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

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">U1</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>, <font color="Maroon">U2</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<font color="Maroon">U2</font>;<br/>





<b>set </b><font color="Maroon">mc</font> =  <a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font>;<br/>
<b>set </b><font color="Maroon">qa</font> =  <a href="msualg_4.html#K14" target="_self">QuotMSAlg</a> <font color="Maroon">U1</font>,<span class="p1">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">qh</font> =  <a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</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/>
<b>assume </b><a NAME="E1:36"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">F</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font>
 ;<br/>

<a NAME="E2:36"/>
for <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>  st  <a href="msualg_1.html#K3">Args</a> <font color="Olive">o</font>,<span class="p1">(<span class="default"><a href="msualg_4.html#K14" target="_self">QuotMSAlg</a> <font color="Maroon">U1</font>,<span class="p2">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <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>,<span class="p1">(<span class="default"><a href="msualg_4.html#K14" target="_self">QuotMSAlg</a> <font color="Maroon">U1</font>,<span class="p2">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">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>,<span class="p3">(<span class="default"><a href="msualg_4.html#K14" target="_self">QuotMSAlg</a> <font color="Maroon">U1</font>,<span class="p4">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span></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>,<font color="Maroon">U2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</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="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E1:36_1"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="msualg_4.html#K14" target="_self">QuotMSAlg</a> <font color="Maroon">U1</font>,<span class="p2">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <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>,<span class="p1">(<span class="default"><a href="msualg_4.html#K14" target="_self">QuotMSAlg</a> <font color="Maroon">U1</font>,<span class="p2">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span>;<br/>

<b>set </b><font color="Maroon">ro</font> =  <a href="msualg_1.html#K2">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="msualg_4.html#K14" target="_self">QuotMSAlg</a> <font color="Maroon">U1</font>,<span class="p2">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="msualg_4.html#K13" target="_self">QuotCharact</a> <span class="p2">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="msualg_4.html#K5" target="_self">.</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="msualg_4.html#K12" target="_self">QuotCharact</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">o</font>
<b>by </b><i/>
;<br/>
<a NAME="E3:36_1"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="msualg_4.html#K14" target="_self">QuotMSAlg</a> <font color="Maroon">U1</font>,<span class="p2">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_4.html#K4" target="_self">Class</a> <span class="p4">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span></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>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D9">MSUALG_1:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font><b> such that </b><br/><a NAME="E5:36_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_4.html#K7" target="_self">#</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>
<a NAME="E6:36_1"/><i><font color="Green">E44</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">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="E7:36_1"/><i><font color="Green">E45</font></i>: 
the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="msualg_1.html#D5">MSUALG_1:def 5</a></i>;<br/>
<a NAME="E8:36_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>
 
;<br/>
<a NAME="E9:36_1"/><i><font color="Green">E48</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="E10:36_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">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>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E50</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_1.html#K1">.</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="msualg_1.html#K2">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="E12:36_1"/><b>then </b><i><font color="Green">E51</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="msualg_1.html#K2">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">da</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">a</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="msualg_1.html#K2">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="E14:36_1"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="msualg_4.html#K19" target="_self">MSHomQuot</a> <font color="Maroon">F</font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E15:36_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K4" target="_self">Class</a> <span class="p3">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span> <a href="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="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K4" target="_self">Class</a> <span class="p3">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> = 
<span class="p1">(<span class="default"><a href="msualg_4.html#K4" target="_self">Class</a> <span class="p2">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_4.html#K4" target="_self">Class</a> <span class="p2">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">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="E17:36_1"/><b>then </b>
 <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="msualg_4.html#K8" target="_self">QuotRes</a> <span class="p2">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_4.html#D1" target="_self">Def1</a>, , <a class="txt" href="msualg_4.html#E1:36"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E18:36_1"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K8" target="_self">QuotRes</a> <span class="p3">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span>,<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/>
<a NAME="E19:36_1"/><i><font color="Green">E68</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</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> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">a</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> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="msualg_3.html#T6">MSUALG_3:6</a></i>;<br/>
<div><i><font color="Green">E75</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:36_1_4"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">y</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">y</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/><a NAME="E3:36_1_4"/><i><font color="Green">E77</font></i>: 
<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> <a href="hidden.html#R1">=</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="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="msualg_4.html#E2:36_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E4:36_1_4"/>
<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="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <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>
 <b>by </b><i><a class="txt" href="msualg_4.html#E2:36_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </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="funct_1.html#K1">.</a> <font color="Maroon">n</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> ;<br/><a NAME="E6:36_1_4"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="msualg_4.html#K6" target="_self">*</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="msualg_4.html#E2:36_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E7:36_1_4"/><b>then </b>
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</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" target="_self">*</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>
 <b>by </b><i><a class="ref" href="msualg_3.html#T6">MSUALG_3:6</a></i>;<br/><b>then reconsider </b><font color="Maroon">an</font> = <font color="Maroon">a</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> <font color="Maroon">s</font> <b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E9:36_1_4"/>
<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_4.html#K2" target="_self">.</a> <font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="msualg_4.html#E2:36_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="msualg_4.html#D3" target="_self">Def3</a>, </i>;<br/><a NAME="E10:36_1_4"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K17" target="_self">MSCng</a> <font color="Maroon">F</font>,<font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, </i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</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">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></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="ref" href="msualg_4.html#D2" target="_self">Def2</a>, <a class="txt" href="msualg_4.html#E2:36_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="msualg_4.html#D3" target="_self">Def3</a>, <a class="ref" href="msualg_3.html#D8">MSUALG_3:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_4.html#K19" target="_self">MSHomQuot</a> <font color="Maroon">F</font>,<font color="Maroon">s</font></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/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">an</font>
<b>by </b><i>, <a class="ref" href="msualg_4.html#D5" target="_self">Def5</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="msualg_4.html#D2" target="_self">Def2</a>, <a class="txt" href="msualg_4.html#E2:36_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="msualg_4.html#D3" target="_self">Def3</a>, <a class="ref" href="msualg_3.html#D8">MSUALG_3:def 8</a></i>
;<br/><b>hence </b><a NAME="E12:36_1_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</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">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E21:36_1"/>
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</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="E22:36_1"/><b>then </b>
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K4" target="_self">Class</a> <span class="p3">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="msualg_4.html#K6" target="_self">*</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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_4.html#K4" target="_self">Class</a> <span class="p4">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span></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>
 
