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<div class="add">

<b>let </b><font color="Maroon">V</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<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="E1:17"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">A</font> is <a href="rlvect_3.html#V1">linearly-independent</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B</font> &amp; <font color="Olive">B</font> is <a href="rlvect_3.html#V1">linearly-independent</a> );<br/>
<b>consider </b><font color="Maroon">Q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:17"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> iff ( <font color="Olive">Z</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</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">Z</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E4:17"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:17_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:17_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E3:17_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">Z</font> is <a href="ordinal1.html#V6">c=-linear</a>
 ;<br/><b>set </b><font color="Maroon">W</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font>;<br/><a NAME="E4:17_1"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</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="E3:17_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E4:17_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:17_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E6:17_1_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>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">W</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> ;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Z</font>;<br/><a NAME="E7:17_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E8:17_1"/><b>then </b><i><font color="Green">E64</font></i>: 
 ex <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B</font> &amp; <font color="Olive">B</font> is <a href="rlvect_3.html#V1">linearly-independent</a> )
 <b>by </b><i/>;<br/><a NAME="E9:17_1"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E10:17_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E11:17_1"/>
<font color="Maroon">W</font> is <a href="rlvect_3.html#V1">linearly-independent</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">l</font> be    <a href="rlvect_2.html#M2">Linear_Combination</a> of <font color="Maroon">W</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_3.html#D1">RLVECT_3:def 1</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:17_1_2"/><i><font color="Green">E66</font></i>: 
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E2:17_1_2"/><i><font color="Green">E67</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> : ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> ) </span>}</span> ;<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="E4:17_1_2"/><i><font color="Green">E68</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">l</font>
 <b>and </b><br/><a NAME="E5:17_1_2"/><i><font color="Green">E69</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>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">l</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon">M</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="rusub_3.html#D1" target="_self">Def1</a>, <a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<b>consider </b><font color="Maroon">F</font> being    <a href="orders_1.html#M1">Choice_Function</a> of <font color="Maroon">M</font>;<br/>
<div><i><font color="Green">E70</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1_2_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">M</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="E3:17_1_2_1"/><i><font color="Green">E71</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">f</font>
 <b>and </b><br/><a NAME="E4:17_1_2_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:17_1_2_1"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">l</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<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="E7:17_1_2_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E8:17_1_2_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="rusub_3.html#D1" target="_self">Def1</a>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i>, , <a class="txt" href="rusub_3.html#E3:17"><i><font color="Green">E28</font></i></a></i>;<br/><a NAME="E10:17_1_2_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> ) </span>}</span> 
 <b>by </b><i>, <a class="txt" href="rusub_3.html#E3:17"><i><font color="Green">E28</font></i></a></i>;<br/><b>hence </b><a NAME="E11:17_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="rusub_3.html#D1" target="_self">Def1</a>, , , </i>;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">S</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>;<br/>
<a NAME="E9:17_1_2"/><i><font color="Green">E75</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> &amp; <font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="txt" href="rusub_3.html#E1:17"><i><font color="Green">E24</font></i></a>, <a class="ref" href="rlvect_3.html#T35">RLVECT_3:35</a></i>;<br/>
<a NAME="E10:17_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<div><i><font color="Green">E84</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2_2"><b>now </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="E1:17_1_2_2"/>
<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>
 ;<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="E3:17_1_2_2"/><i><font color="Green">E85</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">F</font>
 <b>and </b><br/><a NAME="E4:17_1_2_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</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>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:17_1_2_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E7:17_1_2_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="rusub_3.html#E3:17"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:17_1_2_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="rusub_3.html#E1:17"><i><font color="Green">E24</font></i></a>, , <a class="txt" href="rusub_3.html#E3:17"><i><font color="Green">E28</font></i></a>, <a class="txt" href="rusub_3.html#E4:17"><i><font color="Green">E29</font></i></a>, <a class="ref" href="orders_1.html#D1">ORDERS_1:def 1</a></i>;<br/><a NAME="E9:17_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> : ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rusub_3.html#D1" target="_self">Def1</a>, , , <a class="txt" href="rusub_3.html#E1:17_1"><i><font color="Green">E31</font></i></a></i>;<br/><a NAME="E10:17_1_2_2"/><b>then </b>
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E11:17_1_2_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E90</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:17_1_2_3"/>
( <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> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> )
 ;<br/><a NAME="E2:17_1_2_3"/><b>then </b>
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i/>;<br/><a NAME="E3:17_1_2_3"/><b>then </b>
<font color="Maroon">X</font>,<font color="Maroon">Y</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 <b>by </b><i>, <a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a></i>;<br/><b>hence </b><a NAME="E4:17_1_2_3"/>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> or <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E13:17_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="rusub_3.html#D1" target="_self">Def1</a>, , <a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/>
<a NAME="E14:17_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/>
<a NAME="E15:17_1_2"/><b>then </b>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <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="txt" href="rusub_3.html#E3:17"><i><font color="Green">E28</font></i></a>, <a class="ref" href="card_2.html#T81">CARD_2:81</a></i>;<br/>
<a NAME="E16:17_1_2"/><b>then </b>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">B</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E18:17_1_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>and </b><br/><a NAME="E19:17_1_2"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E20:17_1_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">B</font> is <a href="rlvect_3.html#V1">linearly-independent</a>
 <b>by </b><i>, </i>;<br/>
<a NAME="E21:17_1_2"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">l</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_1_2_4"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">l</font>
 ;<br/>

