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

<b>let </b><font color="Maroon" title="c1">S</font> be  <a href="osalg_4.html#V2" title="OSALG_4:attr.2">locally_directed</a> <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon" title="c2">U1</font>, <font color="Maroon" title="c3">U2</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c4">F</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font>;<br/>





<b>set </b><font color="Maroon" title="c5">mc</font> =  <a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font>;<br/>
<b>set </b><font color="Maroon" title="c6">qa</font> =  <a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">qh</font> =  <a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font>;<br/>
<b>set </b><font color="Maroon" title="c8">S1</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font>;<br/>
<b>assume </b><a NAME="E1:56"/><i><font color="Green" title="E31">A1</font></i>: 
( <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font> &amp; <font color="Maroon" title="c4">F</font> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> )
 ;<br/>

<a NAME="E2:56"/>
for <font color="Olive" title="b1">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S</font>  st  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Olive" title="b1">o</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>,<span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/>for <font color="Olive" title="b2">x</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="Olive" title="b1">o</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>,<span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b1">o</font></span>)</span></span>)</span> <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#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b1">o</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>,<span class="p4">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span> <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="Olive" title="b1">o</font>,<font color="Maroon" title="c3">U2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Olive" title="b2">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">o</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S</font>;<br/>

<b>assume </b><a NAME="E1:56_1"/>
 <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c9">o</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>,<span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c10">x</font> be    <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="c9">o</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>,<span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c11">o1</font> = <font color="Maroon" title="c9">o</font> as   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<b>set </b><font color="Maroon" title="c12">ro</font> =  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c9">o</font>;<br/>
<b>set </b><font color="Maroon" title="c13">ar</font> =  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c9">o</font>;<br/>
<i><font color="Green" title="E32">A2</font></i>:  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o</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>,<span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="osalg_4.html#K15" title="OSALG_4:func.15">OSQuotCharact</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="msualg_4.html#K5" title="MSUALG_4:func.5">.</a> <font color="Maroon" title="c9">o</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 href="osalg_4.html#K14" title="OSALG_4:func.14">OSQuotCharact</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c11">o1</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="E4:56_1"/>
 <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c9">o</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>,<span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p4">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span></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="c9">o</font>
 
