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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">C</font> be   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon" title="c2">A</font>;<br/><b>let </b><font color="Maroon" title="c4">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be    <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">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">s</font>;<br/>









<b>set </b><font color="Maroon" title="c7">f</font> = <span class="p1">(<span class="default"><a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font>;<br/>
<b>set </b><font color="Maroon" title="c8">g</font> =  <a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">s</font>;<br/>
<a NAME="E1:45"/><i><font color="Green" title="E17">A1</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">s</font>
 
<b>by </b><i><a class="ref" href="msualg_4.html#D18" title="MSUALG_4:def.18">MSUALG_4:def 18</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:45_1"/><i><font color="Green" title="E18">A2</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font>
 ;<br/> <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c4">s</font></span>)</span>,<font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="ref" href="msualg_4.html#D17" title="MSUALG_4:def.17">MSUALG_4:def 17</a></i>
<br/>.= 
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c4">s</font></span>)</span>,<font color="Maroon" title="c6">y</font>
<b>by </b><i><a class="txt" href="msualg_9.html#E1:45"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="msualg_9.html#E1:45_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="msualg_4.html#D17" title="MSUALG_4:def.17">MSUALG_4:def 17</a></i>
;<br/><b>hence </b><a NAME="E3:45_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c4">s</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T44" title="EQREL_1:th.44">EQREL_1:44</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:45"/><i><font color="Green" title="E18">A3</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c4">s</font>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> = 
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c4">s</font></span>)</span>,<font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="txt" href="msualg_9.html#E1:45"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="msualg_4.html#D17" title="MSUALG_4:def.17">MSUALG_4:def 17</a></i>
<br/>.= 
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c4">s</font></span>)</span>,<font color="Maroon" title="c6">y</font>
<b>by </b><i><a class="txt" href="msualg_9.html#E3:45"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="eqrel_1.html#T44" title="EQREL_1:th.44">EQREL_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font>
<b>by </b><i><a class="txt" href="msualg_9.html#E1:45"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="msualg_4.html#D17" title="MSUALG_4:def.17">MSUALG_4:def 17</a></i>
;<br/>


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