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<b>let </b><font color="Maroon">F</font> be  <a href="group_1.html#V6">finite</a> <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">F</font>;<br/><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">q</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:23"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">V</font> is <a href="matrlin.html#V2">finite-dimensional</a>
 <b>and </b><br/><a NAME="E2:23"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_9.html#K1">dim</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E3:23"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>
 ;<br/>

<a NAME="E4:23"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon">B</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="E6:23"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">B</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_9.html#D1">VECTSP_9:def 1</a></i>;<br/>
<a NAME="E7:23"/><i><font color="Green">E50</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="E8:23"/><i><font color="Green">E51</font></i>: 
 <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</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_7.html#D3">VECTSP_7:def 3</a></i>;<br/>
<a NAME="E9:23"/><i><font color="Green">E52</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , , <a class="ref" href="vectsp_9.html#D2">VECTSP_9:def 2</a></i>;<br/>
<a NAME="E10:23"/>
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="graph_5.html#T5">GRAPH_5:5</a></i>;<br/>
<a NAME="E11:23"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:23_1_1_1"/><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_4.html#K1">(0).</a> <font color="Maroon">V</font>
 <b>by </b><i>, , <a class="ref" href="vectsp_9.html#T33">VECTSP_9:33</a></i>;<br/><a NAME="E3:23_1_1_1"/><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_4.html#K1">(0).</a> <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="E4:23_1_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D3">VECTSP_4:def 3</a></i>;<br/><b>then </b> <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> = 
1
<b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="prepower.html#K6">#Z</a> 0
<b>by </b><i><a class="ref" href="prepower.html#T44">PREPOWER:44</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="uniroots.html#K5">|^</a> 0
<b>by </b><i><a class="ref" href="prepower.html#T46">PREPOWER:46</a></i>
;<br/><b>hence </b><a NAME="E6:23_1_1_1"/>
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:23_1_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="card_4.html#T17">CARD_4:17</a></i>;<br/><b>consider </b><font color="Maroon">bf</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E4:23_1_1_2"/><i><font color="Green">E57</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">bf</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E5:23_1_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">bf</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="finseq_4.html#T73">FINSEQ_4:73</a></i>;<br/><a NAME="E6:23_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">bf</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_4.html#T77">FINSEQ_4:77</a></i>;<br/><a NAME="E7:23_1_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">bf</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E8:23_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon">Vbf</font> = <font color="Maroon">bf</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">cLinB</font> = 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">B</font></span>)</span>;<br/><b>set </b><font color="Maroon">ntocF</font> = <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="funct_1.html#NM1">Function</a>,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">lc</font> being   <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font> st <br/>( ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">lc</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Vbf</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Olive">lc</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Vbf</font></span>)</span> );<br/><a NAME="E10:23_1_1_2"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">x</font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>  ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of 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">B</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>;<br/>

<a NAME="E1:23_1_1_2_1"/>
 ex <font color="Olive">lc</font> being   <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font> st <br/>for <font color="Olive">i</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#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Olive">lc</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Vbf</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_1_1"><b>proof </b></a><div class="add">

<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>  = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><font color="Maroon">Vbf</font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>;<br/>
<a NAME="E1:23_1_1_2_1_1"/><i><font color="Green">E63</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> <font color="Maroon">B</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_1_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> ;<br/>

<b>assume </b><a NAME="E1:23_1_1_2_1_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:23_1_1_2_1_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">Vbf</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , <a class="txt" href="weddwitt.html#E1:23"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T144">FUNCT_1:144</a></i>;<br/>
<a NAME="E4:23_1_1_2_1_1_1"/><i><font color="Green">E66</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">Vbf</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E2:23"><i><font color="Green">E36</font></i></a>, <a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>;<br/>
<a NAME="E5:23_1_1_2_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Vbf</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E2:23"><i><font color="Green">E36</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:23_1_1_2_1_1_1"/><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">Vbf</font> &amp; <font color="Maroon">Vbf</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E7:23_1_1_2_1_1_1"/><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">f</font>
 