<b>by </b><i><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="c14">a</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="c9">o</font>,<font color="Maroon" title="c2">U1</font><b> such that </b><br/><a NAME="E6:56_1"/><i><font color="Green" title="E33">A3</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="osalg_4.html#K9" title="OSALG_4:func.9">#_os</a> <font color="Maroon" title="c14">a</font>
 <b>by </b><i><a class="ref" href="osalg_4.html#T15" target="_self" title="OSALG_4:th.15">Th15</a></i>;<br/>
<a NAME="E7:56_1"/><i><font color="Green" title="E34">A4</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="c9">o</font>,<font color="Maroon" title="c2">U1</font></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="c9">o</font>,<font color="Maroon" title="c2">U1</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon" title="c2">U1</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="c9">o</font>,<font color="Maroon" title="c2">U1</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="E8:56_1"/>
<font color="Maroon" title="c9">o</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="E9:56_1"/><b>then </b>
<font color="Maroon" title="c9">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <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>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>then </b><i><font color="Green" title="E35">A5</font></i>: <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="c9">o</font> = 
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="c9">o</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</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="c9">o</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/>
<a NAME="E11:56_1"/><b>then </b><i><font color="Green" title="E36">A6</font></i>: 
 <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon" title="c2">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c9">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D10" title="MSUALG_1:def.10">MSUALG_1:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">da</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o</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="c14">a</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="c9">o</font></span>)</span> <b>by </b><i><a class="txt" href="osalg_4.html#E10:56_1"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="msualg_1.html#D10" title="MSUALG_1:def.10">MSUALG_1:def 10</a></i>;<br/>
<a NAME="E13:56_1"/><i><font color="Green" title="E37">A7</font></i>: 
<span class="p1">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3: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="c9">o</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="osalg_4.html#K20" title="OSALG_4:func.20">OSHomQuot</a> <font color="Maroon" title="c4">F</font>,<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="c9">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="osalg_4.html#D27" target="_self" title="OSALG_4:def.27">Def27</a></i>;<br/>
<a NAME="E14:56_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon" title="c2">U1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K10" title="OSALG_4:func.10">OSQuotRes</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c9">o</font></span>)</span>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E7:56_1"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="osalg_4.html#E10:56_1"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="osalg_4.html#E11:56_1"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E15:56_1"/><b>then </b><i><font color="Green" title="E38">A8</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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> <span class="p3">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c9">o</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="c9">o</font>,<font color="Maroon" title="c2">U1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c9">o</font>,<font color="Maroon" title="c2">U1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E16:56_1"/><i><font color="Green" title="E39">A9</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c10">x</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="c9">o</font></span>)</span> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c14">a</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="c9">o</font></span>)</span> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">x</font> <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="c9">o</font></span>)</span> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">a</font> <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="c9">o</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="msualg_3.html#T6" title="MSUALG_3:th.6">MSUALG_3:6</a></i>;<br/>
<div><i><font color="Green" title="E40">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:56_1_3"/><i><font color="Green" title="E41">A11</font></i>: 
<font color="Maroon" title="c16">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="c9">o</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c17">n</font> = <font color="Maroon" title="c16">y</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:56_1_3"/><i><font color="Green" title="E42">A12</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="c9">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</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="c9">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font>
 <b>by </b><i><a class="txt" href="osalg_4.html#E1:56_1_3"><i><font color="Green" title="E41">A11</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E4:56_1_3"/>
<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="c9">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</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="c9">o</font></span>)</span>
 <b>by </b><i><a class="txt" href="osalg_4.html#E1:56_1_3"><i><font color="Green" title="E41">A11</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c18">s</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="c9">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> as   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font> ;<br/><b>consider </b><font color="Maroon" title="c19">an</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> <font color="Maroon" title="c18">s</font><b> such that </b><br/><a NAME="E7:56_1_3"/><i><font color="Green" title="E43">A13</font></i>: 
( <font color="Maroon" title="c19">an</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> &amp; <font color="Maroon" title="c10">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c19">an</font> )
 <b>by </b><i><a class="txt" href="osalg_4.html#E6:56_1"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="osalg_4.html#E1:56_1_3"><i><font color="Green" title="E41">A11</font></i></a>, <a class="txt" href="osalg_4.html#E3:56_1_3"><i><font color="Green" title="E42">A12</font></i></a>, <a class="ref" href="osalg_4.html#D15" target="_self" title="OSALG_4:def.15">Def15</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c18">s</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="c10">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font></span>)</span>
<b>by </b><i><a class="txt" href="osalg_4.html#E16:56_1"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="osalg_4.html#E1:56_1_3"><i><font color="Green" title="E41">A11</font></i></a>, <a class="txt" href="osalg_4.html#E3:56_1_3"><i><font color="Green" title="E42">A12</font></i></a>, <a class="ref" href="msualg_3.html#D8" title="MSUALG_3:def.8">MSUALG_3:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="osalg_4.html#K20" title="OSALG_4:func.20">OSHomQuot</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c18">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c19">an</font></span>)</span>
<b>by </b><i><a class="txt" href="osalg_4.html#E7:56_1_3"><i><font color="Green" title="E43">A13</font></i></a>, <a class="ref" href="osalg_4.html#D27" target="_self" title="OSALG_4:def.27">Def27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c18">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">an</font>
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="osalg_4.html#D26" target="_self" title="OSALG_4:def.26">Def26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font>
<b>by </b><i><a class="txt" href="osalg_4.html#E16:56_1"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="osalg_4.html#E1:56_1_3"><i><font color="Green" title="E41">A11</font></i></a>, <a class="txt" href="osalg_4.html#E3:56_1_3"><i><font color="Green" title="E42">A12</font></i></a>, <a class="txt" href="osalg_4.html#E7:56_1_3"><i><font color="Green" title="E43">A13</font></i></a>, <a class="ref" href="msualg_3.html#D8" title="MSUALG_3:def.8">MSUALG_3:def 8</a></i>
;<br/><b>hence </b><a NAME="E9:56_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E18:56_1"/>
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c9">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</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="c9">o</font>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D6" title="MSUALG_1:def.6">MSUALG_1:def 6</a></i>;<br/>
<a NAME="E19:56_1"/><b>then </b>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K7" title="OSALG_4:func.7">OSClass</a> <span class="p3">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="msualg_4.html#K6" title="MSUALG_4:func.6">*</a> <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="c9">o</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p4">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span></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="c9">o</font>
 
