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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">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">U0</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">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="c4">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4" title="EQUATION:func.4">Equations</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s</font>;<br/><b>let </b><font color="Maroon" title="c5">R</font> be   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon" title="c2">U0</font>;<br/>









<b>assume </b><a NAME="E1:51"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c2">U0</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c4">e</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">h</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="equation.html#K3" title="EQUATION:func.3">TermAlg</a> <font color="Maroon" title="c1">S</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c5">R</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="equation.html#D5" target="_self" title="EQUATION:def.5">EQUATION:def 5</a><br/>

<b>assume </b><a NAME="E2:51"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c6">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="equation.html#K3" title="EQUATION:func.3">TermAlg</a> <font color="Maroon" title="c1">S</font>, <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c5">R</font>
 ;<br/>

<a NAME="E3:51"/>
 <a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c5">R</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon" title="c2">U0</font>, <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c5">R</font>
 
<b>by </b><i><a class="ref" href="msualg_4.html#T3" title="MSUALG_4:th.3">MSUALG_4:3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">h0</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="equation.html#K3" title="EQUATION:func.3">TermAlg</a> <font color="Maroon" title="c1">S</font></span>)</span>,<font color="Maroon" title="c2">U0</font><b> such that </b><br/><a NAME="E5:51"/><i><font color="Green" title="E27">A3</font></i>: 
<font color="Maroon" title="c7">h0</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="equation.html#K3" title="EQUATION:func.3">TermAlg</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">U0</font>
 <b>and </b><br/><a NAME="E6:51"/><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c6">h</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <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">U0</font>,<font color="Maroon" title="c5">R</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c7">h0</font>
 <b>by </b><i><a class="txt" href="equation.html#E2:51"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="equation.html#T26" target="_self" title="EQUATION:th.26">Th26</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">n</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">U0</font>,<font color="Maroon" title="c5">R</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font>;<br/>
<a NAME="E7:51"/><i><font color="Green" title="E29">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h0</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="equation.html#K3" title="EQUATION:func.3">TermAlg</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:51"/><b>then </b><i><font color="Green" title="E30">A6</font></i>: 
<font color="Maroon" title="c4">e</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h0</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="equation.html#T31" target="_self" title="EQUATION:th.31">Th31</a></i>;<br/>
<a NAME="E9:51"/><i><font color="Green" title="E31">A7</font></i>: 
<font color="Maroon" title="c4">e</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h0</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span>
 
<b>by </b><i><a class="txt" href="equation.html#E7:51"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="equation.html#T32" target="_self" title="EQUATION:th.32">Th32</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c5">R</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">h0</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E6:51"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</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">U0</font>,<font color="Maroon" title="c5">R</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">h0</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E8:51"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</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">U0</font>,<font color="Maroon" title="c5">R</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">h0</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E1:51"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="equation.html#E5:51"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="equation.html#D5" target="_self" title="EQUATION:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c5">R</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">h0</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E9:51"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">e</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E6:51"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></i>
;<br/>


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