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

<b>set </b><font color="Maroon" title="c4">A</font> =  <a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">O1</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> as   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="ref" href="osalg_1.html#T17" title="OSALG_1:th.17">OSALG_1:17</a></i>;<br/>
<b>let </b><font color="Maroon" title="c6">o1</font>, <font color="Maroon" title="c7">o2</font> be   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_1.html#D23" title="OSALG_1:def.23">OSALG_1:def 23</a><br/>

<b>assume </b><a NAME="E2:68_1_1"/><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon" title="c6">o1</font> <a href="osalg_1.html#R6" title="OSALG_1:pred.6">&lt;=</a> <font color="Maroon" title="c7">o2</font>
 ;<br/>

<a NAME="E3:68_1_1"/><i><font color="Green" title="E38">A2</font></i>: 
( <font color="Maroon" title="c6">o1</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c7">o2</font> &amp;  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c6">o1</font> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c7">o2</font> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c6">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c7">o2</font> )
 
<b>by </b><i><a class="txt" href="osalg_4.html#E2:68_1_1"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="osalg_1.html#D22" title="OSALG_1:def.22">OSALG_1:def 22</a></i>;<br/>
<a NAME="E4:68_1_1"/><i><font color="Green" title="E39">A3</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c7">o2</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c7">o2</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o1</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c6">o1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:68_1_1"/><i><font color="Green" title="E40">A4</font></i>: 
(  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c6">o1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c7">o2</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span> &amp;  <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c6">o1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c7">o2</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="osalg_4.html#E2:68_1_1"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="osalg_1.html#T26" title="OSALG_1:th.26">OSALG_1:26</a></i>;<br/>
<a NAME="E6:68_1_1"/><b>then </b><i><font color="Green" title="E41">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o1</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c7">o2</font>,<span class="p3">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c6">o1</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E4:68_1_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E7:68_1_1"/>
(  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K15" title="OSALG_4:func.15">OSQuotCharact</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="msualg_4.html#K5" title="MSUALG_4:func.5">.</a> <font color="Maroon" title="c6">o1</font> &amp;  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c7">o2</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K15" title="OSALG_4:func.15">OSQuotCharact</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="msualg_4.html#K5" title="MSUALG_4:func.5">.</a> <font color="Maroon" title="c7">o2</font> )
 
<b>by </b><i><a class="ref" href="msualg_1.html#D11" title="MSUALG_1:def.11">MSUALG_1:def 11</a></i>;<br/>
<a NAME="E8:68_1_1"/><b>then </b><i><font color="Green" title="E42">A6</font></i>: 
(  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="osalg_4.html#K14" title="OSALG_4:func.14">OSQuotCharact</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c6">o1</font> &amp;  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c7">o2</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="osalg_4.html#K14" title="OSALG_4:func.14">OSQuotCharact</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c7">o2</font> )
 
<b>by </b><i><a class="ref" href="osalg_4.html#D21" target="_self" title="OSALG_4:def.21">Def21</a></i>;<br/>
<a NAME="E9:68_1_1"/><i><font color="Green" title="E43">A7</font></i>: 
<font color="Maroon" title="c5">O1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c6">o1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">O1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c7">o2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E3:68_1_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="osalg_1.html#D19" title="OSALG_1:def.19">OSALG_1:def 19</a></i>;<br/>
<a NAME="E10:68_1_1"/>
( <font color="Maroon" title="c6">o1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c7">o2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S</font> )
 
;<br/>
<a NAME="E11:68_1_1"/><b>then </b><i><font color="Green" title="E44">A8</font></i>: 
( <font color="Maroon" title="c6">o1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c7">o2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">S</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:68_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c6">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c7">o2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E3:68_1_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="osalg_1.html#D7" title="OSALG_1:def.7">OSALG_1:def 7</a></i>;<br/>
<a NAME="E13:68_1_1"/><b>then </b><i><font color="Green" title="E45">A9</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c6">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c7">o2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E14:68_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o1</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o1</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c7">o2</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:68_1_1_1"/><i><font color="Green" title="E46">A10</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o1</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:68_1_1_1"/><i><font color="Green" title="E47">A11</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c6">o1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E1:68_1_1_1"><i><font color="Green" title="E46">A10</font></i></a></i>;<br/>
<a NAME="E3:68_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c7">o2</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E5:68_1_1"><i><font color="Green" title="E40">A4</font></i></a></i>;<br/>
<a NAME="E4:68_1_1_1"/><b>then </b><i><font color="Green" title="E48">A12</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">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" title="OSALG_4:func.7">OSClass</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">o1</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">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" title="OSALG_4:func.7">OSClass</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">o2</font> )
 
