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

<b>let </b><font color="Maroon">U1</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>, <font color="Maroon">U2</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>, <font color="Maroon">U3</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>, <font color="Maroon">U4</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/>

<b>assume </b><a NAME="E1:13"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">U1</font> is    <a href="unialg_2.html#M3">SubAlgebra</a> of <font color="Maroon">U2</font> &amp; <font color="Maroon">U3</font> is    <a href="unialg_2.html#M3">SubAlgebra</a> of <font color="Maroon">U4</font> &amp; <font color="Maroon">U2</font>,<font color="Maroon">U4</font> <a href="unialg_2.html#R1">are_similar</a>  )
 ;<br/>

<a NAME="E2:13"/><b>then </b><i><font color="Green">E26</font></i>: 
( <font color="Maroon">U1</font>,<font color="Maroon">U2</font> <a href="unialg_2.html#R1">are_similar</a>  &amp; <font color="Maroon">U3</font>,<font color="Maroon">U4</font> <a href="unialg_2.html#R1">are_similar</a>  )
 
<b>by </b><i><a class="ref" href="unialg_2.html#T16">UNIALG_2:16</a></i>;<br/>
<a NAME="E3:13"/><b>then </b>
<font color="Maroon">U1</font>,<font color="Maroon">U4</font> <a href="unialg_2.html#R1">are_similar</a> 
 
<b>by </b><i>, <a class="ref" href="unialg_2.html#T4">UNIALG_2:4</a></i>;<br/>
<a NAME="E4:13"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">U1</font>,<font color="Maroon">U3</font> <a href="unialg_2.html#R1">are_similar</a> 
 
<b>by </b><i>, <a class="ref" href="unialg_2.html#T4">UNIALG_2:4</a></i>;<br/>
<a NAME="E5:13"/><b>then </b><i><font color="Green">E39</font></i>: 
( <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U1</font>,<font color="Maroon">U3</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U3</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">U1</font>,<font color="Maroon">U3</font></span>)</span> #) &amp; <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U4</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">U2</font>,<font color="Maroon">U4</font></span>)</span> #) )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:13"/><i><font color="Green">E41</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U2</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U3</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U4</font> )
 
<b>by </b><i>, <a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E7:13"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U3</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U4</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>thus </b><a NAME="E8:13"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U1</font>,<font color="Maroon">U3</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a><br/>

<b>let </b><font color="Maroon">B</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span>;<br/>

<b>assume </b><a NAME="E9:13"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U1</font>,<font color="Maroon">U3</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">B1</font> = 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="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B3</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U3</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U4</font> <b>by </b><i>, <a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E12:13"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">B1</font> is <a href="unialg_2.html#V1">opers_closed</a> &amp; <font color="Maroon">B3</font> is <a href="unialg_2.html#V1">opers_closed</a> )
 
<b>by </b><i>, <a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E13:13"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">o</font> being  <a href="unialg_2.html#NM5">operation</a> of <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span> holds  <font color="Maroon">B</font> <a href="unialg_2.html#R2">is_closed_on</a> <font color="Olive">o</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be   <a href="unialg_2.html#NM5">operation</a> of <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span>;<br/><b>let </b><font color="Maroon">s</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">B</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="E1:13_1"/><i><font color="Green">E53</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">o</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">s0</font> = <font color="Maroon">s</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U3</font></span><a href="zfmisc_1.html#K2">:]</a></span> <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/>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:13_1"/><i><font color="Green">E57</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 <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span> &amp; the <a href="unialg_1.html#U1">charact</a> of <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E5:13_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">n</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> the <a href="unialg_1.html#U1">charact</a> of <span class="p2"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span></span>)</span> &amp; the <a href="unialg_1.html#U1">charact</a> of <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:13_1"/><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></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>, , </i>;<br/>
<a NAME="E7:13_1"/><b>then </b><i><font color="Green">E60</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>
 
<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">o2</font> = 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> as   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U2</font> <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:13_1"/>
 <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U4</font> <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_2.html#T3">UNIALG_2:3</a></i>;<br/>
<a NAME="E10:13_1"/><b>then </b>
<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">U4</font>
 
<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">o4</font> = the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U4</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">U4</font> <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:13_1"/>
( <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> 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> &amp; <font color="Maroon">o4</font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 
;<br/>
<a NAME="E13:13_1"/><b>then </b><i><font color="Green">E62</font></i>: 
(  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font> &amp;  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o4</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="pralg_1.html#K3" target="_self">[[:</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o4</font></span><a href="pralg_1.html#K3" target="_self">:]]</a></span> )
 