<a NAME="E2:17_1_2_4"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="rusub_3.html#D1" target="_self">Def1</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">X</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>;<br/>
<a NAME="E3:17_1_2_4"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="rusub_3.html#E1:17"><i><font color="Green">E24</font></i></a>, , <a class="ref" href="orders_1.html#D1">ORDERS_1:def 1</a></i>;<br/>
<a NAME="E4:17_1_2_4"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> ) </span>}</span> 
 
<b>by </b><i>, <a class="txt" href="rusub_3.html#E1:17_1"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E5:17_1_2_4"/><b>then </b><i><font color="Green">E97</font></i>: 
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 
<b>by </b><i><a class="txt" href="rusub_3.html#E2:17_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E6:17_1_2_4"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <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#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:17_1_2_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:17_1_2"/><b>then </b>
<font color="Maroon">l</font> is    <a href="rlvect_2.html#M2">Linear_Combination</a> of <font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="rusub_3.html#E4:17"><i><font color="Green">E29</font></i></a>, <a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E23:17_1_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="rlvect_3.html#D1">RLVECT_3:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E12:17_1"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:17"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E8:17"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Olive">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">X</font> holds <br/> not <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font>
 <b>by </b><i>, <a class="ref" href="orders_1.html#T177">ORDERS_1:177</a></i>;<br/>
<b>consider </b><font color="Maroon">B</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E10:17"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E11:17"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E12:17"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">B</font> is <a href="rlvect_3.html#V1">linearly-independent</a>
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">B</font>
;<br/>

<b>thus </b><a NAME="E13:17"/>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> is <a href="rlvect_3.html#V1">linearly-independent</a> )
 <b>by </b><i>, </i>;<br/>

<a NAME="E14:17"/><i><font color="Green">E103</font></i>: 
 <a href="rusub_1.html#K2">(Omega).</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="bhsp_1.html#G1">UNITSTR</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font>,the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">V</font> #)
 
<b>by </b><i><a class="ref" href="rusub_1.html#D3">RUSUB_1:def 3</a></i>;<br/>
<b>assume </b><a NAME="E15:17"/>
 <a href="rusub_3.html#K1" target="_self">Lin</a> <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="bhsp_1.html#G1">UNITSTR</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font>,the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">V</font> #)
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E17:17"/><i><font color="Green">E104</font></i>: 
( ( <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="rusub_3.html#K1" target="_self">Lin</a> <font color="Maroon">B</font> &amp; not <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="bhsp_1.html#G1">UNITSTR</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font>,the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">V</font> #) ) or ( <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="bhsp_1.html#G1">UNITSTR</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font>,the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">V</font> #) &amp; not <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="rusub_3.html#K1" target="_self">Lin</a> <font color="Maroon">B</font> ) )
 <b>by </b><i>, <a class="ref" href="rusub_1.html#T25">RUSUB_1:25</a></i>;<br/>
<a NAME="E18:17"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> is <a href="rlvect_3.html#V1">linearly-independent</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">l</font> be    <a href="rlvect_2.html#M2">Linear_Combination</a> of <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_3.html#D1">RLVECT_3:def 1</a><br/>

<b>assume </b><a NAME="E1:17_2"/><i><font color="Green">E106</font></i>: 
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_2_1_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">l</font> or  not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">l</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_1"><b>suppose </b></a><a NAME="E1:17_2_1_1_1"/><i><font color="Green">E107</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">l</font>
 ;<br/><div class="add"><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:17_2_1_1_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E4:17_2_1_1_1"/><i><font color="Green">E109</font></i>: 
for <font color="Olive">u</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">u</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S6">FUNCT_2:sch 6</a>();<br/></i><b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.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#T11">FUNCT_2:11</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>assume </b><a NAME="E1:17_2_1_1_1_1"/>
not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E2:17_2_1_1_1_1"/><b>then </b>
(  not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">l</font> or <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:17_2_1_1_1_1"/><b>then </b>
( ( <font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> ) or <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:17_2_1_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</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="E8:17_2_1_1_1"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_2_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:17_2_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">u</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:17_2_1_1_1_2"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E5:17_2_1_1_1_2"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>
<a NAME="E6:17_2_1_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font>
 
<b>by </b><i>, , <a class="txt" href="rusub_3.html#E3:17"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E7:17_2_1_1_1_2"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v1</font> where <font color="Olive">v1</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 </span>}</span> 
 