<b>by </b><i><a class="txt" href="osalg_4.html#E2:68_1_1_1"><i><font color="Green" title="E47">A11</font></i></a>, <a class="ref" href="msualg_1.html#D9" title="MSUALG_1:def.9">MSUALG_1:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">a1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c6">o1</font>,<font color="Maroon" title="c2">U1</font><b> such that </b><br/><a NAME="E6:68_1_1_1"/><i><font color="Green" title="E49">A13</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">R</font> <a href="osalg_4.html#K9" title="OSALG_4:func.9">#_os</a> <font color="Maroon" title="c9">a1</font>
 <b>by </b><i><a class="ref" href="osalg_4.html#T15" target="_self" title="OSALG_4:th.15">Th15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">a2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c7">o2</font>,<font color="Maroon" title="c2">U1</font><b> such that </b><br/><a NAME="E8:68_1_1_1"/><i><font color="Green" title="E50">A14</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">R</font> <a href="osalg_4.html#K9" title="OSALG_4:func.9">#_os</a> <font color="Maroon" title="c10">a2</font>
 <b>by </b><i><a class="txt" href="osalg_4.html#E4:68_1_1_1"><i><font color="Green" title="E48">A12</font></i></a>, <a class="ref" href="osalg_4.html#T15" target="_self" title="OSALG_4:th.15">Th15</a></i>;<br/>
<a NAME="E9:68_1_1_1"/>
for <font color="Olive" title="b1">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">a1</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">a2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c7">o2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">y</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:68_1_1_1_1"/><i><font color="Green" title="E51">A15</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">a1</font>
 ;<br/>

<a NAME="E2:68_1_1_1_1"/><i><font color="Green" title="E52">A16</font></i>: 
( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c6">o1</font></span>)</span> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c7">o2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="osalg_4.html#E13:68_1_1"><i><font color="Green" title="E45">A9</font></i></a>, <a class="txt" href="osalg_4.html#E1:68_1_1_1_1"><i><font color="Green" title="E51">A15</font></i></a>, <a class="ref" href="msualg_6.html#T2" title="MSUALG_6:th.2">MSUALG_6:2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">z1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c6">o1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span><b> such that </b><br/><a NAME="E4:68_1_1_1_1"/><i><font color="Green" title="E53">A17</font></i>: 
( <font color="Maroon" title="c12">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="osalg_4.html#K9" title="OSALG_4:func.9">#_os</a> <font color="Maroon" title="c9">a1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c12">z1</font> )
 <b>by </b><i><a class="ref" href="osalg_4.html#D15" target="_self" title="OSALG_4:def.15">Def15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">z2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c7">o2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span><b> such that </b><br/><a NAME="E6:68_1_1_1_1"/><i><font color="Green" title="E54">A18</font></i>: 
( <font color="Maroon" title="c13">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="osalg_4.html#K9" title="OSALG_4:func.9">#_os</a> <font color="Maroon" title="c10">a2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c13">z2</font> )
 <b>by </b><i><a class="txt" href="osalg_4.html#E2:68_1_1_1_1"><i><font color="Green" title="E52">A16</font></i></a>, <a class="ref" href="osalg_4.html#D15" target="_self" title="OSALG_4:def.15">Def15</a></i>;<br/>
<a NAME="E7:68_1_1_1_1"/><i><font color="Green" title="E55">A19</font></i>: 
( <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c6">o1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c6">o1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c7">o2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c7">o2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> )
 
