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

<b>let </b><font color="Maroon">S</font> be  <a href="osalg_4.html#V2" target="_self">locally_directed</a> <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">U1</font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">S</font>, <font color="Maroon">U2</font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">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="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">F</font>;<br/>
<b>set </b><font color="Maroon">qa</font> =  <a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">U1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">F</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">qh</font> =  <a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">F</font>;<br/>
<b>assume </b><a NAME="E1:57"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">F</font> <a href="msualg_3.html#R2">is_epimorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font> &amp; <font color="Maroon">F</font> is <a href="osalg_3.html#V1">order-sorted</a> )
 ;<br/>

<a NAME="E2:57"/><b>then </b><i><font color="Green">E42</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> &amp; <font color="Maroon">F</font> is <a href="msualg_3.html#V2">"onto"</a> )
 
<b>by </b><i><a class="ref" href="msualg_3.html#D10">MSUALG_3:def 10</a></i>;<br/>
<a NAME="E3:57"/><b>then </b>
 <a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">F</font> <a href="msualg_3.html#R3">is_monomorphism</a>  <a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">U1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">U2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:57"/><b>then </b><i><font color="Green">E44</font></i>: 
(  <a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">F</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">U1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">U2</font> &amp;  <a href="osalg_4.html#K21" target="_self">OSHomQuot</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#D11">MSUALG_3:def 11</a></i>;<br/>
<b>set </b><font color="Maroon">Sq</font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">U1</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span>;<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>set </b><font color="Maroon">S2</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font>;<br/>
<a NAME="E5:57"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/> <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><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="osalg_4.html#K21" target="_self">OSHomQuot</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="E1:57_1"/>
<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="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E3:57_1"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</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="osalg_4.html#K20" target="_self">OSHomQuot</a> <font color="Maroon">F</font>,<font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:57_1"/><b>then </b><i><font color="Green">E61</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">U1</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <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>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>thus </b><a NAME="E5:57_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<a NAME="E6:57_1"/><i><font color="Green">E62</font></i>: 
 <a href="relset_1.html#K5">rng</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="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="msualg_3.html#D3">MSUALG_3:def 3</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E7:57_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:57_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</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> &amp; <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_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <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> <font color="Maroon">s</font> <b>by </b><i><a class="ref" href="osalg_4.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E11:57_1"/><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</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> <span class="p1">(<span class="default"><a href="osalg_4.html#K8" target="_self">OSClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , , <a class="ref" href="osalg_4.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E12:57_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">U1</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="osalg_4.html#K6" target="_self">OSClass</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E13:57_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:57"/><b>then </b>
 <a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">F</font> is <a href="msualg_3.html#V2">"onto"</a>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D3">MSUALG_3:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:57"/>
 <a href="osalg_4.html#K21" target="_self">OSHomQuot</a> <font color="Maroon">F</font> <a href="msualg_3.html#R4">is_isomorphism</a>  <a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">U1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">U2</font>
 <b>by </b><i>, <a class="ref" href="msualg_3.html#T13">MSUALG_3:13</a></i>;<br/>


</div>
