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

<b>let </b><font color="Maroon">R</font> be   <a href="vectsp_2.html#NM3">Skew-Field</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">R</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/>





<a NAME="E1:25"/><i><font color="Green">E32</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/>
<b>assume </b><a NAME="E2:25"/><i><font color="Green">E33</font></i>: 
 <a href="mod_3.html#K1" target="_self">Lin</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font>
 ;<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="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Olive">B</font> is <a href="lmod_5.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="E4:25"/><i><font color="Green">E34</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="E5:25"/>
(  <a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</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;  <a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp;  <a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> is <a href="lmod_5.html#V1">linearly-independent</a> )
 
<b>by </b><i><a class="ref" href="lmod_5.html#T4">LMOD_5:4</a>, <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E6:25"/><b>then </b><i><font color="Green">E35</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/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_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="E1:25_1"/>
<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="E2:25_1"/><i><font color="Green">E38</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="E3:25_1"/><i><font color="Green">E39</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="E4:25_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="25_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="E1:25_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="E3:25_1_1"/><i><font color="Green">E40</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="E4:25_1_1"/><i><font color="Green">E42</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="E5:25_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="E6:25_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/><a NAME="E6:25_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_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="E1:25_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</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="E3:25_1_2"/><i><font color="Green">E44</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="E4:25_1_2"/><i><font color="Green">E45</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="E5:25_1_2"/>
 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="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Olive">B</font> is <a href="lmod_5.html#V1">linearly-independent</a> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E6:25_1_2"/>
<font color="Maroon">x</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="E7:25_1"/>
<font color="Maroon">W</font> is <a href="lmod_5.html#V1">linearly-independent</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">l</font> be    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">W</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lmod_5.html#D1">LMOD_5:def 1</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:25_1_3"/><i><font color="Green">E46</font></i>: 
 <a href="vectsp_6.html#K4">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="E2:25_1_3"/><i><font color="Green">E47</font></i>: 
 <a href="vectsp_6.html#K1">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="E4:25_1_3"/><i><font color="Green">E48</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l</font>
 <b>and </b><br/><a NAME="E5:25_1_3"/><i><font color="Green">E70</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="vectsp_6.html#K1">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="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">E71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1_3_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="E3:25_1_3_1"/><i><font color="Green">E72</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="E4:25_1_3_1"/><i><font color="Green">E73</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="E5:25_1_3_1"/>
 <a href="vectsp_6.html#K1">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="vectsp_6.html#D7">VECTSP_6:def 7</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="E7:25_1_3_1"/><i><font color="Green">E90</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="E8:25_1_3_1"/><i><font color="Green">E92</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/><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="mod_3.html#E2:25"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E10:25_1_3_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="mod_3.html#E1:25"><i><font color="Green">E32</font></i></a>, <a class="txt" href="mod_3.html#E2:25"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E11:25_1_3_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="mod_3.html#T6" target="_self">Th6</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="E9:25_1_3"/><i><font color="Green">E93</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="ref" href="rlvect_3.html#T35">RLVECT_3:35</a></i>;<br/>
<a NAME="E10:25_1_3"/><b>then </b><i><font color="Green">E94</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">E95</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_3_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="E1:25_1_3_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="E3:25_1_3_2"/><i><font color="Green">E96</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="E4:25_1_3_2"/><i><font color="Green">E97</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="E6:25_1_3_2"/><i><font color="Green">E98</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="E7:25_1_3_2"/><i><font color="Green">E99</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="mod_3.html#E2:25"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:25_1_3_2"/><i><font color="Green">E100</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="mod_3.html#E2:25"><i><font color="Green">E33</font></i></a>, <a class="txt" href="mod_3.html#E4:25"><i><font color="Green">E34</font></i></a>, <a class="ref" href="orders_1.html#D1">ORDERS_1:def 1</a></i>;<br/><a NAME="E9:25_1_3_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="txt" href="mod_3.html#E6:25"><i><font color="Green">E35</font></i></a>, </i>;<br/><a NAME="E10:25_1_3_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="E11:25_1_3_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">E101</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_3_3"><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>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:25_1_3_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="E2:25_1_3_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><a class="txt" href="mod_3.html#E1:25"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E3:25_1_3_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="E4:25_1_3_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="E13:25_1_3"/>
 <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="finset_1.html#T26">FINSET_1:26</a></i>;<br/>
<a NAME="E14:25_1_3"/><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="E15:25_1_3"/><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="ref" href="mod_3.html#T6" target="_self">Th6</a>, <a class="txt" href="mod_3.html#E2:25"><i><font color="Green">E33</font></i></a>, <a class="ref" href="card_2.html#T81">CARD_2:81</a></i>;<br/>
<a NAME="E16:25_1_3"/><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><a class="txt" href="mod_3.html#E1:25"><i><font color="Green">E32</font></i></a></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="E18:25_1_3"/><i><font color="Green">E102</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="E19:25_1_3"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E20:25_1_3"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">B</font> is <a href="lmod_5.html#V1">linearly-independent</a>
 <b>by </b><i>, </i>;<br/>
<a NAME="E21:25_1_3"/>
 <a href="vectsp_6.html#K1">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="25_1_3_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="E1:25_1_3_4"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l</font>
 ;<br/>

