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<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>;<br/><b>let </b><font color="Maroon">C1</font> be   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">A</font>, <font color="Maroon">C2</font> be   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">G</font> be   <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>,<span class="p1">(<span class="default"><a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">A</font>,<font color="Maroon">C2</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:47"/><i><font color="Green">E39</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">G</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> <font color="Maroon">C2</font>,<font color="Olive">xx</font>
 ;<br/>

<b>set </b><font color="Maroon">sL</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><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="E1:47_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="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> ;<br/><a NAME="E3:47_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">c</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">G</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">c</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>,<font color="Maroon">C2</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="Olive">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</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="E1:47_1_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">c</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="E2:47_1_1"/><b>then </b>
 <a href="msualg_4.html#K3">Class</a> <font color="Maroon">C1</font>,<font color="Maroon">c</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="E3:47_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="msualg_4.html#K3">Class</a> <font color="Maroon">C1</font>,<font color="Maroon">c</font> is    <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">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">G</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">c</font> = 
<span class="p1">(<span class="default"><font color="Maroon">G</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">c</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">G</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">c</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">G</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">c</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> <font color="Maroon">C2</font>,<font color="Maroon">c</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>,<font color="Maroon">C2</font>,<font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</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>,<font color="Maroon">C2</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">c</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">G</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">G</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>,<font color="Maroon">C2</font></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>hence </b><a NAME="E3:47"/>
<font color="Maroon">G</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> <a href="pboole.html#R6">=</a>  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">A</font>,<font color="Maroon">C2</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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