<b>by </b><i>, , <a class="txt" href="pralg_1.html#E1:13"><i><font color="Green">E23</font></i></a>, </i>;<br/>
<a NAME="E14:13_1"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="pralg_1.html#K3" target="_self">[[:</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o4</font></span><a href="pralg_1.html#K3" target="_self">:]]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U4</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E15:13_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">s0</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U3</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E16:13_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">s0</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U4</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E17:13_1"/><b>then </b>
<font color="Maroon">s0</font> is    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U4</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">ss</font> = <font color="Maroon">s0</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U4</font></span><a href="zfmisc_1.html#K2">:]</a></span> <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="E19:13_1"/>
<font color="Maroon">ss</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 <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U4</font></span><a href="zfmisc_1.html#K2">:]</a></span> <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">o2</font> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E20:13_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">ss</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="pralg_1.html#K3" target="_self">[[:</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o4</font></span><a href="pralg_1.html#K3" target="_self">:]]</a></span>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">s1</font> =  <a href="pralg_1.html#K1" target="_self">pr1</a> <font color="Maroon">s0</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">B1</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/>
<b>reconsider </b><font color="Maroon">s3</font> =  <a href="pralg_1.html#K2" target="_self">pr2</a> <font color="Maroon">s0</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">B3</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="E23:13_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">B1</font> <a href="unialg_2.html#R2">is_closed_on</a> <font color="Maroon">o2</font> &amp; <font color="Maroon">B3</font> <a href="unialg_2.html#R2">is_closed_on</a> <font color="Maroon">o4</font> )
 
<b>by </b><i>, <a class="ref" href="unialg_2.html#D5">UNIALG_2:def 5</a></i>;<br/>
<a NAME="E24:13_1"/>
(  <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">s0</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s3</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s0</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E25:13_1"/><b>then </b><i><font color="Green">E66</font></i>: 
( <font color="Maroon">o2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">o4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B3</font> )
 
<b>by </b><i>, , , <a class="ref" href="unialg_2.html#D4">UNIALG_2:def 4</a></i>;<br/>
<a NAME="E26:13_1"/>
<font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">o2</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K1" target="_self">pr1</a> <font color="Maroon">ss</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">o4</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K2" target="_self">pr2</a> <font color="Maroon">ss</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, <a class="txt" href="pralg_1.html#E2:13"><i><font color="Green">E26</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E27:13_1"/>
<font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:13"/><i><font color="Green">E68</font></i>: 
 <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <span class="p2"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span>,<font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D7">UNIALG_2:def 7</a></i>;<br/>
<a NAME="E15:13"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <span class="p2"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span>,<font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E16:13"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="finseq_1.html#K3">len</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#K3">len</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <span class="p2"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span>,<font color="Maroon">B</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E17:13"/><i><font color="Green">E70</font></i>: 
 <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U1</font>,<font color="Maroon">U3</font></span><a href="pralg_1.html#K5" target="_self">:]</a></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>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E18:13"/>
 <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="E19:13"/><b>then </b>
 <a href="finseq_1.html#K3">len</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#K3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E20:13"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U1</font>,<font color="Maroon">U3</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <span class="p2"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span>,<font color="Maroon">B</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E21:13"/><i><font color="Green">E72</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> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U3</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">U3</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">U4</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">U4</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U1</font>,<font color="Maroon">U3</font></span><a href="pralg_1.html#K5" target="_self">:]</a></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> the <a href="unialg_1.html#U1">charact</a> of <span class="p2"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U1</font>,<font color="Maroon">U3</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></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> the <a href="unialg_1.html#U1">charact</a> of <span class="p2"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <span class="p2"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span>,<font color="Maroon">B</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_2.html#K3">Opers</a> <span class="p3"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span>,<font color="Maroon">B</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <font color="Maroon">U2</font>,<font color="Maroon">B1</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_2.html#K3">Opers</a> <font color="Maroon">U2</font>,<font color="Maroon">B1</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <font color="Maroon">U4</font>,<font color="Maroon">B3</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_2.html#K3">Opers</a> <font color="Maroon">U4</font>,<font color="Maroon">B3</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E22:13"/>
 <a href="unialg_1.html#K4">signature</a> <font color="Maroon">U4</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="E23:13"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U4</font> <a href="hidden.html#R1">=</a>  <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>
 