<a NAME="E2:25_1_3_4"/><b>then </b><i><font color="Green">E105</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="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="E3:25_1_3_4"/><i><font color="Green">E106</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="ref" href="orders_1.html#D1">ORDERS_1:def 1</a></i>;<br/>
<a NAME="E4:25_1_3_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>, </i>;<br/>
<a NAME="E5:25_1_3_4"/><b>then </b><i><font color="Green">E107</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="mod_3.html#E2:25_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E6:25_1_3_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="E7:25_1_3_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="txt" href="mod_3.html#E3:25_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:25_1_3"/><b>then </b>
<font color="Maroon">l</font> is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="mod_3.html#E4:25"><i><font color="Green">E34</font></i></a>, <a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/>
<b>hence </b><a NAME="E23:25_1_3"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="mod_3.html#E6:25"><i><font color="Green">E35</font></i></a>, <a class="ref" href="lmod_5.html#D1">LMOD_5:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:25_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="E9:25"/><i><font color="Green">E108</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="E10:25"/><i><font color="Green">E109</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="mod_3.html#T2" target="_self">Th2</a>, <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="E12:25"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E13:25"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E14:25"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">B</font> is <a href="lmod_5.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="E15:25"/>
( <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">B</font> is <a href="lmod_5.html#V1">linearly-independent</a> )
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="mod_3.html#D2" target="_self">MOD_3:def 2</a><br/>

<b>assume </b><a NAME="E16:25"/><i><font color="Green">E113</font></i>: 
 <a href="mod_3.html#K1" target="_self">Lin</a> <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</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="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font> #)
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_2"/><i><font color="Green">E114</font></i>: 
for <font color="Olive">v</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>  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="rlvect_1.html#R1">in</a>  <a href="mod_3.html#K1" target="_self">Lin</a> <font color="Maroon">B</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/><b>assume </b><a NAME="E1:25_2_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="mod_3.html#K1" target="_self">Lin</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">l</font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:25_2_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">l</font>
 <b>by </b><i/>;<br/><a NAME="E4:25_2_1"/><i><font color="Green">E116</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="mod_3.html#K1" target="_self">Lin</a> <font color="Maroon">B</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_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="E1:25_2_1_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> ;<br/>
<a NAME="E3:25_2_1_1"/>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/>
<a NAME="E4:25_2_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a>  <a href="mod_3.html#K1" target="_self">Lin</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="mod_3.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E5:25_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="mod_3.html#K1" target="_self">Lin</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">F</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="mod_3.html#K1" target="_self">Lin</a> <font color="Maroon">B</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">l</font> as    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">F</font> <b>by </b><i>, <a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/><a NAME="E7:25_2_1"/>
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i/>;<br/><a NAME="E8:25_2_1"/><b>then </b>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="mod_3.html#K1" target="_self">Lin</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:25_2_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="mod_3.html#K1" target="_self">Lin</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="mod_3.html#E1"><i><font color="Green">Lemma18</font></i></a>, </i>;<br/></div><b>end;</b></div><a NAME="E3:25_2"/><b>then </b>
 <a href="mod_3.html#K1" target="_self">Lin</a> <font color="Maroon">A</font> is    <a href="vectsp_4.html#M1">Subspace</a> of  <a href="mod_3.html#K1" target="_self">Lin</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T36">VECTSP_4:36</a></i>;<br/><b>hence </b><a NAME="E4:25_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="vectsp_4.html#T33">VECTSP_4:33</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">v</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E19:25"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E20:25"/><i><font color="Green">E119</font></i>: 
not <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="mod_3.html#K1" target="_self">Lin</a> <font color="Maroon">B</font>
 ;<br/>
<a NAME="E21:25"/><i><font color="Green">E120</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="lmod_5.html#V1">linearly-independent</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">l</font> be    <a href="vectsp_6.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="lmod_5.html#D1">LMOD_5:def 1</a><br/>