<b>by </b><i><a class="txt" href="osalg_4.html#E2:68_1_1_1_1"><i><font color="Green" title="E52">A16</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">s1</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c6">o1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font>, <font color="Maroon" title="c15">s2</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c7">o2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> as   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="osalg_4.html#E2:68_1_1_1_1"><i><font color="Green" title="E52">A16</font></i></a>, <a class="ref" href="partfun1.html#T27" title="PARTFUN1:th.27">PARTFUN1:27</a></i>;<br/>
<a NAME="E9:68_1_1_1_1"/><i><font color="Green" title="E56">A20</font></i>: 
<font color="Maroon" title="c14">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">s2</font>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E3:68_1_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="osalg_4.html#E2:68_1_1_1_1"><i><font color="Green" title="E52">A16</font></i></a>, <a class="ref" href="osalg_1.html#D7" title="OSALG_1:def.7">OSALG_1:def 7</a></i>;<br/>
<a NAME="E10:68_1_1_1_1"/><b>then </b>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c6">o1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c7">o2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E7:68_1_1_1_1"><i><font color="Green" title="E55">A19</font></i></a>, <a class="ref" href="osalg_1.html#D19" title="OSALG_1:def.19">OSALG_1:def 19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">z12</font> = <font color="Maroon" title="c12">z1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c7">o2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E12:68_1_1_1_1"/>
 <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c13">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c16">z12</font>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E6:68_1_1_1"><i><font color="Green" title="E49">A13</font></i></a>, <a class="txt" href="osalg_4.html#E8:68_1_1_1"><i><font color="Green" title="E50">A14</font></i></a>, <a class="txt" href="osalg_4.html#E4:68_1_1_1_1"><i><font color="Green" title="E53">A17</font></i></a>, <a class="txt" href="osalg_4.html#E6:68_1_1_1_1"><i><font color="Green" title="E54">A18</font></i></a>, <a class="txt" href="osalg_4.html#E7:68_1_1_1_1"><i><font color="Green" title="E55">A19</font></i></a>, <a class="txt" href="osalg_4.html#E9:68_1_1_1_1"><i><font color="Green" title="E56">A20</font></i></a>, <a class="ref" href="osalg_4.html#T5" target="_self" title="OSALG_4:th.5">Th5</a></i>;<br/>
<b>hence </b><a NAME="E13:68_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">a2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c7">o2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="osalg_4.html#E4:68_1_1_1_1"><i><font color="Green" title="E53">A17</font></i></a>, <a class="txt" href="osalg_4.html#E6:68_1_1_1_1"><i><font color="Green" title="E54">A18</font></i></a>, <a class="ref" href="osalg_4.html#T13" target="_self" title="OSALG_4:th.13">Th13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:68_1_1_1"/><b>then </b><i><font color="Green" title="E51">A21</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o1</font>,<font color="Maroon" title="c2">U1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c7">o2</font>,<font color="Maroon" title="c2">U1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c7">o2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E2:68_1_1"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="osalg_4.html#D28" target="_self" title="OSALG_4:def.28">Def28</a></i>;<br/>
<a NAME="E11:68_1_1_1"/><i><font color="Green" title="E52">A22</font></i>: 
( <span class="p1">(<span class="default"><a href="osalg_4.html#K14" title="OSALG_4:func.14">OSQuotCharact</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c6">o1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="osalg_4.html#K9" title="OSALG_4:func.9">#_os</a> <font color="Maroon" title="c9">a1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K10" title="OSALG_4:func.10">OSQuotRes</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c6">o1</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o1</font>,<font color="Maroon" title="c2">U1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a1</font> &amp; <span class="p1">(<span class="default"><a href="osalg_4.html#K14" title="OSALG_4:func.14">OSQuotCharact</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c7">o2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="osalg_4.html#K9" title="OSALG_4:func.9">#_os</a> <font color="Maroon" title="c10">a2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K10" title="OSALG_4:func.10">OSQuotRes</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c7">o2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c7">o2</font>,<font color="Maroon" title="c2">U1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a2</font> )
 
<b>by </b><i><a class="txt" href="osalg_4.html#E4:68_1_1_1"><i><font color="Green" title="E48">A12</font></i></a>, <a class="txt" href="osalg_4.html#E6:68_1_1_1"><i><font color="Green" title="E49">A13</font></i></a>, <a class="txt" href="osalg_4.html#E8:68_1_1_1"><i><font color="Green" title="E50">A14</font></i></a>, <a class="ref" href="osalg_4.html#D20" target="_self" title="OSALG_4:def.20">Def20</a></i>;<br/>
 <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c6">o1</font>,<font color="Maroon" title="c2">U1</font> = 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">o1</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D10" title="MSUALG_1:def.10">MSUALG_1:def 10</a></i>
<br/>.= 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">o1</font></span>)</span>
<b>by </b><i><a class="txt" href="osalg_4.html#E11:68_1_1"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c6">o1</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D7" title="MSUALG_1:def.7">MSUALG_1:def 7</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">da1</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o1</font>,<font color="Maroon" title="c2">U1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c6">o1</font></span>)</span> ;<br/>
<a NAME="E14:68_1_1_1"/>
<font color="Maroon" title="c9">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c6">o1</font>,<font color="Maroon" title="c2">U1</font>
 
