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

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x3</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x4</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>









<b>thus </b><a NAME="E1:91"/>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K8" target="_self">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font>,<font color="Maroon">x4</font></span><a href="finseq_4.html#K8" target="_self">*&gt;</a></span> implies (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 4 &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">x3</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 4 <a href="hidden.html#R1">=</a> <font color="Maroon">x4</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:91_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K8" target="_self">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font>,<font color="Maroon">x4</font></span><a href="finseq_4.html#K8" target="_self">*&gt;</a></span>
 ;<br/>

<b>hence </b> <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x4</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
3 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x4</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
<br/>.= 
3 <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
<br/>.= 
4

;<br/><br/>

<b>set </b><font color="Maroon">p3</font> = <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="finseq_1.html#K11">*&gt;</a></span>;<br/>
<a NAME="E3:91_1"/><i><font color="Green">E33</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="finseq_1.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default">1,2,3</span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T1">FINSEQ_3:1</a>, <a class="ref" href="finseq_3.html#T30">FINSEQ_3:30</a></i>;<br/>
<a NAME="E4:91_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="finseq_1.html#K11">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x4</font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E5:91_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="finseq_1.html#K11">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T11" target="_self">Th11</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<b>hence </b><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="finseq_1.html#K11">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#D3" target="_self">Def3</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">x1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/><br/>

<a NAME="E7:91_1"/>
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="finseq_1.html#K11">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T11" target="_self">Th11</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<b>hence </b><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 2 = 
<span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="finseq_1.html#K11">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2
<b>by </b><i><a class="ref" href="finseq_4.html#D3" target="_self">Def3</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">x2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/><br/>

<a NAME="E9:91_1"/>
3 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="finseq_1.html#K11">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T11" target="_self">Th11</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<b>hence </b><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 3 = 
<span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="finseq_1.html#K11">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 3
<b>by </b><i><a class="ref" href="finseq_4.html#D3" target="_self">Def3</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">x3</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/><br/>

<a NAME="E11:91_1"/>
1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</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/>
<a NAME="E12:91_1"/><b>then </b><i><font color="Green">E36</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x4</font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<b>thus </b><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 4 = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="finseq_1.html#K11">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x4</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#T10" target="_self">Th10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="finseq_1.html#K11">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x4</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x4</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T17" target="_self">Th17</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">x4</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>
;<br/>


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

<b>assume </b><a NAME="E2:91"/><i><font color="Green">E37</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 4 &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">x3</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 4 <a href="hidden.html#R1">=</a> <font color="Maroon">x4</font> )
 ;<br/>

<b>then </b><i><font color="Green">E38</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x4</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/>
<a NAME="E4:91"/><i><font color="Green">E39</font></i>: 
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> <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="finseq_1.html#K11">*&gt;</a></span> holds <br/><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="finseq_1.html#K11">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_3"><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="E1:91_3"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="finseq_1.html#K11">*&gt;</a></span>
 ;<br/>

<a NAME="E2:91_3"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="finseq_1.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a> 3
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E3:91_3"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default">1,2,3</span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T17" target="_self">Th17</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="finseq_3.html#T1">FINSEQ_3:1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_3_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 3 )
 </span><b>by </b><i>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_3_1_1"><b>suppose </b></a><a NAME="E1:91_3_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:91_3_1_1"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="finseq_1.html#K11">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T10" target="_self">Th10</a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_3_1_2"><b>suppose </b></a><a NAME="E1:91_3_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:91_3_1_2"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="finseq_1.html#K11">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T10" target="_self">Th10</a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_3_1_3"><b>suppose </b></a><a NAME="E1:91_3_1_3"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:91_3_1_3"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="finseq_1.html#K11">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T10" target="_self">Th10</a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:91"/>
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> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x4</font></span><a href="finseq_1.html#K9">*&gt;</a></span> holds <br/><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x4</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_4"><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="E1:91_4"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x4</font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 ;<br/>

<a NAME="E2:91_4"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E3:91_4"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> = 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x4</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T10" target="_self">Th10</a>, <a class="ref" href="finseq_4.html#T17" target="_self">Th17</a>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:91"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K8" target="_self">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font>,<font color="Maroon">x4</font></span><a href="finseq_4.html#K8" target="_self">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_4.html#T11" target="_self">Th11</a>, <a class="ref" href="finseq_4.html#D3" target="_self">Def3</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>


</div>
