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

<b>let </b><font color="Maroon">GF</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">GF</font>;<br/>



<b>assume </b><a NAME="E1:23"/>
<font color="Maroon">V</font> is <a href="matrlin.html#V2">finite-dimensional</a>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:23"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">A</font> is    <a href="vectsp_7.html#M1">Basis</a> of <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">B</font> be    <a href="vectsp_7.html#M1">Basis</a> of <font color="Maroon">V</font>;<br/>

<b>consider </b><font color="Maroon">p</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E5:23"/><i><font color="Green">E40</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T73">FINSEQ_1:73</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <b>by </b><i>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">Car</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Olive">L</font></span>)</span> where <font color="Olive">L</font> is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font> :  ex <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">C</font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Olive">L</font></span>)</span> where <font color="Olive">L</font> is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font> :  ex <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) </span>}</span> ;<br/>
<a NAME="E7:23"/><i><font color="Green">E42</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Olive">L</font></span>)</span> where <font color="Olive">L</font> is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font> :  ex <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_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="E1:23_1"/>
<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"> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Olive">L</font></span>)</span> where <font color="Olive">L</font> is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font> :  ex <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">R</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:23_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E4:23_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Olive">L</font></span>)</span> where <font color="Olive">L</font> is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font> :  ex <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">L</font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E6:23_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">R</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="E7:23_1"/>
 ex <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> )
 <b>by </b><i/>;<br/>
<a NAME="E8:23_1"/>
<font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <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="E9:23_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">C</font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Olive">L</font></span>)</span> where <font color="Olive">L</font> is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font> :  ex <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</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="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:23"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">B</font> is <a href="vectsp_7.html#V1">linearly-independent</a>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#D3">VECTSP_7:def 3</a></i>;<br/>
<a NAME="E10:23"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">C</font> is <a href="vectsp_7.html#V1">linearly-independent</a>
 
<b>by </b><i>, <a class="ref" href="vectsp_7.html#T2">VECTSP_7:2</a></i>;<br/>
<a NAME="E11:23"/>
for <font color="Olive">v</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> holds <br/> ( <font color="Olive">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">V</font> iff <font color="Olive">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">C</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>proof </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/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:23_2_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">V</font>
 ;<br/><a NAME="E2:23_2_1"/><b>then </b>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</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> #)
 <b>by </b><i><a class="ref" href="vectsp_4.html#D4">VECTSP_4:def 4</a></i>;<br/><a NAME="E3:23_2_1"/><b>then </b>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="vectsp_7.html#D3">VECTSP_7:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">LA</font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E5:23_2_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">LA</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T12">VECTSP_7:12</a></i>;<br/><a NAME="E6:23_2_1"/>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">LA</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="vectsp_7.html#K1">Lin</a> <font color="Maroon">C</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w</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="E1:23_2_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">LA</font>
 ;<br/>

<a NAME="E2:23_2_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">LA</font> &amp;  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">LA</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/>
<b>reconsider </b><font color="Maroon">w'</font> = <font color="Maroon">w</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/>
<a NAME="E4:23_2_1_1"/>
<font color="Maroon">w'</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">LB</font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E6:23_2_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">LB</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T12">VECTSP_7:12</a></i>;<br/>
<a NAME="E7:23_2_1_1"/>
 ex <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:23_2_1_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E3:23_2_1_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="txt" href="vectsp_9.html#E5"><i><font color="Green">Lemma30</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="vectsp_9.html#E3:23"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">i</font>
;<br/>

<b>thus </b><a NAME="E5:23_2_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="txt" href="vectsp_9.html#E3:23"><i><font color="Green">E32</font></i></a>, <a class="ref" href="vectsp_9.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:23_2_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">LB</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Olive">L</font></span>)</span> where <font color="Olive">L</font> is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font> :  ex <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E9:23_2_1_1"/>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">LB</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_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="E1:23_2_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">LB</font>
 ;<br/>

<b>hence </b><a NAME="E2:23_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E3:23"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:23_2_1_1"/><b>then </b>
<font color="Maroon">LB</font> is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/>
<a NAME="E11:23_2_1_1"/><b>then </b>
<font color="Maroon">w</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_7.html#T12">VECTSP_7:12</a></i>;<br/>
<b>hence </b><a NAME="E12:23_2_1_1"/>
<font color="Maroon">w</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="vectsp_7.html#K1">Lin</a> <font color="Maroon">C</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>then consider </b><font color="Maroon">LC</font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E8:23_2_1"/><i><font color="Green">E75</font></i>: 
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">LA</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">LC</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E9:23_2_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="vectsp_7.html#T12">VECTSP_7:12</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:23_2"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E3:23_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <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/>
<a NAME="E4:23_2"/><b>then </b>
<font color="Maroon">v</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="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">V</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#D4">VECTSP_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:23_2"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:23"/><b>then </b>
 <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T38">VECTSP_4:38</a></i>;<br/>
<a NAME="E13:23"/><b>then </b>
 <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> #) <a href="hidden.html#R1">=</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#D4">VECTSP_4:def 4</a></i>;<br/>
<a NAME="E14:23"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">C</font> is    <a href="vectsp_7.html#M1">Basis</a> of <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_7.html#D3">VECTSP_7:def 3</a></i>;<br/>
<a NAME="E15:23"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_3"/>
not <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:23_3"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E4:23_3"/><i><font color="Green">E78</font></i>: 
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">D</font> = <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">C</font>;<br/>
<a NAME="E5:23_3"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">C</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> ;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">C</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i>, <a class="txt" href="vectsp_9.html#E5"><i><font color="Green">Lemma30</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<font color="Maroon">C</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">C</font></span>)</span> = 
<font color="Maroon">C</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<font color="Maroon">B</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>
<a NAME="E9:23_3"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">D</font>
 