<b>by </b><i><a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E24:13"/><b>then </b><i><font color="Green">E73</font></i>: 
 <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U4</font> <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="E25:13"/>
 <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">U3</font>
 
<b>by </b><i>, <a class="ref" href="unialg_2.html#D2">UNIALG_2:def 2</a></i>;<br/>
<a NAME="E26:13"/><b>then </b>
 <a href="finseq_1.html#K3">len</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#K3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E27:13"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="finseq_1.html#K3">len</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#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U3</font>
 
<b>by </b><i><a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E28:13"/><i><font color="Green">E76</font></i>: 
 <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#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <font color="Maroon">U2</font>,<font color="Maroon">B1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D7">UNIALG_2:def 7</a></i>;<br/>
<a NAME="E29:13"/><i><font color="Green">E77</font></i>: 
 <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U4</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <font color="Maroon">U4</font>,<font color="Maroon">B3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D7">UNIALG_2:def 7</a></i>;<br/>
<a NAME="E30:13"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</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> the <a href="unialg_1.html#U1">charact</a> of <span class="p2"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U1</font>,<font color="Maroon">U3</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span></span>)</span> holds <br/>the <a href="unialg_1.html#U1">charact</a> of <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U1</font>,<font color="Maroon">U3</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <span class="p2"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:13_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">n</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> the <a href="unialg_1.html#U1">charact</a> of <span class="p2"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U1</font>,<font color="Maroon">U3</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">o2</font> = 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> as   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U2</font> <b>by </b><i>, , <a class="txt" href="pralg_1.html#E1:13"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">o4</font> = the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U4</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">U4</font> <b>by </b><i>, , <a class="txt" href="pralg_1.html#E1:13"><i><font color="Green">E23</font></i></a>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">o24</font> = the <a href="unialg_1.html#U1">charact</a> of <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as   <a href="unialg_2.html#NM5">operation</a> of <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span> <b>by </b><i>, , <a class="txt" href="pralg_1.html#E1:13"><i><font color="Green">E23</font></i></a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>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="txt" href="pralg_1.html#E1:13"><i><font color="Green">E23</font></i></a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">o3</font> = the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U3</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">U3</font> <b>by </b><i>, <a class="txt" href="pralg_1.html#E1:13"><i><font color="Green">E23</font></i></a>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:13_2"/>
( <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> 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> &amp; <font color="Maroon">o4</font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U4</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 
;<br/>
<a NAME="E8:13_2"/><b>then </b><i><font color="Green">E107</font></i>: 
( <font color="Maroon">o24</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="pralg_1.html#K3" target="_self">[[:</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o4</font></span><a href="pralg_1.html#K3" target="_self">:]]</a></span> &amp;  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o24</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o4</font> &amp;  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o24</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="txt" href="pralg_1.html#E1:13"><i><font color="Green">E23</font></i></a>, , </i>;<br/>
<a NAME="E9:13_2"/><i><font color="Green">E108</font></i>: 
<span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <font color="Maroon">U2</font>,<font color="Maroon">B1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">B1</font>
 
<b>by </b><i>, , <a class="txt" href="pralg_1.html#E1:13"><i><font color="Green">E23</font></i></a>, <a class="txt" href="pralg_1.html#E2:13"><i><font color="Green">E26</font></i></a>, , <a class="ref" href="unialg_2.html#D7">UNIALG_2:def 7</a></i>;<br/>
<a NAME="E10:13_2"/><i><font color="Green">E109</font></i>: 
<span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <font color="Maroon">U4</font>,<font color="Maroon">B3</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o4</font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">B3</font>
 
<b>by </b><i>, , <a class="txt" href="pralg_1.html#E1:13"><i><font color="Green">E23</font></i></a>, , , , <a class="ref" href="unialg_2.html#D7">UNIALG_2:def 7</a></i>;<br/>
<a NAME="E11:13_2"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">B1</font>
 
<b>by </b><i>, , <a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E12:13_2"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">o3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o4</font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">B3</font>
 
<b>by </b><i>, , <a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E13:13_2"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">B</font> <a href="unialg_2.html#R2">is_closed_on</a> <font color="Maroon">o24</font> &amp; <font color="Maroon">B1</font> <a href="unialg_2.html#R2">is_closed_on</a> <font color="Maroon">o2</font> &amp; <font color="Maroon">B3</font> <a href="unialg_2.html#R2">is_closed_on</a> <font color="Maroon">o4</font> )
 