;<br/>
<a NAME="E15:68_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o1</font>,<font color="Maroon" title="c2">U1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E53">A23</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K10" title="OSALG_4:func.10">OSQuotRes</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c6">o1</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o1</font>,<font color="Maroon" title="c2">U1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a1</font> = 
<span class="p1">(<span class="default"><a href="osalg_4.html#K10" title="OSALG_4:func.10">OSQuotRes</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c6">o1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">da1</font>
<b>by </b><i><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
 <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c11">da1</font>
<b>by </b><i><a class="ref" href="osalg_4.html#D16" target="_self" title="OSALG_4:def.16">Def16</a></i>
;<br/>
 <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c7">o2</font>,<font color="Maroon" title="c2">U1</font> = 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">o2</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D10" title="MSUALG_1:def.10">MSUALG_1:def 10</a></i>
<br/>.= 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">o2</font></span>)</span>
<b>by </b><i><a class="txt" href="osalg_4.html#E11:68_1_1"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c7">o2</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D7" title="MSUALG_1:def.7">MSUALG_1:def 7</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">da2</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c7">o2</font>,<font color="Maroon" title="c2">U1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c7">o2</font></span>)</span> ;<br/>
<a NAME="E19:68_1_1_1"/>
<font color="Maroon" title="c10">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c7">o2</font>,<font color="Maroon" title="c2">U1</font>
 
;<br/>
<a NAME="E20:68_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c7">o2</font>,<font color="Maroon" title="c2">U1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E54">A24</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K10" title="OSALG_4:func.10">OSQuotRes</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c7">o2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c7">o2</font>,<font color="Maroon" title="c2">U1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a2</font> = 
<span class="p1">(<span class="default"><a href="osalg_4.html#K10" title="OSALG_4:func.10">OSQuotRes</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c7">o2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">da2</font>
<b>by </b><i><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
 <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c12">da2</font>
<b>by </b><i><a class="ref" href="osalg_4.html#D16" target="_self" title="OSALG_4:def.16">Def16</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c13">da12</font> = <font color="Maroon" title="c11">da1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c7">o2</font></span>)</span> <b>by </b><i><a class="txt" href="osalg_4.html#E9:68_1_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
 <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c11">da1</font> = 
 <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c13">da12</font>
<b>by </b><i><a class="txt" href="osalg_4.html#E3:68_1_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="osalg_4.html#T5" target="_self" title="OSALG_4:th.5">Th5</a></i>
<br/>.= 
 <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c12">da2</font>
<b>by </b><i><a class="txt" href="osalg_4.html#E10:68_1_1_1"><i><font color="Green" title="E51">A21</font></i></a>, <a class="ref" href="osalg_4.html#T13" target="_self" title="OSALG_4:th.13">Th13</a></i>
;<br/>
<b>hence </b><a NAME="E24:68_1_1_1"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o1</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c7">o2</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="osalg_4.html#E8:68_1_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="txt" href="osalg_4.html#E6:68_1_1_1"><i><font color="Green" title="E49">A13</font></i></a>, <a class="txt" href="osalg_4.html#E8:68_1_1_1"><i><font color="Green" title="E50">A14</font></i></a>, <a class="txt" href="osalg_4.html#E11:68_1_1_1"><i><font color="Green" title="E52">A22</font></i></a>, <a class="txt" href="osalg_4.html#E16:68_1_1_1"><i><font color="Green" title="E53">A23</font></i></a>, <a class="txt" href="osalg_4.html#E21:68_1_1_1"><i><font color="Green" title="E54">A24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:68_1_1"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c7">o2</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c6">o1</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="osalg_4.html#E6:68_1_1"><i><font color="Green" title="E41">A5</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>


</div>