<b>by </b><i><a class="ref" href="msafree.html#T1" title="MSAFREE:th.1">MSAFREE:1</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o</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>,<span class="p3">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> = 
<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> <span class="p3">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c9">o</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="c9">o</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="c14">a</font>
<b>by </b><i><a class="txt" href="osalg_4.html#E3:56_1"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="osalg_4.html#E6:56_1"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="osalg_4.html#D20" target="_self" title="OSALG_4:def.20">Def20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="osalg_4.html#K10" title="OSALG_4:func.10">OSQuotRes</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c9">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">da</font>
<b>by </b><i><a class="txt" href="osalg_4.html#E7:56_1"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="osalg_4.html#E15:56_1"><i><font color="Green" title="E38">A8</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
 <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c15">da</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>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c9">o</font></span>)</span></span>)</span> <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#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o</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>,<span class="p4">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c9">o</font></span>)</span></span>)</span> <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#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o</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="c14">a</font></span>)</span>
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="osalg_4.html#E13:56_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="osalg_4.html#D26" target="_self" title="OSALG_4:def.26">Def26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon" title="c3">U2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c14">a</font></span>)</span>
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="msualg_3.html#D9" title="MSUALG_3:def.9">MSUALG_3:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E22:56_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c9">o</font></span>)</span></span>)</span> <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#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o</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>,<span class="p4">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <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="c9">o</font>,<font color="Maroon" title="c3">U2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c10">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="osalg_4.html#E16:56_1"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="osalg_4.html#E17:56_1"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:56"/>
 <a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c3">U2</font>
 <b>by </b><i><a class="ref" href="msualg_3.html#D9" title="MSUALG_3:def.9">MSUALG_3:def 9</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D11" title="MSUALG_3:def.11">MSUALG_3:def 11</a><br/>

<a NAME="E4:56"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> holds <br/><span class="p1">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>set </b><font color="Maroon" title="c10">f</font> = <span class="p1">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>;<br/>
<b>assume </b><a NAME="E1:56_2"/>
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c11">s</font> = <font color="Maroon" title="c9">i</font> as   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E3:56_2"/><i><font color="Green" title="E32">A14</font></i>: 
<span class="p1">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_4.html#K20" title="OSALG_4:func.20">OSHomQuot</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c11">s</font>
 
<b>by </b><i><a class="ref" href="osalg_4.html#D27" target="_self" title="OSALG_4:def.27">Def27</a></i>;<br/>
<a NAME="E4:56_2"/>
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <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#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x1</font>, <font color="Maroon" title="c13">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:56_2_1"/><i><font color="Green" title="E33">A15</font></i>: 
( <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> &amp; <font color="Maroon" title="c13">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font> <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#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x2</font> )
 ;<br/>

<a NAME="E2:56_2_1"/><b>then </b>
( <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K7" title="OSALG_4:func.7">OSClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s</font> &amp; <font color="Maroon" title="c13">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K7" title="OSALG_4:func.7">OSClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s</font> )
 
<b>by </b><i><a class="txt" href="osalg_4.html#E3:56_2"><i><font color="Green" title="E32">A14</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:56_2_1"/><b>then </b><i><font color="Green" title="E34">A16</font></i>: 
( <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c11">s</font> &amp; <font color="Maroon" title="c13">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c11">s</font> )
 
<b>by </b><i><a class="ref" href="osalg_4.html#D13" target="_self" title="OSALG_4:def.13">Def13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">a1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:56_2_1"/><i><font color="Green" title="E35">A17</font></i>: 
( <font color="Maroon" title="c14">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c11">s</font> &amp; <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K5" title="OSALG_4:func.5">CompClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <font color="Maroon" title="c11">s</font></span>)</span></span>)</span>,<font color="Maroon" title="c14">a1</font> )
 <b>by </b><i><a class="ref" href="osalg_4.html#D12" target="_self" title="OSALG_4:def.12">Def12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">a2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:56_2_1"/><i><font color="Green" title="E36">A18</font></i>: 
( <font color="Maroon" title="c15">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c11">s</font> &amp; <font color="Maroon" title="c13">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K5" title="OSALG_4:func.5">CompClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <font color="Maroon" title="c11">s</font></span>)</span></span>)</span>,<font color="Maroon" title="c15">a2</font> )
 <b>by </b><i><a class="txt" href="osalg_4.html#E3:56_2_1"><i><font color="Green" title="E34">A16</font></i></a>, <a class="ref" href="osalg_4.html#D12" target="_self" title="OSALG_4:def.12">Def12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">a1</font> = <font color="Maroon" title="c14">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> <font color="Maroon" title="c11">s</font> <b>by </b><i><a class="txt" href="osalg_4.html#E5:56_2_1"><i><font color="Green" title="E35">A17</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">a2</font> = <font color="Maroon" title="c15">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> <font color="Maroon" title="c11">s</font> <b>by </b><i><a class="txt" href="osalg_4.html#E7:56_2_1"><i><font color="Green" title="E36">A18</font></i></a></i>;<br/>
<a NAME="E10:56_2_1"/><i><font color="Green" title="E37">A19</font></i>: 
( <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c16">a1</font> &amp; <font color="Maroon" title="c13">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c17">a2</font> )
 
