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

<b>set </b><font color="Maroon">Ca</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <font color="Maroon">R</font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>;<br/>
<b>set </b><font color="Maroon">Cr</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <font color="Maroon">R</font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>;<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>for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">A</font>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="osalg_4.html#K9" target="_self">#_os</a> <font color="Olive">a</font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K10" target="_self">OSQuotRes</a> <font color="Maroon">R</font>,<font color="Maroon">o</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font>;<br/>
<a NAME="E1:45_1_1"/><i><font color="Green">E33</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <font color="Maroon">R</font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <font color="Maroon">R</font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</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="45_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="E1:45_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <font color="Maroon">R</font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:45_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="osalg_4.html#K9" target="_self">#_os</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">y</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K10" target="_self">OSQuotRes</a> <font color="Maroon">R</font>,<font color="Maroon">o</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>;<br/>

<a NAME="E4:45_1_1_1"/><i><font color="Green">E44</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">A</font> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:45_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>
 
;<br/>
<b>set </b><font color="Maroon">ro</font> =  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o</font>;<br/>
<b>set </b><font color="Maroon">ar</font> =  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font>;<br/>
<a NAME="E6:45_1_1_1"/><i><font color="Green">E61</font></i>: 
 <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="osalg_4.html#K3" target="_self">CComp</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:45_1_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E62</font></i>: <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> = 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>
;<br/>
<a NAME="E9:45_1_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <font color="Maroon">R</font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E65</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <font color="Maroon">R</font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> = 
<span class="p1">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>
;<br/>
<a NAME="E11:45_1_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K10" target="_self">OSQuotRes</a> <font color="Maroon">R</font>,<font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K10" target="_self">OSQuotRes</a> <font color="Maroon">R</font>,<font color="Maroon">o</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:45_1_1_1"/><i><font color="Green">E68</font></i>: 
 <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="msualg_1.html#D10">MSUALG_1:def 10</a></i>;<br/>
<a NAME="E13:45_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="osalg_4.html#K10" target="_self">OSQuotRes</a> <font color="Maroon">R</font>,<font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K10" target="_self">OSQuotRes</a> <font color="Maroon">R</font>,<font color="Maroon">o</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E14:45_1_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><a href="osalg_4.html#K10" target="_self">OSQuotRes</a> <font color="Maroon">R</font>,<font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E15:45_1_1_1"/><i><font color="Green">E71</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K10" target="_self">OSQuotRes</a> <font color="Maroon">R</font>,<font color="Maroon">o</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<b>hence </b><a NAME="E16:45_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <font color="Maroon">R</font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i>, , <a class="txt" href="osalg_4.html#E1:45_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>

<b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E17:45_1_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="osalg_4.html#K9" target="_self">#_os</a> <font color="Maroon">b</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">da</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>, <font color="Maroon">db</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span> <b>by </b><i>, <a class="ref" href="msualg_1.html#D10">MSUALG_1:def 10</a></i>;<br/>
<a NAME="E19:45_1_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">da</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="osalg_4.html#K4" target="_self">-carrier_of</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K3" target="_self">CComp</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E75</font></i>: <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><a href="osalg_4.html#K10" target="_self">OSQuotRes</a> <font color="Maroon">R</font>,<font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
 <a href="osalg_4.html#K8" target="_self">OSClass</a> <font color="Maroon">R</font>,<font color="Maroon">da</font>
<b>by </b><i/>
<br/>.= 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K5" target="_self">CompClass</a> <font color="Maroon">R</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K3" target="_self">CComp</a> <span class="p3">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">da</font>

;<br/>
<i><font color="Green">E86</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K10" target="_self">OSQuotRes</a> <font color="Maroon">R</font>,<font color="Maroon">o</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><a href="osalg_4.html#K10" target="_self">OSQuotRes</a> <font color="Maroon">R</font>,<font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">db</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
 <a href="osalg_4.html#K8" target="_self">OSClass</a> <font color="Maroon">R</font>,<font color="Maroon">db</font>
<b>by </b><i/>
<br/>.= 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K5" target="_self">CompClass</a> <font color="Maroon">R</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K3" target="_self">CComp</a> <span class="p3">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">db</font>

;<br/>
<a NAME="E22:45_1_1_1"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="msualg_4.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:45_1_1_1_5"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:45_1_1_1_5"/><i><font color="Green">E88</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="msualg_3.html#T6">MSUALG_3:6</a></i>;<br/>
<b>then consider </b><font color="Maroon">ya</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E4:45_1_1_1_5"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">ya</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="osalg_4.html#K9" target="_self">#_os</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="osalg_4.html#K8" target="_self">OSClass</a> <font color="Maroon">R</font>,<font color="Maroon">ya</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">yb</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E6:45_1_1_1_5"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">yb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="osalg_4.html#K9" target="_self">#_os</a> <font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="osalg_4.html#K8" target="_self">OSClass</a> <font color="Maroon">R</font>,<font color="Maroon">yb</font> )
 <b>by </b><i>, <a class="ref" href="osalg_4.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>thus </b><a NAME="E7:45_1_1_1_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="msualg_4.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , , <a class="txt" href="osalg_4.html#E3:45_1_1_1"><i><font color="Green">E42</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:45_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">da</font>,<font color="Maroon">db</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="msualg_4.html#K2">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_4.html#D6">MSUALG_4:def 6</a></i>;<br/>
<a NAME="E24:45_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">da</font>,<font color="Maroon">db</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="osalg_4.html#K5" target="_self">CompClass</a> <font color="Maroon">R</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K3" target="_self">CComp</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E25:45_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K10" target="_self">OSQuotRes</a> <font color="Maroon">R</font>,<font color="Maroon">o</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
 <b>by </b><i>, , , <a class="ref" href="eqrel_1.html#T44">EQREL_1:44</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:45_1_1"/><i><font color="Green">E92</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <font color="Maroon">R</font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <font color="Maroon">R</font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <font color="Maroon">R</font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</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>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <font color="Maroon">R</font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <font color="Maroon">R</font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E5:45_1_1"/>
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">A</font>  st <font color="Maroon">R</font> <a href="osalg_4.html#K9" target="_self">#_os</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <font color="Maroon">R</font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="osalg_4.html#K9" target="_self">#_os</a> <font color="Olive">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K10" target="_self">OSQuotRes</a> <font color="Maroon">R</font>,<font color="Maroon">o</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font>
 <b>by </b><i/>;<br/>


</div>