<b>by </b><i><a class="txt" href="rusub_3.html#E3:17"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E8:17_2_1_1_1_2"/><b>then </b>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">l</font> &amp;  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">l</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/>
<a NAME="E9:17_2_1_1_1_2"/><b>then </b>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> or <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E10:17_2_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, , <a class="txt" href="rusub_3.html#E3:17"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="rlvect_2.html#M2">Linear_Combination</a> of <font color="Maroon">B</font> <b>by </b><i><a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = 0;<br/><b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E11:17_2_1_1_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span>
 <b>and </b><br/><a NAME="E12:17_2_1_1_1"/><i><font color="Green">E113</font></i>: 
for <font color="Olive">u</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> holds <br/><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">u</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S6">FUNCT_2:sch 6</a>();<br/></i><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.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#T11">FUNCT_2:11</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>assume </b><a NAME="E1:17_2_1_1_1_3"/>
not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E2:17_2_1_1_1_3"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:17_2_1_1_1_3"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="rusub_3.html#E3:17"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</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="E16:17_2_1_1_1"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_2_1_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:17_2_1_1_1_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 </span>}</span> 
 
;<br/>
<a NAME="E3:17_2_1_1_1_4"/><b>then </b>
 ex <font color="Olive">u</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">u</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
;<br/>
<a NAME="E4:17_2_1_1_1_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="txt" href="rusub_3.html#E3:17"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:17_2_1_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as    <a href="rlvect_2.html#M2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> <b>by </b><i><a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/><a NAME="E18:17_2_1_1_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">f</font> <a href="rlvect_2.html#K10">-</a> <font color="Maroon">g</font> <a href="rlvect_2.html#R1">=</a> <font color="Maroon">l</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_2.html#D11">RLVECT_2:def 11</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_2_1_1_1_5_1_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> or <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">u</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:17_2_1_1_1_5_1_1_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 ;<br/><div class="add"><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rlvect_2.html#K10">-</a> <font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_2.html#T79">RLVECT_2:79</a></i>
<br/>.= 
<font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font>
<b>by </b><i>, , </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:17_2_1_1_1_5_1_1_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">u</font>
 ;<br/><div class="add"><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rlvect_2.html#K10">-</a> <font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_2.html#T79">RLVECT_2:79</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span> <a href="real_1.html#K5">-</a> 0
<b>by </b><i><a class="txt" href="rusub_3.html#E3:17"><i><font color="Green">E28</font></i></a>, </i>
<br/>.= 
<font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font>

;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:17_2_1_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rlvect_2.html#K10">-</a> <font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E19:17_2_1_1_1"/><i><font color="Green">E117</font></i>: 
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="rlvect_2.html#T50">RLVECT_2:50</a></i>;<br/><a NAME="E20:17_2_1_1_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">f</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="rusub_3.html#E4:17"><i><font color="Green">E29</font></i></a>, <a class="ref" href="rlvect_3.html#T4">RLVECT_3:4</a></i>;<br/><b>then </b> <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="ref" href="rlsub_2.html#T78">RLSUB_2:78</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><a NAME="E22:17_2_1_1_1"/><b>then </b><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="rusub_3.html#K1" target="_self">Lin</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/><a NAME="E23:17_2_1_1_1"/>
<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="txt" href="rusub_3.html#E1:17"><i><font color="Green">E24</font></i></a>, <a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>;<br/><a NAME="E24:17_2_1_1_1"/><b>then </b><i><font color="Green">E119</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font>
<b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/><b>hence </b><a NAME="E26:17_2_1_1_1"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , <a class="ref" href="rlvect_1.html#T3">RLVECT_1:3</a>, <a class="ref" href="rusub_1.html#T15">RUSUB_1:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_2"><b>suppose </b></a><a NAME="E1:17_2_1_1_2"/><i><font color="Green">E120</font></i>: 
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">l</font>
 ;<br/><div class="add"><a NAME="E2:17_2_1_1_2"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">l</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_2_1_1_2_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">l</font>
 ;<br/>

<a NAME="E2:17_2_1_1_2_1"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">l</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/>
<a NAME="E3:17_2_1_1_2_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:17_2_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="rusub_3.html#E1:17"><i><font color="Green">E24</font></i></a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:17_2_1_1_2"/><b>then </b>
<font color="Maroon">l</font> is    <a href="rlvect_2.html#M2">Linear_Combination</a> of <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/><b>hence </b><a NAME="E4:17_2_1_1_2"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , <a class="ref" href="rlvect_3.html#D1">RLVECT_3:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:17_2"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E19:17"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E20:17"/><b>then </b><i><font color="Green">E122</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> &amp; not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#T3">RLVECT_1:3</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E21:17"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E22:17"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E23:17"/><b>then </b>
<font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 
<b>by </b><i>, <a class="ref" href="rusub_3.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E24:17"/>
contradiction
 <b>by </b><i>, , , <a class="txt" href="rusub_3.html#E1:17"><i><font color="Green">E24</font></i></a></i>;<br/>


</div>