<b>by </b><i>, <a class="ref" href="pralg_1.html#T1" target="_self">Th1</a>, <a class="ref" href="unialg_2.html#D5">UNIALG_2:def 5</a></i>;<br/>
<a NAME="E14:13_2"/><b>then </b><i><font color="Green">E113</font></i>: 
(  <a href="unialg_1.html#K3">arity</a> <span class="p1">(<span class="default"><font color="Maroon">o2</font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">B1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font> &amp;  <a href="unialg_1.html#K3">arity</a> <span class="p1">(<span class="default"><font color="Maroon">o4</font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">B3</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o4</font> )
 
<b>by </b><i><a class="ref" href="unialg_2.html#T8">UNIALG_2:8</a></i>;<br/>
<a NAME="E15:13_2"/><b>then </b><i><font color="Green">E114</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="pralg_1.html#K3" target="_self">[[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">o2</font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">B1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">o4</font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">B3</font></span>)</span></span><a href="pralg_1.html#K3" target="_self">:]]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font>
 
<b>by </b><i>, , , </i>;<br/>
<i><font color="Green">E115</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="pralg_1.html#K3" target="_self">[[:</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o4</font></span><a href="pralg_1.html#K3" target="_self">:]]</a></span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o24</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2"><a href="pralg_1.html#K3" target="_self">[[:</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o4</font></span><a href="pralg_1.html#K3" target="_self">:]]</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="unialg_2.html#T1">UNIALG_2:1</a></i>
;<br/>
<div><i><font color="Green">E116</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:13_2_2"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">x</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">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E2:13_2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">B</font> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">s</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">B</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E4:13_2_2"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <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="E5:13_2_2"/>
( <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U4</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E6:13_2_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U4</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">s0</font> = <font color="Maroon">s</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U4</font></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E8:13_2_2"/><i><font color="Green">E119</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="pralg_1.html#K3" target="_self">[[:</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o4</font></span><a href="pralg_1.html#K3" target="_self">:]]</a></span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o24</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="pralg_1.html#K3" target="_self">[[:</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o4</font></span><a href="pralg_1.html#K3" target="_self">:]]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/><i><font color="Green">E120</font></i>: <span class="p1">(<span class="default"><span class="p2"><a href="pralg_1.html#K3" target="_self">[[:</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o4</font></span><a href="pralg_1.html#K3" target="_self">:]]</a></span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o24</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1"><a href="pralg_1.html#K3" target="_self">[[:</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o4</font></span><a href="pralg_1.html#K3" target="_self">:]]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s0</font>
<b>by </b><i>, <a class="txt" href="pralg_1.html#E5:13"><i><font color="Green">E39</font></i></a>, <a class="ref" href="pralg_1.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">o2</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K1" target="_self">pr1</a> <font color="Maroon">s0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">o4</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K2" target="_self">pr2</a> <font color="Maroon">s0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="txt" href="pralg_1.html#E5:13"><i><font color="Green">E39</font></i></a>, <a class="ref" href="pralg_1.html#D2" target="_self">Def2</a>, <a class="txt" href="pralg_1.html#E6:13"><i><font color="Green">E41</font></i></a>, </i>
;<br/><b>reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">s</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B1</font>,<font color="Maroon">B3</font></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">s11</font> =  <a href="pralg_1.html#K1" target="_self">pr1</a> <font color="Maroon">s1</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">B1</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/><b>reconsider </b><font color="Maroon">s12</font> =  <a href="pralg_1.html#K2" target="_self">pr2</a> <font color="Maroon">s1</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">B3</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="E13:13_2_2"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K1" target="_self">pr1</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> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K1" target="_self">pr1</a> <font color="Maroon">s0</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s0</font> )
 <b>by </b><i/>;<br/><a NAME="E14:13_2_2"/><b>then </b>
<font color="Maroon">s11</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">B1</font> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="pralg_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E15:13_2_2"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">s11</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">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B1</font>
 ;<br/><i><font color="Green">E124</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">o2</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">o2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B1</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</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">U2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B1</font></span>)</span>
<b>by </b><i><a class="ref" href="unialg_2.html#T2">UNIALG_2:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B1</font>
<b>by </b><i><a class="ref" href="unialg_2.html#T1">UNIALG_2:1</a></i>
;<br/><a NAME="E17:13_2_2"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K2" target="_self">pr2</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> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K2" target="_self">pr2</a> <font color="Maroon">s0</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s0</font> )
 <b>by </b><i/>;<br/><a NAME="E18:13_2_2"/><b>then </b>