<b>by </b><i>, <a class="ref" href="weddwitt.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_2.html#T144">FINSEQ_2:144</a></i>;<br/>
<a NAME="E8:23_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>hence </b><a NAME="E9:23_1_1_2_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<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">F</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E3:23"><i><font color="Green">E37</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">lc</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">B</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E3:23_1_1_2_1_1"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">lc</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">y</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2">FUNCT_2:sch 2</a>();<br/></i>
<b>set </b><font color="Maroon">ll</font> = <font color="Maroon">lc</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span>)</span>;<br/>
<a NAME="E4:23_1_1_2_1_1"/><i><font color="Green">E68</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E5:23_1_1_2_1_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">lc</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">lc</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>
<a NAME="E6:23_1_1_2_1_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">lc</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:23_1_1_2_1_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">lc</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>;<br/>
<a NAME="E8:23_1_1_2_1_1"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">lc</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E9:23_1_1_2_1_1"/><i><font color="Green">E70</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E10:23_1_1_2_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E11:23_1_1_2_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E4:23"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:23_1_1_2_1_1"/><i><font color="Green">E72</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">lc</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">lc</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/>
<a NAME="E13:23_1_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">lc</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="weddwitt.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E14:23_1_1_2_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">lc</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E15:23_1_1_2_1_1"/><b>then </b>
<font color="Maroon">lc</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span>)</span> is   <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">F</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E3:23"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E16:23_1_1_2_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">lc</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span>)</span> is    <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">F</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E17:23_1_1_2_1_1"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">i</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#K4">dom</a> <font color="Maroon">f</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">lc</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Vbf</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:23_1_1_2_1_1_2"/><i><font color="Green">E76</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">f</font>
 ;<br/>

<a NAME="E2:23_1_1_2_1_1_2"/><i><font color="Green">E77</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">Vbf</font>
 
<b>by </b><i>, , <a class="ref" href="weddwitt.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_2.html#T144">FINSEQ_2:144</a></i>;<br/>
<a NAME="E3:23_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">Vbf</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E5:23_1_1_2_1_1_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Vbf</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>
<a NAME="E6:23_1_1_2_1_1_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">Vbf</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">ee</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:23_1_1_2_1_1_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">Vbf</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">ee</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T159">FUNCT_1:159</a></i>;<br/>
<a NAME="E9:23_1_1_2_1_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Vbf</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="weddwitt.html#T3" target="_self">Th3</a>, <a class="txt" href="weddwitt.html#E6:23"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E10:23_1_1_2_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">ee</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E11:23_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ee</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T24">ZFMISC_1:24</a></i>;<br/>
<a NAME="E12:23_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">Vbf</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/>
<a NAME="E13:23_1_1_2_1_1_2"/>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E14:23_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">lc</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E2:23"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/>
<a NAME="E15:23_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">lc</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><font color="Maroon">Vbf</font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="weddwitt.html#E1:23"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:23_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">lc</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Vbf</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:23_1_1_2_1_1"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>  st not <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">lc</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/>

<b>assume </b><a NAME="E1:23_1_1_2_1_1_3"/>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:23_1_1_2_1_1_3"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:23_1_1_2_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">lc</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E2:23"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/>
<b>hence </b><a NAME="E4:23_1_1_2_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">lc</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="weddwitt.html#T3" target="_self">Th3</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">L</font> = <font color="Maroon">lc</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span>)</span> 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="E20:23_1_1_2_1_1"/>
for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>  st <font color="Olive">v</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="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/>

<b>assume </b><a NAME="E1:23_1_1_2_1_1_4"/><i><font color="Green">E88</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><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_2_1_1_4_1"/>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E2:23_1_1_2_1_1_4_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">lc</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:23_1_1_2_1_1_4_1"/>
contradiction
 <b>by </b><i><a class="ref" href="weddwitt.html#T3" target="_self">Th3</a>, <a class="ref" href="vectsp_6.html#T20">VECTSP_6:20</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:23_1_1_2_1_1_4"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E21:23_1_1_2_1_1"/><b>then </b>
 <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>
 
