<?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">A</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">B</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">A</font>,<font color="Maroon">B</font>;<br/>





<b>assume </b><a NAME="E1:49"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">F</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
 ;<br/>

<b>let </b><font color="Maroon">C1</font> be   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E2:49"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">C1</font> <a href="pboole.html#R2">c=</a>  <a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font>
 ;<br/>

<b>set </b><font color="Maroon">G</font> =  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">A</font>,<font color="Maroon">C1</font>;<br/>
<b>set </b><font color="Maroon">I</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<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="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<font color="Maroon">C1</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">sF</font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> ex <font color="Olive">xx</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> st <br/>( <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K3">Class</a> <font color="Maroon">C1</font>,<font color="Olive">xx</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K3">Class</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">xx</font> );<br/>
<a NAME="E3:49"/><i><font color="Green">E43</font></i>: 
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/>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="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<font color="Maroon">C1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</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="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>,<font color="Olive">i</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_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>assume </b><a NAME="E1:49_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="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> ;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E3:49_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="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<font color="Maroon">C1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font><b> such that </b><br/><a NAME="E5:49_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K3">Class</a> <font color="Maroon">C1</font>,<font color="Maroon">x1</font>
 <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">y</font> =  <a href="msualg_4.html#K3">Class</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">x1</font>;<br/>

<a NAME="E6:49_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">y</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">MSCng</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>thus </b><a NAME="E7:49_1"/>
<font color="Maroon">y</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="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="msualg_4.html#D8">MSUALG_4:def 8</a></i>;<br/>

<b>thus </b><a NAME="E8:49_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">x</font>,<font color="Maroon">i</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">C12</font> being    <a href="pboole.html#M3">ManySortedFunction</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<font color="Maroon">C1</font></span>)</span>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E5:49"/><i><font color="Green">E50</font></i>: 
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/> ex <font color="Olive">f</font> being  <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="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<font color="Maroon">C1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C12</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; ( 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="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<font color="Maroon">C1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">x</font>,<font color="Olive">i</font>] ) )
 <b>from </b><i><a class="ref" href="mssubfam.html#S1">MSSUBFAM:sch 1</a>();<br/></i>
<a NAME="E6:49"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><br/> for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<font color="Maroon">C1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font><br/> for <font color="Olive">xx</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K3">Class</a> <font color="Maroon">C1</font>,<font color="Olive">xx</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">C12</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K3">Class</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span>,<font color="Olive">xx</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<font color="Maroon">C1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>;<br/><b>let </b><font color="Maroon">xx</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>;<br/>





<b>assume </b><a NAME="E1:49_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K3">Class</a> <font color="Maroon">C1</font>,<font color="Maroon">xx</font>
 ;<br/>

<b>consider </b><font color="Maroon">f</font> being   <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="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<font color="Maroon">C1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font><b> such that </b><br/><a NAME="E3:49_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">C12</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E4:49_2"/><i><font color="Green">E70</font></i>: 
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="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<font color="Maroon">C1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">x</font>,<font color="Maroon">i</font>]
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">s</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>, <font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font><b> such that </b><br/><a NAME="E6:49_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E7:49_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K3">Class</a> <font color="Maroon">C1</font>,<font color="Maroon">x1</font>
 <b>and </b><br/><a NAME="E8:49_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K3">Class</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">x1</font>
 <b>by </b><i/>;<br/>
<a NAME="E9:49_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">C1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E10:49_2"/>
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s</font></span>)</span>,<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"><font color="Maroon">C1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s</font></span>)</span>,<font color="Maroon">xx</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E11:49_2"/><b>then </b>
<font color="Maroon">xx</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s</font></span>)</span>,<font color="Maroon">x1</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<a NAME="E12:49_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">xx</font>,<font color="Maroon">x1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<a NAME="E13:49_2"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">xx</font> <a href="hidden.html#R2">in</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">MSCng</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s</font></span>)</span>,<font color="Maroon">x1</font>
 
<b>by </b><i>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<b>thus </b><a NAME="E14:49_2"/>
<span class="p1">(<span class="default"><font color="Maroon">C12</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K3">Class</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">xx</font>
 <b>by </b><i>, , , , <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:49"/><b>then </b>
<font color="Maroon">C12</font> <a href="msualg_3.html#R2">is_epimorphism</a>  <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<font color="Maroon">C1</font>, <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:49"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">C12</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<font color="Maroon">C1</font>, <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D10">MSUALG_3:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">H</font> = <span class="p1">(<span class="default"><a href="msualg_4.html#K20">MSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">C12</font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<font color="Maroon">C1</font></span>)</span>,<font color="Maroon">B</font> ;<br/>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

<a NAME="E10:49"/>
 <a href="msualg_4.html#K20">MSHomQuot</a> <font color="Maroon">F</font> <a href="msualg_3.html#R3">is_monomorphism</a>  <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="msualg_4.html#T4">MSUALG_4:4</a></i>;<br/>
<a NAME="E11:49"/><b>then </b>
 <a href="msualg_4.html#K20">MSHomQuot</a> <font color="Maroon">F</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D11">MSUALG_3:def 11</a></i>;<br/>
<b>hence </b><a NAME="E12:49"/>
<font color="Maroon">H</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<font color="Maroon">C1</font>,<font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="msualg_3.html#T10">MSUALG_3:10</a></i>;<br/>

<div><i><font color="Green">E109</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:49_3"/>
<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="E3:49_3"/><i><font color="Green">E110</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C12</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">A</font>,<font color="Maroon">C1</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">A</font>,<span class="p3">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>;<br/>

<a NAME="E1:49_3_1"/>
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s</font></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E2:49_3_1"/><b>then </b>
 <a href="msualg_4.html#K3">Class</a> <font color="Maroon">C1</font>,<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s</font></span>)</span>
 
;<br/>
<a NAME="E3:49_3_1"/><b>then </b><i><font color="Green">E111</font></i>: 
 <a href="msualg_4.html#K3">Class</a> <font color="Maroon">C1</font>,<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K4">Class</a> <font color="Maroon">C1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="msualg_4.html#D8">MSUALG_4:def 8</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C12</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">A</font>,<font color="Maroon">C1</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">C12</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">A</font>,<font color="Maroon">C1</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C12</font> <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"><span class="p2">(<span class="default"><a href="msualg_4.html#K15">MSNat_Hom</a> <font color="Maroon">A</font>,<font color="Maroon">C1</font>,<font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_4.html#D18">MSUALG_4:def 18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C12</font> <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"><a href="msualg_4.html#K3">Class</a> <font color="Maroon">C1</font>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_4.html#D17">MSUALG_4:def 17</a></i>
<br/>.= 
 <a href="msualg_4.html#K3">Class</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">x</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_4.html#K15">MSNat_Hom</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="msualg_4.html#D17">MSUALG_4:def 17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">A</font>,<span class="p3">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="msualg_4.html#D18">MSUALG_4:def 18</a></i>
;<br/>


</div><b>end;</b></div><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">C12</font> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">A</font>,<font color="Maroon">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">C12</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">A</font>,<font color="Maroon">C1</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>
;<br/></div><b>end;</b></div>
<b>thus </b><font color="Maroon">F</font> = 
<span class="p1">(<span class="default"><a href="msualg_4.html#K20">MSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">F</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_4.html#K20">MSHomQuot</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><font color="Maroon">C12</font> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">A</font>,<font color="Maroon">C1</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">A</font>,<font color="Maroon">C1</font></span>)</span>
<b>by </b><i><a class="ref" href="autalg_1.html#T13">AUTALG_1:13</a></i>
;<br/>


</div>