<b>by </b><i>, <a class="ref" href="msafree.html#T1">MSAFREE:1</a></i>;<br/>
<b>then </b><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="msualg_4.html#K14" target="_self">QuotMSAlg</a> <font color="Maroon">U1</font>,<span class="p3">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K8" target="_self">QuotRes</a> <span class="p3">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span>,<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">a</font>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_4.html#K8" target="_self">QuotRes</a> <span class="p2">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">da</font>
<b>by </b><i><a class="ref" href="msualg_4.html#D1" target="_self">Def1</a>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
 <a href="msualg_4.html#K3" target="_self">Class</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">da</font>
<b>by </b><i/>
<br/>.= 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K17" target="_self">MSCng</a> <font color="Maroon">F</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>,<font color="Maroon">da</font>
<b>by </b><i>, </i>
;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">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>,<span class="p3">(<span class="default"><a href="msualg_4.html#K14" target="_self">QuotMSAlg</a> <font color="Maroon">U1</font>,<span class="p4">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">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">a</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, <a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E25:36_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">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>,<span class="p3">(<span class="default"><a href="msualg_4.html#K14" target="_self">QuotMSAlg</a> <font color="Maroon">U1</font>,<span class="p4">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span></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>,<font color="Maroon">U2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="msualg_4.html#D2" target="_self">Def2</a>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:36"/>
 <a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msualg_4.html#K14" target="_self">QuotMSAlg</a> <font color="Maroon">U1</font>,<span class="p1">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">U2</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#D11">MSUALG_3:def 11</a><br/>

<a NAME="E4:36"/>
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/><span class="p1">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_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>set </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>;<br/>
<b>assume </b><a NAME="E1:36_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="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E3:36_2"/><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K19" target="_self">MSHomQuot</a> <font color="Maroon">F</font>,<font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:36_2"/>
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x2</font> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:36_2_1"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/>

<a NAME="E2:36_2_1"/><b>then </b><i><font color="Green">E82</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K4" target="_self">Class</a> <span class="p2">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K4" target="_self">Class</a> <span class="p2">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span></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="E3:36_2_1"/><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_4.html#K2" target="_self">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K17" target="_self">MSCng</a> <font color="Maroon">F</font>,<font color="Maroon">s</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:36_2_1"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_4.html#K2" target="_self">.</a> <font color="Maroon">s</font></span>)</span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_4.html#K2" target="_self">.</a> <font color="Maroon">s</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="msualg_4.html#D1" target="_self">Def1</a>, </i>;<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="E6:36_2_1"/><i><font color="Green">E85</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">x1</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_4.html#K2" target="_self">.</a> <font color="Maroon">s</font></span>)</span>,<font color="Maroon">a1</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">a2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:36_2_1"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">a2</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">x2</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_4.html#K2" target="_self">.</a> <font color="Maroon">s</font></span>)</span>,<font color="Maroon">a2</font> )
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a1</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/>
<b>reconsider </b><font color="Maroon">a2</font> = <font color="Maroon">a2</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="E11:36_2_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a2</font> )
 
<b>by </b><i>, , , , , </i>;<br/>
<a NAME="E12:36_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_4.html#K17" target="_self">MSCng</a> <font color="Maroon">F</font>,<font color="Maroon">s</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E13:36_2_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , , <a class="ref" href="eqrel_1.html#T44">EQREL_1:44</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:36_2"/>
<span class="p1">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:36"/>
 <a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">F</font> is <a href="msualg_3.html#V1">"1-1"</a>
 <b>by </b><i><a class="ref" href="msualg_3.html#T1">MSUALG_3:1</a></i>;<br/>


</div>
