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

<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">cqa</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>;<br/>
<b>set </b><font color="Maroon" title="c8">u1</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>set </b><font color="Maroon" title="c9">u2</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">U2</font>;<br/>
<a NAME="E1:54_1_2"/><i><font color="Green" title="E31">A7</font></i>: 
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="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c5">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>let </b><font color="Maroon" title="c10">H</font>, <font color="Maroon" title="c11">G</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 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="c5">s</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:54_1_2"/><i><font color="Green" title="E32">A8</font></i>: 
for <font color="Olive" title="b1">a</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="c5">s</font> holds  <font color="Maroon" title="c10">H</font> <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="Olive" title="b1">a</font></span>)</span> <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="c5">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font>
 <b>and </b><br/><a NAME="E3:54_1_2"/><i><font color="Green" title="E33">A9</font></i>: 
for <font color="Olive" title="b1">a</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="c5">s</font> holds  <font color="Maroon" title="c11">G</font> <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="Olive" title="b1">a</font></span>)</span> <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="c5">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font>
 ;<br/>

<a NAME="E4:54_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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 <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="c5">s</font> holds <br/><font color="Maroon" title="c10">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:54_1_2_1"/>
<font color="Maroon" title="c12">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 <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="c5">s</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:54_1_2_1"/><i><font color="Green" title="E34">A10</font></i>: 
( <font color="Maroon" title="c13">y</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="c5">s</font> &amp; <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">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="c5">s</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">y</font> )
 <b>by </b><i><a class="txt" href="osalg_4.html#E1:54_1_2"><i><font color="Green" title="E31">A7</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="c14">y1</font> = <font color="Maroon" title="c13">y</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="c5">s</font> <b>by </b><i><a class="txt" href="osalg_4.html#E3:54_1_2_1"><i><font color="Green" title="E34">A10</font></i></a></i>;<br/>
<a NAME="E5:54_1_2_1"/><i><font color="Green" title="E35">A11</font></i>: 
 <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="c14">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E3:54_1_2_1"><i><font color="Green" title="E34">A10</font></i></a></i>;<br/>
<b>hence </b><font color="Maroon" title="c10">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> = 
<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="c5">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y1</font>
<b>by </b><i><a class="txt" href="osalg_4.html#E2:54_1_2"><i><font color="Green" title="E32">A8</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c11">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
<b>by </b><i><a class="txt" href="osalg_4.html#E3:54_1_2"><i><font color="Green" title="E33">A9</font></i></a>, <a class="txt" href="osalg_4.html#E5:54_1_2_1"><i><font color="Green" title="E35">A11</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:54_1_2"/>
<font color="Maroon" title="c10">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">G</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>;<br/>


</div>
