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

<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:9"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="rlvect_3.html#V1" title="RLVECT_3:attr.1">linearly-independent</a>
 <b>and </b><br/><a NAME="E2:9"/><i><font color="Green" title="E9">A2</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="c2">A</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>) <b>-&gt; </b>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<b>consider </b><font color="Maroon" title="c3">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E4:9"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>and </b><br/><a NAME="E5:9"/><i><font color="Green" title="E11">A4</font></i>: 
for <font color="Olive" title="b1">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">v</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S6" title="FUNCT_2:sch.6">FUNCT_2:sch 6</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c3">f</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<a NAME="E7:9"/>
 ex <font color="Olive" title="b1">T</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> st <br/>for <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st not <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">T</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c5">T</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<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 <font color="Maroon" title="c1">V</font>;<br/>

<b>assume </b><a NAME="E1:9_1"/>
not <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">T</font>
 ;<br/>

<a NAME="E2:9_1"/><b>then </b>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:9_1"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="rlvect_3.html#E5:9"><i><font color="Green" title="E11">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c4">f</font> as    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rlvect_2.html#D5" title="RLVECT_2:def.5">RLVECT_2:def 5</a></i>;<br/>
<a NAME="E9:9"/><i><font color="Green" title="E12">A5</font></i>: 
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:9_2"/>
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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="E1:9_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:9_2_1"/><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> <font color="Maroon" title="c6">x</font>
 <b>and </b><br/><a NAME="E4:9_2_1"/><i><font color="Green" title="E14">A7</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>
<a NAME="E5:9_2_1"/>
<font color="Maroon" title="c7">v</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="txt" href="rlvect_3.html#E5:9"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="rlvect_3.html#E4:9_2_1"><i><font color="Green" title="E14">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:9_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E3:9_2_1"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c6">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="E2:9_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<a NAME="E3:9_2"/><b>then </b>
( <font color="Maroon" title="c6">x</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> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:9_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="txt" href="rlvect_3.html#E4:9"><i><font color="Green" title="E10">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:9"/><b>then </b>
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="rlvect_3.html#E2:9"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c5">f</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#D8" title="RLVECT_2:def.8">RLVECT_2:def 8</a></i>;<br/>
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c6">f</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_3.html#E9:9"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="rlvect_2.html#T53" title="RLVECT_2:th.53">RLVECT_2:53</a></i>
<br/>.= 
 <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_1.html#T23" title="RLVECT_1:th.23">RLVECT_1:23</a></i>
;<br/>
<b>hence </b><a NAME="E13:9"/>
contradiction
 <b>by </b><i><a class="txt" href="rlvect_3.html#E1:9"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="rlvect_3.html#E9:9"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="rlvect_3.html#D1" target="_self" title="RLVECT_3:def.1">Def1</a></i>;<br/>


</div>