<font color="Maroon">s12</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">B3</font> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o4</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="pralg_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E19:13_2_2"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">s12</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">o4</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B3</font>
 ;<br/><i><font color="Green">E127</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">o4</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o4</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B3</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">o4</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o4</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B3</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o4</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">U4</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o4</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B3</font></span>)</span>
<b>by </b><i><a class="ref" href="unialg_2.html#T2">UNIALG_2:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o4</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B3</font>
<b>by </b><i><a class="ref" href="unialg_2.html#T1">UNIALG_2:1</a></i>
;<br/><span class="p1"><a href="pralg_1.html#K3" target="_self">[[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">o2</font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">B1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">o4</font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">B3</font></span>)</span></span><a href="pralg_1.html#K3" target="_self">:]]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">o2</font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">B1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K1" target="_self">pr1</a> <font color="Maroon">s1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">o4</font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">B3</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K2" target="_self">pr2</a> <font color="Maroon">s1</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="pralg_1.html#D1" target="_self">Def1</a>, <a class="txt" href="pralg_1.html#E5:13"><i><font color="Green">E39</font></i></a>, <a class="ref" href="pralg_1.html#D2" target="_self">Def2</a>, </i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">o2</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s11</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">o4</font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">B3</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K2" target="_self">pr2</a> <font color="Maroon">s1</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="unialg_2.html#D6">UNIALG_2:def 6</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">o2</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K1" target="_self">pr1</a> <font color="Maroon">s1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">o4</font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">B3</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K2" target="_self">pr2</a> <font color="Maroon">s1</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="pralg_1.html#E9:13"><i><font color="Green">E43</font></i></a>, <a class="txt" href="pralg_1.html#E12:13"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">o2</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K1" target="_self">pr1</a> <font color="Maroon">s1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">o4</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o4</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B3</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K2" target="_self">pr2</a> <font color="Maroon">s1</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="unialg_2.html#D6">UNIALG_2:def 6</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">o2</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K1" target="_self">pr1</a> <font color="Maroon">s1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">o4</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K2" target="_self">pr2</a> <font color="Maroon">s1</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="pralg_1.html#D3" target="_self">Def3</a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/><b>hence </b><a NAME="E22:13_2_2"/>
<span class="p1"><a href="pralg_1.html#K3" target="_self">[[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">o2</font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">B1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">o4</font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">B3</font></span>)</span></span><a href="pralg_1.html#K3" target="_self">:]]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="pralg_1.html#K3" target="_self">[[:</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o4</font></span><a href="pralg_1.html#K3" target="_self">:]]</a></span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o24</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="pralg_1.html#E7:13"><i><font color="Green">E42</font></i></a></i>;<br/></div><b>end;</b></div>
<b>thus </b><span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <span class="p2"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">o24</font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">B</font>
<b>by </b><i>, <a class="txt" href="pralg_1.html#E1:13"><i><font color="Green">E23</font></i></a>, , <a class="ref" href="unialg_2.html#D7">UNIALG_2:def 7</a></i>
<br/>.= 
<span class="p1"><a href="pralg_1.html#K3" target="_self">[[:</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o4</font></span><a href="pralg_1.html#K3" target="_self">:]]</a></span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o24</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="unialg_2.html#D6">UNIALG_2:def 6</a></i>
<br/>.= 
<span class="p1"><a href="pralg_1.html#K3" target="_self">[[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">o2</font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">B1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">o4</font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">B3</font></span>)</span></span><a href="pralg_1.html#K3" target="_self">:]]</a></span>
<b>by </b><i><a class="ref" href="pralg_1.html#D1" target="_self">Def1</a>, , <a class="txt" href="pralg_1.html#E4:13"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>
<br/>.= 
the <a href="unialg_1.html#U1">charact</a> of <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U1</font>,<font color="Maroon">U3</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i>, , <a class="txt" href="pralg_1.html#E1:13"><i><font color="Green">E23</font></i></a>, , , , </i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E31:13"/>
( the <a href="unialg_1.html#U1">charact</a> of <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U1</font>,<font color="Maroon">U3</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="unialg_2.html#K3">Opers</a> <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U2</font>,<font color="Maroon">U4</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span>,<font color="Maroon">B</font> &amp; <font color="Maroon">B</font> is <a href="unialg_2.html#V1">opers_closed</a> )
 <b>by </b><i><a class="ref" href="pralg_1.html#T1" target="_self">Th1</a>, , <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a>, <a class="ref" href="unialg_2.html#D5">UNIALG_2:def 5</a></i>;<br/>


</div>