<b>by </b><i><a class="txt" href="osalg_4.html#E5:56_2_1"><i><font color="Green" title="E35">A17</font></i></a>, <a class="txt" href="osalg_4.html#E7:56_2_1"><i><font color="Green" title="E36">A18</font></i></a></i>;<br/>
<a NAME="E11:56_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">a1</font> <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#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c16">a1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">a2</font> <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#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c17">a2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="osalg_4.html#E3:56_2"><i><font color="Green" title="E32">A14</font></i></a>, <a class="ref" href="osalg_4.html#D26" target="_self" title="OSALG_4:def.26">Def26</a></i>;<br/>
<a NAME="E12:56_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">a1</font>,<font color="Maroon" title="c17">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c11">s</font>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56_2_1"><i><font color="Green" title="E33">A15</font></i></a>, <a class="txt" href="osalg_4.html#E5:56_2_1"><i><font color="Green" title="E35">A17</font></i></a>, <a class="txt" href="osalg_4.html#E7:56_2_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="ref" href="msualg_4.html#D19" title="MSUALG_4:def.19">MSUALG_4:def 19</a></i>;<br/>
<a NAME="E13:56_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">a1</font>,<font color="Maroon" title="c17">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c11">s</font>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="msualg_4.html#D20" title="MSUALG_4:def.20">MSUALG_4:def 20</a></i>;<br/>
<a NAME="E14:56_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">a1</font>,<font color="Maroon" title="c17">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c11">s</font>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="osalg_4.html#D25" target="_self" title="OSALG_4:def.25">Def25</a></i>;<br/>
<b>hence </b><a NAME="E15:56_2_1"/>
<font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x2</font>
 <b>by </b><i><a class="txt" href="osalg_4.html#E10:56_2_1"><i><font color="Green" title="E37">A19</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>
<b>hence </b><a NAME="E5:56_2"/>
<span class="p1">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:56"/>
 <a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a>
 <b>by </b><i><a class="ref" href="msualg_3.html#T1" title="MSUALG_3:th.1">MSUALG_3:1</a></i>;<br/>

<b>thus </b><a NAME="E6:56"/>
 <a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">s1</font>, <font color="Maroon" title="c10">s2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_3.html#D1" title="OSALG_3:def.1">OSALG_3:def 1</a><br/>

<b>assume </b><a NAME="E1:56_3"/><i><font color="Green" title="E32">A20</font></i>: 
<font color="Maroon" title="c9">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">s2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c11">a1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E2:56_3"/><i><font color="Green" title="E33">A21</font></i>: 
<font color="Maroon" title="c11">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"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c9">s1</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c12">sqa</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <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>,<span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> as   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c13">S1O</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/>
<a NAME="E5:56_3"/><i><font color="Green" title="E34">A22</font></i>: 
<font color="Maroon" title="c13">S1O</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">S1O</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s2</font>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56_3"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="osalg_1.html#D18" title="OSALG_1:def.18">OSALG_1:def 18</a></i>;<br/>
<a NAME="E6:56_3"/><i><font color="Green" title="E35">A23</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c9">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <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>,<span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s1</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c10">s2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <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>,<span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s2</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="E7:56_3"/>
<font color="Maroon" title="c12">sqa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">sqa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s2</font>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56_3"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="osalg_1.html#D18" title="OSALG_1:def.18">OSALG_1:def 18</a></i>;<br/>
<b>hence </b><a NAME="E8:56_3"/>
<font color="Maroon" title="c11">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"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c10">s2</font></span>)</span>
 <b>by </b><i><a class="txt" href="osalg_4.html#E2:56_3"><i><font color="Green" title="E33">A21</font></i></a>, <a class="txt" href="osalg_4.html#E6:56_3"><i><font color="Green" title="E35">A23</font></i></a></i>;<br/>

