"><b>suppose </b></a><a NAME="E1:21_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_2"><b>suppose </b></a><a NAME="E1:21_1_1_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1_1_2"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:21_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">L1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:21"/><i><font color="Green">E41</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<a NAME="E8:21"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/><font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:21_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:21_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_1"><b>suppose </b></a><a NAME="E1:21_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><a NAME="E2:21_2_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:21_2_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E4:21_2_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_2"><b>suppose </b></a><a NAME="E1:21_2_1_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><a NAME="E2:21_2_1_1_2"/><b>then </b>
<font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:21_2_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:21_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:21"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E10:21"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">L1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E12:21"/>
for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>  st not <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">L1</font> = <font color="Maroon">L1</font> as    <a href="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">V</font> <b>by </b><i>, <a class="ref" href="rlvect_2.html#D5">RLVECT_2:def 5</a></i>;<br/>
<a NAME="E14:21"/>
for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:21_3"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> ;<br/>
<a NAME="E3:21_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:21_3"/>
<font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, , <a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>;<br/>
<b>hence </b><a NAME="E5:21_3"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font>
 <b>by </b><i>, , <a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:21"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E16:21"/>
for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font> holds <br/><font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:21_4"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> ;<br/>
<a NAME="E3:21_4"/>
<font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>;<br/>
<b>hence </b><a NAME="E4:21_4"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E17:21"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">L1</font>
;<br/>

<b>thus </b><a NAME="E18:21"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