;<br/>
<b>hence </b><a NAME="E10:23_3"/>
contradiction
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:23"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">L</font> being   <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font> st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Olive">L</font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E17:23"/><i><font color="Green">E81</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:23_4"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>
 <b>and </b><br/><a NAME="E2:23_4"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">y1</font>]
 <b>and </b><br/><a NAME="E3:23_4"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">y2</font>]
 ;<br/>

<b>consider </b><font color="Maroon">L1</font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E5:23_4"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L1</font>
 <b>and </b><br/><a NAME="E6:23_4"/><i><font color="Green">E86</font></i>: 
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">L2</font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E8:23_4"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">y2</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L2</font>
 <b>and </b><br/><a NAME="E9:23_4"/><i><font color="Green">E88</font></i>: 
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E3:23"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E10:23_4"/>
(  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp;  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L2</font> <a href="tarski.html#R1">c=</a> <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="E11:23_4"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i>, <a class="ref" href="vectsp_9.html#T5" target="_self">Th5</a>, , , , <a class="ref" href="matrlin.html#T9">MATRLIN:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:23"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> holds <br/> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:23_5"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>
 ;<br/>

<a NAME="E2:23_5"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:23_5"/><b>then </b>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E4:23_5"/><b>then </b>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">L</font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E6:23_5"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T12">VECTSP_7:12</a></i>;<br/>
<a NAME="E7:23_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>, <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font>]
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E3:23"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:23_5"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Olive">x</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E19:23"/>
 ex <font color="Olive">q</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K4">dom</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>] ) )
 
<b>from </b><i><a class="ref" href="finseq_1.html#S1">FINSEQ_1:sch 1</a>(<a class="txt" href="vectsp_9.html#E5"><i><font color="Green">Lemma30</font></i></a>, );<br/></i>
<b>then consider </b><font color="Maroon">q</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E21:23"/><i><font color="Green">E93</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>] ) )
 ;<br/>
<a NAME="E22:23"/><i><font color="Green">E95</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E3:23"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_6"><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="E1:23_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font>
 ;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:23_6"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E4:23_6"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">L</font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E7:23_6"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">x</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="E8:23_6"/><i><font color="Green">E137</font></i>: 
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E3:23"><i><font color="Green">E32</font></i></a>, , </i>;<br/><b>thus </b><a NAME="E9:23_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Olive">L</font></span>)</span> where <font color="Olive">L</font> is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font> :  ex <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="vectsp_9.html#T5" target="_self">Th5</a>, , , </i>;<br/></div><b>end;</b></div>
<a NAME="E24:23"/><b>then </b><i><font color="Green">E138</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Olive">L</font></span>)</span> where <font color="Olive">L</font> is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font> :  ex <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_7"><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="E1:23_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Olive">L</font></span>)</span> where <font color="Olive">L</font> is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font> :  ex <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">L</font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E3:23_7"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">x</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="E4:23_7"/><i><font color="Green">E140</font></i>: 
 ex <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> )
 ;<br/><b>consider </b><font color="Maroon">i</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:23_7"/><i><font color="Green">E141</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i/>;<br/><a NAME="E7:23_7"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">x</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>] )
 <b>by </b><i><a class="txt" href="vectsp_9.html#E3:23"><i><font color="Green">E32</font></i></a>, <a class="ref" href="vectsp_9.html#T5" target="_self">Th5</a>, , </i>;<br/><a NAME="E8:23_7"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E5"><i><font color="Green">Lemma30</font></i></a>, <a class="txt" href="vectsp_9.html#E3:23"><i><font color="Green">E32</font></i></a>, <a class="ref" href="vectsp_9.html#T5" target="_self">Th5</a>, </i>;<br/><b>hence </b><a NAME="E9:23_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T5" target="_self">Th5</a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E26:23"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Olive">L</font></span>)</span> where <font color="Olive">L</font> is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font> :  ex <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</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="E27:23"/><b>then </b><i><font color="Green">E142</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Olive">L</font></span>)</span> where <font color="Olive">L</font> is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font> :  ex <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E28:23"/>
for <font color="Olive">R</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">R</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Olive">L</font></span>)</span> where <font color="Olive">L</font> is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font> :  ex <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) </span>}</span>  holds <br/><font color="Olive">R</font> is <a href="finset_1.html#V1">finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_8"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:23_8"/>
<font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Olive">L</font></span>)</span> where <font color="Olive">L</font> is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font> :  ex <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">L</font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E3:23_8"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">R</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="E4:23_8"/>
 ex <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> )
 ;<br/>
<b>thus </b><a NAME="E5:23_8"/>
<font color="Maroon">R</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E29:23"/>
<font color="Maroon">B</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, , <a class="ref" href="finset_1.html#T25">FINSET_1:25</a></i>;<br/>


</div>
