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

<a NAME="E1:20_1_1"/><i><font color="Green">E34</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U2</font> <b>by </b><i><a class="ref" href="relat_1.html#T152">RELAT_1:152</a></i>;<br/>
<b>take </b><font color="Maroon">B</font> =  <a href="unialg_2.html#K4">UniAlgSetClosed</a> <font color="Maroon">A</font>;<br/>

<a NAME="E3:20_1_1"/>
<font color="Maroon">A</font> is <a href="unialg_2.html#V1">opers_closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o2</font> be   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_2.html#D5">UNIALG_2:def 5</a><br/>

<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:20_1_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U2</font> &amp; the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E3:20_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">U1</font>,<font color="Maroon">U2</font> <a href="unialg_2.html#R1">are_similar</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:20_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K4">signature</a> <font color="Maroon">U2</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D2">UNIALG_2:def 2</a></i>;<br/>
<a NAME="E5:20_1_1_1"/><i><font color="Green">E47</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U2</font> )
 
<b>by </b><i><a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E6:20_1_1_1"/><i><font color="Green">E48</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U2</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U2</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U2</font></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="unialg_2.html#D2">UNIALG_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">o1</font> = the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U1</font> <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>let </b><font color="Maroon">s</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_2.html#D4">UNIALG_2:def 4</a><br/>

<b>assume </b><a NAME="E8:20_1_1_1"/><i><font color="Green">E49</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font>
 ;<br/>

<a NAME="E9:20_1_1_1"/><i><font color="Green">E50</font></i>: 
( <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font> &amp; <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font> )
 
<b>by </b><i>, , , <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E10:20_1_1_1"/><i><font color="Green">E61</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#K4">dom</a> <font color="Maroon">s</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> &amp; <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="20_1_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="E1:20_1_1_1_1"/><i><font color="Green">E63</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">s</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x0</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E3:20_1_1_1_1"/>
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:20_1_1_1_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <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">U1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x0</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E6:20_1_1_1_1"/>
( <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">U1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>] )
 <b>by </b><i><a class="ref" href="alg_1.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">s1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E12:20_1_1_1"/><i><font color="Green">E72</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> &amp; ( 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#K4">dom</a> <font color="Maroon">s</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<a NAME="E13:20_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">s1</font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">s1</font> as   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">U1</font> <b>by </b><i>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">s1</font> as    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E17:20_1_1_1"/><i><font color="Green">E73</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E18:20_1_1_1"/><b>then </b>
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font> </span>}</span> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E19:20_1_1_1"/><b>then </b>
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#D4">FINSEQ_2:def 4</a></i>;<br/>
<a NAME="E20:20_1_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">o1</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#T2">UNIALG_2:2</a></i>;<br/>
<a NAME="E21:20_1_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">s1</font></span>)</span>
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E22:20_1_1_1"/><i><font color="Green">E76</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">s1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E23:20_1_1_1"/><i><font color="Green">E77</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">s1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">s1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:20_1_1_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font></span>)</span>
 ;<br/><a NAME="E2:20_1_1_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:20_1_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:20_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="txt" href="alg_1.html#E1:20"><i><font color="Green">E33</font></i></a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E25:20_1_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">s1</font>
 
<b>by </b><i><a class="ref" href="alg_1.html#T1" target="_self">Th1</a>, , <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>
<a NAME="E26:20_1_1_1"/>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">o1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> &amp; <font color="Maroon">o1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">o1</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E27:20_1_1_1"/>
<font color="Maroon">o2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="txt" href="alg_1.html#E1:20"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:20_1_1"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#G1">UAStr</a>(# <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <font color="Maroon">U2</font>,<font color="Maroon">A</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="unialg_2.html#D9">UNIALG_2:def 9</a></i>;<br/>
<b>hence </b><a NAME="E5:20_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>
 ;<br/>


</div>