<a NAME="E9:56_3"/>
<font color="Maroon" title="c11">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K7" title="OSALG_4:func.7">OSClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s1</font>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E2:56_3"><i><font color="Green" title="E33">A21</font></i></a></i>;<br/>
<a NAME="E10:56_3"/><b>then </b>
<font color="Maroon" title="c11">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c9">s1</font>
 
<b>by </b><i><a class="ref" href="osalg_4.html#D13" target="_self" title="OSALG_4:def.13">Def13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:56_3"/><i><font color="Green" title="E36">A24</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c9">s1</font>
 <b>and </b><br/><a NAME="E13:56_3"/><i><font color="Green" title="E37">A25</font></i>: 
<font color="Maroon" title="c11">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K5" title="OSALG_4:func.5">CompClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <font color="Maroon" title="c9">s1</font></span>)</span></span>)</span>,<font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="ref" href="osalg_4.html#D12" target="_self" title="OSALG_4:def.12">Def12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">x1</font> = <font color="Maroon" title="c14">x</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> <font color="Maroon" title="c9">s1</font> <b>by </b><i><a class="txt" href="osalg_4.html#E12:56_3"><i><font color="Green" title="E36">A24</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">x2</font> = <font color="Maroon" title="c14">x</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> <font color="Maroon" title="c10">s2</font> <b>by </b><i><a class="txt" href="osalg_4.html#E5:56_3"><i><font color="Green" title="E34">A22</font></i></a>, <a class="txt" href="osalg_4.html#E12:56_3"><i><font color="Green" title="E36">A24</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">s3</font> = <font color="Maroon" title="c9">s1</font>, <font color="Maroon" title="c18">s4</font> = <font color="Maroon" title="c10">s2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E17:56_3"/>
<font color="Maroon" title="c15">x1</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"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c17">s3</font></span>)</span>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E12:56_3"><i><font color="Green" title="E36">A24</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E18:56_3"/><b>then </b><i><font color="Green" title="E38">A26</font></i>: 
( <font color="Maroon" title="c15">x1</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"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c18">s4</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c17">s3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c18">s4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x1</font> )
 
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="osalg_4.html#E1:56_3"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="osalg_3.html#D1" title="OSALG_3:def.1">OSALG_3:def 1</a></i>;<br/>
<a NAME="E19:56_3"/><i><font color="Green" title="E39">A27</font></i>: 
<font color="Maroon" title="c11">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c16">x2</font>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56_3"><i><font color="Green" title="E32">A20</font></i></a>, <a class="txt" href="osalg_4.html#E13:56_3"><i><font color="Green" title="E37">A25</font></i></a>, <a class="ref" href="osalg_4.html#T5" target="_self" title="OSALG_4:th.5">Th5</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c9">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">a1</font> = 
<span class="p1">(<span class="default"><a href="osalg_4.html#K20" title="OSALG_4:func.20">OSHomQuot</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c9">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c15">x1</font></span>)</span>
<b>by </b><i><a class="txt" href="osalg_4.html#E13:56_3"><i><font color="Green" title="E37">A25</font></i></a>, <a class="ref" href="osalg_4.html#D27" target="_self" title="OSALG_4:def.27">Def27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c10">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x1</font>
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="osalg_4.html#E18:56_3"><i><font color="Green" title="E38">A26</font></i></a>, <a class="ref" href="osalg_4.html#D26" target="_self" title="OSALG_4:def.26">Def26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="osalg_4.html#K20" title="OSALG_4:func.20">OSHomQuot</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c10">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c16">x2</font></span>)</span>
<b>by </b><i><a class="txt" href="osalg_4.html#E1:56"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="osalg_4.html#D26" target="_self" title="OSALG_4:def.26">Def26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c10">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">a1</font>
<b>by </b><i><a class="txt" href="osalg_4.html#E19:56_3"><i><font color="Green" title="E39">A27</font></i></a>, <a class="ref" href="osalg_4.html#D27" target="_self" title="OSALG_4:def.27">Def27</a></i>
;<br/>


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


</div>