<b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>
<a NAME="E22:23_1_1_2_1_1"/><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="E23:23_1_1_2_1_1"/>
 ex <font color="Olive">lc</font> being   <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font> st <br/>for <font color="Olive">i</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#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Olive">lc</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Vbf</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 <b>by </b><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">B</font><b> such that </b><br/><a NAME="E3:23_1_1_2_1"/><i><font color="Green">E89</font></i>: 
for <font color="Olive">i</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#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">lc</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Vbf</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 ;<br/>
<a NAME="E4:23_1_1_2_1"/>
 ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> st <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">lc</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Vbf</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E5:23_1_1_2_1"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of 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">B</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f</font>,<font color="Olive">y</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">G</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>,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">B</font></span>)</span><b> such that </b><br/><a NAME="E12:23_1_1_2"/><i><font color="Green">E102</font></i>: 
for <font color="Olive">tup</font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">tup</font>,<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">tup</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i><a NAME="E13:23_1_1_2"/><i><font color="Green">E103</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E14:23_1_1_2"/><i><font color="Green">E104</font></i>: 
for <font color="Olive">L</font> being   <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font>  ex <font color="Olive">nt</font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font> st <br/>for <font color="Olive">i</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#K4">dom</a> <font color="Olive">nt</font> holds <br/><font color="Olive">nt</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Vbf</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_2"><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>;<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>  = <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Vbf</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:23_1_1_2_2"/><i><font color="Green">E105</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="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_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> ;<br/>

<b>assume </b><a NAME="E1:23_1_1_2_2_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:23_1_1_2_2_1"/>
<font color="Maroon">Vbf</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Vbf</font>
 
<b>by </b><i>, <a class="txt" href="weddwitt.html#E4:23"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon">vv</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:23_1_1_2_2_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">Vbf</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vv</font>
 ;<br/>
<a NAME="E5:23_1_1_2_2_1"/>
<font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">vv</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>
 
;<br/>
<b>hence </b><a NAME="E6:23_1_1_2_2_1"/>
<font color="Maroon">H<sub>1</sub></font>(<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">F</font>
 <b>by </b><i><a class="ref" href="weddwitt.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E3:23_1_1_2_2"/><i><font color="Green">E108</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="finseq_1.html#K2">Seg</a> <font color="Maroon">n</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_2.html#S2">FUNCT_2:sch 2</a>(<a class="txt" href="weddwitt.html#E3:23"><i><font color="Green">E37</font></i></a>);<br/></i>
<a NAME="E4:23_1_1_2_2"/><i><font color="Green">E111</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:23_1_1_2_2"/><i><font color="Green">E112</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>
 
;<br/>
<a NAME="E6:23_1_1_2_2"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E7:23_1_1_2_2"/><b>then </b>
<font color="Maroon">f</font> is <a href="finseq_1.html#V1">FinSequence-like</a>
 
<b>by </b><i><a class="ref" href="weddwitt.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">f</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font> <b>by </b><i>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E9:23_1_1_2_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">ff</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="weddwitt.html#T2" target="_self">Th2</a>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:23_1_1_2_2"/><b>then </b>
<font color="Maroon">ff</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<b>hence </b><a NAME="E11:23_1_1_2_2"/>
 ex <font color="Olive">nt</font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font> st <br/>for <font color="Olive">i</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#K4">dom</a> <font color="Olive">nt</font> holds <br/><font color="Olive">nt</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Vbf</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E4:23"><i><font color="Green">E38</font></i></a>, <a class="ref" href="weddwitt.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:23_1_1_2"/>
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</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">B</font></span>)</span> holds <br/> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:23_1_1_2_3"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">c</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">B</font></span>)</span>
 ;<br/>

