<?xml version="1.0"?>
<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> , <font color="Maroon">x3</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x4</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:18"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K2">{</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="enumset1.html#K2">}</a></span> iff  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>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K8">&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">*&gt;</a></span> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K8">&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">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:18_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K2">{</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="enumset1.html#K2">}</a></span> implies  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>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K8">&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">*&gt;</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K8">&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">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_1_1"/><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K8">&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">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">1,2,3,4</span><a href="enumset1.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T2">FINSEQ_3:2</a>, <a class="ref" href="finseq_4.html#T92">FINSEQ_4:92</a></i>;<br/>
<b>assume </b><a NAME="E2:18_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K2">{</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="enumset1.html#K2">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x3</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x4</font> )
 </span><b>by </b><i>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_1"><b>suppose </b></a><a NAME="E1:18_1_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 ;<br/><div class="add"><b>take </b>
1
;<br/><b>thus </b><a NAME="E2:18_1_1_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K8">&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">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><b>thus </b><a NAME="E3:18_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K8">&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">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T91">FINSEQ_4:91</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_2"><b>suppose </b></a><a NAME="E1:18_1_1_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/><div class="add"><b>take </b>
2
;<br/><b>thus </b><a NAME="E2:18_1_1_1_2"/>
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K8">&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">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><b>thus </b><a NAME="E3:18_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K8">&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">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T91">FINSEQ_4:91</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_3"><b>suppose </b></a><a NAME="E1:18_1_1_1_3"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x3</font>
 ;<br/><div class="add"><b>take </b>
3
;<br/><b>thus </b><a NAME="E2:18_1_1_1_3"/>
3 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K8">&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">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><b>thus </b><a NAME="E3:18_1_1_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K8">&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">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 3
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T91">FINSEQ_4:91</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_4"><b>suppose </b></a><a NAME="E1:18_1_1_1_4"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x4</font>
 ;<br/><div class="add"><b>take </b>
4
;<br/><b>thus </b><a NAME="E2:18_1_1_1_4"/>
4 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K8">&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">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><b>thus </b><a NAME="E3:18_1_1_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K8">&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">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 4
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T91">FINSEQ_4:91</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>given </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:18_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K8">&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">*&gt;</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K8">&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">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 ;<br/>

<a NAME="E3:18_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 4
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T92">FINSEQ_4:92</a></i>;<br/>
<a NAME="E4:18_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 4 )
 
<b>by </b><i><a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a>, <a class="ref" href="finseq_3.html#T2">FINSEQ_3:2</a></i>;<br/>
<a NAME="E5:18_1"/><b>then </b>
( <span class="p1"><a href="finseq_4.html#K8">&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">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> or <span class="p1"><a href="finseq_4.html#K8">&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">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> or <span class="p1"><a href="finseq_4.html#K8">&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">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x3</font> or <span class="p1"><a href="finseq_4.html#K8">&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">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x4</font> )
 
<b>by </b><i><a class="ref" href="finseq_4.html#T91">FINSEQ_4:91</a></i>;<br/>
<b>hence </b><a NAME="E6:18_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K2">{</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="enumset1.html#K2">}</a></span>
 <b>by </b><i>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:18"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_4.html#K8">&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">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K2">{</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="enumset1.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div>
