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

<b>let </b><font color="Maroon">S</font> be  <a href="osalg_4.html#V2" target="_self">locally_directed</a> <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">U1</font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">R</font> be   <a href="osalg_4.html#NM4">OSCongruence</a> of <font color="Maroon">U1</font>;<br/>





<b>set </b><font color="Maroon">F</font> =  <a href="osalg_4.html#K18" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">QA</font> =  <a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">S1</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>;<br/>
<a NAME="E1:51"/>
for <font color="Olive">o</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>  st  <a href="msualg_1.html#K3">Args</a> <font color="Olive">o</font>,<font color="Maroon">U1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Olive">o</font>,<font color="Maroon">U1</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Olive">o</font></span>)</span></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="Olive">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="msualg_3.html#K6">#</a> <font color="Olive">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E1:51_1"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">x</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">U1</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/>
<i><font color="Green">E33</font></i>:  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="osalg_4.html#K15" target="_self">OSQuotCharact</a> <font color="Maroon">R</font></span>)</span> <a href="msualg_4.html#K5">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
<br/>.= 
 <a href="osalg_4.html#K14" target="_self">OSQuotCharact</a> <font color="Maroon">R</font>,<font color="Maroon">o</font>
<b>by </b><i/>
;<br/>
<a NAME="E3:51_1"/><i><font color="Green">E42</font></i>: 
(  <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#K18" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="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">x</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/>
<a NAME="E4:51_1"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="osalg_4.html#K9" target="_self">#_os</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">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="msualg_4.html#K6">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E5:51_1"/>
 <a href="relat_1.html#K1">dom</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="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:51_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K7" target="_self">OSClass</a> <font color="Maroon">R</font></span>)</span>
 
;<br/>
<a NAME="E7:51_1"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="osalg_4.html#K9" target="_self">#_os</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E8:51_1"/>
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>;<br/>
<a NAME="E9:51_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<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> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</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="msualg_4.html#K6">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="msafree.html#T1">MSAFREE:1</a></i>;<br/>
<a NAME="E10:51_1"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</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> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="osalg_4.html#K9" target="_self">#_os</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:51_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</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>
 ;<br/>

<b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">a</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>set </b><font color="Maroon">Fo</font> =  <a href="osalg_4.html#K17" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font>,<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>;<br/>
<b>set </b><font color="Maroon">s</font> = <span class="p1">(<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>;<br/>
<a NAME="E3:51_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="msualg_4.html#K6">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E2:51_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="msualg_4.html#K6">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</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="funct_1.html#K1">.</a> <font color="Maroon">n</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">U1</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>by </b><i><a class="txt" href="osalg_4.html#E2:51_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">xn</font> = <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</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>by </b><i>, <a class="ref" href="msualg_3.html#T6">MSUALG_3:6</a></i>;<br/>
<b>consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</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="E7:51_1_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</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">x</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">z</font> )
 <b>by </b><i><a class="txt" href="osalg_4.html#E2:51_1"><i><font color="Green">E33</font></i></a>, </i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<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></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="txt" href="osalg_4.html#E2:51_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="msualg_3.html#D8">MSUALG_3:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="osalg_4.html#K17" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font>,<span class="p2">(<span class="default"><span class="p3">(<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></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">xn</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="osalg_4.html#K9" target="_self">#_os</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E11:51_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="osalg_4.html#K18" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="msualg_3.html#K6">#</a> <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">x</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E12:51_1"/><i><font color="Green">E70</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">U1</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">U1</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">U1</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">U1</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:51_1"/><i><font color="Green">E71</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E14:51_1"/><i><font color="Green">E73</font></i>: 
(  <a href="relset_1.html#K4">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K5">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E15:51_1"/>
 <a href="relset_1.html#K5">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E16:51_1"/><b>then </b>
 <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">U1</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="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<b>then </b><i><font color="Green">E74</font></i>: <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</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">U1</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">U1</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="E18:51_1"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</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/>
<b>reconsider </b><font color="Maroon">dx</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</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="E20:51_1"/>
 <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">U1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <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>
 
<b>by </b><i>, , , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E21:51_1"/><b>then </b><i><font color="Green">E86</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">U1</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">U1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> = 
<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">U1</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="osalg_4.html#E2:51_1"><i><font color="Green">E33</font></i></a>, </i>
<br/>.= 
<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">dx</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">dx</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="osalg_4.html#K17" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font>,<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> <a href="funct_2.html#K3">.</a> <font color="Maroon">dx</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="msualg_3.html#K1">.</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> <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">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E23:51_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="msualg_3.html#K1">.</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> <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">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:51"/>
 <a href="osalg_4.html#K18" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U1</font>, <a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D10">MSUALG_3:def 10</a><br/>

<a NAME="E3:51"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/> <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:51_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><a href="osalg_4.html#K18" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E4:51_2"/><i><font color="Green">E87</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="osalg_4.html#K6" target="_self">OSClass</a> <font color="Maroon">R</font>,<font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:51_2"/><i><font color="Green">E88</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:51_2"/>
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> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2_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="E1:51_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:51_2_1"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <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> <font color="Maroon">s</font></span>)</span></span>)</span>,<font color="Maroon">a1</font> )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">a2</font> = <font color="Maroon">a1</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <b>by </b><i/>;<br/>
<a NAME="E5:51_2_1"/><i><font color="Green">E91</font></i>: 
 <a href="osalg_4.html#K8" target="_self">OSClass</a> <font color="Maroon">R</font>,<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:51_2_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:51_2_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#R1">=</a>  <a href="osalg_4.html#K17" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font>,<font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:51_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:51_2"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:51_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:51"/>
 <a href="osalg_4.html#K18" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font> is <a href="msualg_3.html#V2">"onto"</a>
 <b>by </b><i><a class="ref" href="msualg_3.html#D3">MSUALG_3:def 3</a></i>;<br/>

<b>thus </b><a NAME="E5:51"/>
 <a href="osalg_4.html#K18" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font> is <a href="osalg_3.html#V1">order-sorted</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">s2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_3.html#D1">OSALG_3:def 1</a><br/>

<b>assume </b><a NAME="E1:51_3"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s2</font>
 ;<br/>

<b>let </b><font color="Maroon">a1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:51_3"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <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#K18" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s1</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">S2</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> as   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="osalg_1.html#T17">OSALG_1:17</a></i>;<br/>
<a NAME="E4:51_3"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font>
 
<b>by </b><i>, <a class="ref" href="osalg_1.html#D18">OSALG_1:def 18</a></i>;<br/>
<a NAME="E5:51_3"/><i><font color="Green">E97</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#K18" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:51_3"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:51_3"/>
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <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#K18" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s2</font></span>)</span>
 <b>by </b><i>, </i>;<br/>

<b>reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">a1</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">b2</font> = <font color="Maroon">a1</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> <b>by </b><i>, </i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> = 
<span class="p1">(<span class="default"><a href="osalg_4.html#K17" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font>,<font color="Maroon">s1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font>
<b>by </b><i/>
<br/>.= 
 <a href="osalg_4.html#K8" target="_self">OSClass</a> <font color="Maroon">R</font>,<font color="Maroon">b1</font>
<b>by </b><i/>
<br/>.= 
 <a href="osalg_4.html#K8" target="_self">OSClass</a> <font color="Maroon">R</font>,<font color="Maroon">b2</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="osalg_4.html#K17" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font>,<font color="Maroon">s2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b2</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K18" target="_self">OSNat_Hom</a> <font color="Maroon">U1</font>,<font color="Maroon">R</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font>
<b>by </b><i/>
;<br/>


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


</div>