<a NAME="E2:23_1_1_2_3"/>
<font color="Maroon">c</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><a class="txt" href="weddwitt.html#E3:23"><i><font color="Green">E37</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></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="E4:23_1_1_2_3"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">c</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="E5:23_1_1_2_3"/>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Vbf</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/>
<a NAME="E6:23_1_1_2_3"/><b>then </b><i><font color="Green">E118</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Vbf</font></span>)</span> <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_9.html#T7">VECTSP_9:7</a></i>;<br/>
<b>consider </b><font color="Maroon">t</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E8:23_1_1_2_3"/><i><font color="Green">E119</font></i>: 
for <font color="Olive">i</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#K4">dom</a> <font color="Maroon">t</font> holds <br/><font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Vbf</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E2:23"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">lc</font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E10:23_1_1_2_3"/><i><font color="Green">E120</font></i>: 
for <font color="Olive">i</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#K4">dom</a> <font color="Maroon">t</font> holds <br/><font color="Maroon">lc</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Vbf</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 <b>and </b><br/><a NAME="E11:23_1_1_2_3"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">lc</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Vbf</font></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E12:23_1_1_2_3"/>
for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> holds  <font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lc</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_2_3_1_1_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Vbf</font> or  not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Vbf</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_2_3_1_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Vbf</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:23_1_1_2_3_1_1_1_1"/><i><font color="Green">E122</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Vbf</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/><a NAME="E4:23_1_1_2_3_1_1_1_1"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Vbf</font> &amp; <font color="Maroon">Vbf</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E5:23_1_1_2_3_1_1_1_1"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="weddwitt.html#T1" target="_self">Th1</a>, , <a class="ref" href="finseq_2.html#T144">FINSEQ_2:144</a></i>;<br/><a NAME="E6:23_1_1_2_3_1_1_1_1"/><b>then </b>
<font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Vbf</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:23_1_1_2_3_1_1_1_1"/>
<font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lc</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="weddwitt.html#T3" target="_self">Th3</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_2_3_1_1_1_2"/><i><font color="Green">E125</font></i>: 
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Vbf</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_3_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_2_3_1_1_1_2_1"/><i><font color="Green">E126</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/><a NAME="E2:23_1_1_2_3_1_1_1_2_1"/>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">l</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Vbf</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="E3:23_1_1_2_3_1_1_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="weddwitt.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div><a NAME="E3:23_1_1_2_3_1_1_1_2"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">l</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">F</font>
 <b>by </b><i><a class="ref" href="vectsp_6.html#T20">VECTSP_6:20</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_3_1_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_2_3_1_1_1_2_2"/><i><font color="Green">E129</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">lc</font>
 ;<br/><a NAME="E2:23_1_1_2_3_1_1_1_2_2"/>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">lc</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Vbf</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="E3:23_1_1_2_3_1_1_1_2_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:23_1_1_2_3_1_1_1_2"/>
<font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lc</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="weddwitt.html#T3" target="_self">Th3</a>, <a class="ref" href="vectsp_6.html#T20">VECTSP_6:20</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:23_1_1_2_3_1"/>
<font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lc</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E13:23_1_1_2_3"/><b>then </b>
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Vbf</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="vectsp_6.html#D10">VECTSP_6:def 10</a></i>;<br/>
<b>hence </b><a NAME="E14:23_1_1_2_3"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> )
 <b>by </b><i><a class="txt" href="weddwitt.html#E4:23"><i><font color="Green">E38</font></i></a>, <a class="ref" href="weddwitt.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:23_1_1_2"/><b>then </b><i><font color="Green">E130</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</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">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T16">FUNCT_2:16</a></i>;<br/><a NAME="E17:23_1_1_2"/>
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x2</font> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:23_1_1_2_4"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E2:23_1_1_2_4"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E3:23_1_1_2_4"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">t1</font> = <font color="Maroon">x1</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font> <b>by </b><i><a class="txt" href="weddwitt.html#E4:23"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">t2</font> = <font color="Maroon">x2</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font> <b>by </b><i><a class="ref" href="weddwitt.html#T2" target="_self">Th2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t1</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">B</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t2</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">B</font></span>)</span> ;<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="E9:23_1_1_2_4"/><i><font color="Green">E136</font></i>: 
( ( for <font color="Olive">i</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#K4">dom</a> <font color="Maroon">t1</font> holds <br/><font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Vbf</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) &amp; <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Vbf</font></span>)</span> )
 <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="E11:23_1_1_2_4"/><i><font color="Green">E137</font></i>: 
( ( for <font color="Olive">i</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#K4">dom</a> <font color="Maroon">t2</font> holds <br/><font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Vbf</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) &amp; <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Vbf</font></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E12:23_1_1_2_4"/>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Vbf</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/>
<a NAME="E13:23_1_1_2_4"/><b>then </b><i><font color="Green">E138</font></i>: 
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Vbf</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="vectsp_9.html#T7">VECTSP_9:7</a></i>;<br/>
<a NAME="E14:23_1_1_2_4"/>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L2</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Vbf</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/>
<a NAME="E15:23_1_1_2_4"/><b>then </b>
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Vbf</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="vectsp_9.html#T7">VECTSP_9:7</a></i>;<br/>
<a NAME="E16:23_1_1_2_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L1</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">L2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</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="E17:23_1_1_2_4"/><b>then </b><i><font color="Green">E139</font></i>: 
 <a href="vectsp_6.html#K4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K8">-</a> <font color="Maroon">L2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#T80">VECTSP_6:80</a></i>;<br/>
<a NAME="E18:23_1_1_2_4"/>
<font color="Maroon">L1</font> <a href="vectsp_6.html#K8">-</a> <font color="Maroon">L2</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#T75">VECTSP_6:75</a></i>;<br/>
<a NAME="E19:23_1_1_2_4"/><b>then </b><i><font color="Green">E140</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="vectsp_6.html#K8">-</a> <font color="Maroon">L2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="weddwitt.html#T3" target="_self">Th3</a>, <a class="ref" href="vectsp_7.html#D1">VECTSP_7:def 1</a></i>;<br/>
<a NAME="E20:23_1_1_2_4"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E21:23_1_1_2_4"/>
for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> holds  <font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/>

