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

<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">cqa</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">u1</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>;<br/>
<b>set </b><font color="Maroon">u2</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font>;<br/>
<a NAME="E1:54_1_2"/><i><font color="Green">E65</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">U1</font>,<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>let </b><font color="Maroon">H</font> be   <a href="funct_2.html#NM1">Function</a> of 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>,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>, <font color="Maroon">G</font> be   <a href="funct_2.html#NM1">Function</a> of 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>,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>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:54_1_2"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> holds  <font color="Maroon">H</font> <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="Olive">a</font></span>)</span> <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="Olive">a</font>
 <b>and </b><br/><a NAME="E3:54_1_2"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> holds  <font color="Maroon">G</font> <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="Olive">a</font></span>)</span> <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="Olive">a</font>
 ;<br/>

<a NAME="E4:54_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">U1</font>,<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> holds <br/><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:54_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_4.html#K16" target="_self">QuotOSAlg</a> <font color="Maroon">U1</font>,<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>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:54_1_2_1"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K5" target="_self">CompClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" target="_self">OSCng</a> <font color="Maroon">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="osalg_4.html#K3" target="_self">CComp</a> <font color="Maroon">s</font></span>)</span></span>)</span>,<font color="Maroon">y</font> )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <b>by </b><i/>;<br/>
<a NAME="E5:54_1_2_1"/><i><font color="Green">E71</font></i>: 
 <a href="osalg_4.html#K8" 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">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<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">y1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="osalg_4.html#E1:54"><i><font color="Green">E33</font></i></a>, </i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:54_1_2"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>


</div>