<b>assume </b><a NAME="E1:25_3"/><i><font color="Green">E121</font></i>: 
 <a href="vectsp_6.html#K4">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="25_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_3_1_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l</font> or  not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_3_1_1_1"><b>suppose </b></a><a NAME="E1:25_3_1_1_1"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l</font>
 ;<br/><div class="add"><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> = <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>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:25_3_1_1_1"/><i><font color="Green">E123</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>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E4:25_3_1_1_1"/><i><font color="Green">E124</font></i>: 
for <font color="Olive">u</font> being  <a href="vectsp_1.html#NM5">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>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <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="25_3_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/><b>assume </b><a NAME="E1:25_3_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="vectsp_6.html#K1">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="E2:25_3_1_1_1_1"/><b>then </b>
(  not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">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="E3:25_3_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>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> &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="E4:25_3_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>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="txt" href="mod_3.html#E2:25"><i><font color="Green">E33</font></i></a>, <a class="txt" href="mod_3.html#E4:25"><i><font color="Green">E34</font></i></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="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="vectsp_6.html#D4">VECTSP_6:def 4</a></i>;<br/><a NAME="E8:25_3_1_1_1"/>
 <a href="vectsp_6.html#K1">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="25_3_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="E1:25_3_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:25_3_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="vectsp_1.html#NM5">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>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">u</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:25_3_1_1_1_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E5:25_3_1_1_1_2"/><i><font color="Green">E126</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>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 ;<br/>
<a NAME="E6:25_3_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="mod_3.html#E2:25"><i><font color="Green">E33</font></i></a>, <a class="txt" href="mod_3.html#E4:25"><i><font color="Green">E34</font></i></a>, </i>;<br/>
<a NAME="E7:25_3_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="vectsp_1.html#NM5">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>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E8:25_3_1_1_1_2"/><b>then </b>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l</font> &amp;  <a href="vectsp_6.html#K1">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="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/>
<a NAME="E9:25_3_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="E10:25_3_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="mod_3.html#E2:25"><i><font color="Green">E33</font></i></a>, <a class="txt" href="mod_3.html#E6:25"><i><font color="Green">E35</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="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font> <b>by </b><i><a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> =  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>;<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>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E11:25_3_1_1_1"/><i><font color="Green">E127</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="rlvect_1.html#K5">-</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="E12:25_3_1_1_1"/><i><font color="Green">E128</font></i>: 
for <font color="Olive">u</font> being  <a href="vectsp_1.html#NM5">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>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <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="25_3_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/><b>assume </b><a NAME="E1:25_3_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="E2:25_3_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="E3:25_3_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>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="vectsp_6.html#D4">VECTSP_6:def 4</a></i>;<br/><a NAME="E16:25_3_1_1_1"/>
 <a href="vectsp_6.html#K1">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="25_3_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="E1:25_3_1_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:25_3_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="vectsp_1.html#NM5">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>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> </span>}</span> 
 
;<br/>
<a NAME="E3:25_3_1_1_1_4"/><b>then </b>
 ex <font color="Olive">u</font> being  <a href="vectsp_1.html#NM5">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>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> )
 