<b>reconsider </b><font color="Maroon">LL</font> = <font color="Maroon">L1</font> <a href="vectsp_6.html#K8">-</a> <font color="Maroon">L2</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#T75">VECTSP_6:75</a></i>;<br/>
<a NAME="E2:23_1_1_2_4_1"/>
<font color="Maroon">LL</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">F</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_6.html#T20">VECTSP_6:20</a></i>;<br/>
<a NAME="E3:23_1_1_2_4_1"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#T73">VECTSP_6:73</a></i>;<br/>
<b>hence </b><a NAME="E4:23_1_1_2_4_1"/>
<font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T84">VECTSP_1:84</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:23_1_1_2_4"/><b>then </b><i><font color="Green">E143</font></i>: 
<font color="Maroon">L1</font> <a href="vectsp_6.html#R1">=</a> <font color="Maroon">L2</font>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D10">VECTSP_6:def 10</a></i>;<br/>
<a NAME="E23:23_1_1_2_4"/><i><font color="Green">E145</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="txt" href="weddwitt.html#E6:23"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_2.html#T144">FINSEQ_2:144</a></i>;<br/>
<a NAME="E24:23_1_1_2_4"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t1</font> holds <br/><font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:23_1_1_2_4_2"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t1</font>
 ;<br/>

<a NAME="E2:23_1_1_2_4_2"/>
<font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Vbf</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:23_1_1_2_4_2"/>
<font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E25:23_1_1_2_4"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


</div><b>end;</b></div><a NAME="E18:23_1_1_2"/><b>then </b>
<font color="Maroon">G</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E19:23_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>,<font color="Maroon">G</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:23"><i><font color="Green">E35</font></i></a>, <a class="ref" href="card_1.html#T60">CARD_1:60</a></i>;<br/><a NAME="E20:23_1_1_2"/><b>then </b><i><font color="Green">E147</font></i>: 
<font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font>,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">B</font></span>)</span> <a href="tarski.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:23"><i><font color="Green">E35</font></i></a>, <a class="txt" href="weddwitt.html#E3:23"><i><font color="Green">E37</font></i></a>, <a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>;<br/><a NAME="E21:23_1_1_2"/><i><font color="Green">E148</font></i>: 
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">n</font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="weddwitt.html#T1" target="_self">Th1</a>, , , <a class="ref" href="card_1.html#T66">CARD_1:66</a>, <a class="ref" href="finseq_1.html#T76">FINSEQ_1:76</a></i>;<br/> <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font></span>)</span> = 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F</font></span>)</span>
<b>by </b><i><a class="ref" href="weddwitt.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_2.html#T111">FINSEQ_2:111</a></i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">n</font>,<font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="weddwitt.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_5.html#T68">FUNCT_5:68</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E23:23_1_1_2"/>
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="txt" href="weddwitt.html#E4:23"><i><font color="Green">E38</font></i></a>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E13:23"/>
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font>
 ;<br/>


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