;<br/>
<a NAME="E4:25_3_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/>;<br/>
<b>hence </b><a NAME="E5:25_3_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="vectsp_6.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="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/><a NAME="E18:25_3_1_1_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">f</font> <a href="vectsp_6.html#K8">-</a> <font color="Maroon">g</font> <a href="vectsp_6.html#R1">=</a> <font color="Maroon">l</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3_1_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_6.html#D10">VECTSP_6:def 10</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3_1_1_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_3_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="E1:25_3_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="25_3_1_1_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:25_3_1_1_1_5_1_1_1"/><i><font color="Green">E130</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="vectsp_6.html#K8">-</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="rlvect_1.html#K6">-</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="vectsp_6.html#T73">VECTSP_6:73</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</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>
<b>by </b><i><a class="txt" href="mod_3.html#E2:25"><i><font color="Green">E33</font></i></a>, <a class="txt" href="mod_3.html#E6:25"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
<br/>.= 
<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="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T30">RLVECT_1:30</a></i>
<br/>.= 
<font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_3_1_1_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:25_3_1_1_1_5_1_1_2"/><i><font color="Green">E132</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="vectsp_6.html#K8">-</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="rlvect_1.html#K6">-</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="vectsp_6.html#T73">VECTSP_6:73</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="rlvect_1.html#K6">-</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="txt" href="mod_3.html#E4:25"><i><font color="Green">E34</font></i></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="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:25_3_1_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="vectsp_6.html#K8">-</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="E19:25_3_1_1_1"/><i><font color="Green">E133</font></i>: 
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</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="vectsp_1.html#K6">*</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="txt" href="mod_3.html#E6:25"><i><font color="Green">E35</font></i></a>, <a class="ref" href="vectsp_6.html#T43">VECTSP_6:43</a></i>;<br/><a NAME="E20:25_3_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="vectsp_6.html#K4">Sum</a> <font color="Maroon">f</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="mod_3.html#T6" target="_self">Th6</a>, , <a class="ref" href="vectsp_6.html#T80">VECTSP_6:80</a></i>;<br/><a NAME="E21:25_3_1_1_1"/><b>then </b>
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</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="vectsp_1.html#K6">*</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="vectsp_1.html#T66">VECTSP_1:66</a></i>;<br/><a NAME="E22:25_3_1_1_1"/><b>then </b><i><font color="Green">E134</font></i>: 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</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="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="mod_3.html#K1" target="_self">Lin</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/><a NAME="E23:25_3_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>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="txt" href="mod_3.html#E1:25"><i><font color="Green">E32</font></i></a>, <a class="ref" href="vectsp_6.html#T20">VECTSP_6:20</a></i>;<br/><a NAME="E24:25_3_1_1_1"/><b>then </b>
 <a href="rlvect_1.html#K5">-</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>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</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="vectsp_2.html#K1">"</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</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="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
;<br/><b>hence </b><a NAME="E26:25_3_1_1_1"/>
 <a href="vectsp_6.html#K1">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="vectsp_4.html#T29">VECTSP_4:29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_3_1_1_2"><b>suppose </b></a><a NAME="E1:25_3_1_1_2"/><i><font color="Green">E135</font></i>: 
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l</font>
 ;<br/><div class="add"><a NAME="E2:25_3_1_1_2"/>
 <a href="vectsp_6.html#K1">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="25_3_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="E1:25_3_1_1_2_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l</font>
 ;<br/>

<a NAME="E2:25_3_1_1_2_1"/>
 <a href="vectsp_6.html#K1">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="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/>
<a NAME="E3:25_3_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="txt" href="mod_3.html#E2:25"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:25_3_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="mod_3.html#E1:25"><i><font color="Green">E32</font></i></a>, <a class="txt" href="mod_3.html#E2:25"><i><font color="Green">E33</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:25_3_1_1_2"/><b>then </b>
<font color="Maroon">l</font> is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/><b>hence </b><a NAME="E4:25_3_1_1_2"/>
 <a href="vectsp_6.html#K1">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="mod_3.html#T6" target="_self">Th6</a>, <a class="ref" href="lmod_5.html#D1">LMOD_5: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="E3:25_3"/>
 <a href="vectsp_6.html#K1">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="E22:25"/>
<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="E23:25"/><b>then </b><i><font color="Green">E137</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="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E24:25"/><i><font color="Green">E138</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="E25:25"/>
<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="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E26:25"/><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="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E27:25"/><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>, </i>;<br/>
<b>hence </b><a NAME="E28:25"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="mod_3.html#T6" target="_self">Th6</a>, <a class="txt" href="mod_3.html#E1:25"><i><font color="Green">E32</font></i></a></i>;<br/>


</div>
