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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">U0</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">s</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">e</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4" target="_self">Equations</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>;<br/><b>let </b><font color="Maroon">R</font> be   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">U0</font>;<br/>









<b>assume </b><a NAME="E1:51"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">U0</font> <a href="equation.html#R1" target="_self">|=</a> <font color="Maroon">e</font>
 ;<br/>

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

<b>assume </b><a NAME="E2:51"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">h</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="equation.html#K3" target="_self">TermAlg</a> <font color="Maroon">S</font>, <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">U0</font>,<font color="Maroon">R</font>
 ;<br/>

<a NAME="E3:51"/>
 <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">U0</font>,<font color="Maroon">R</font> <a href="msualg_3.html#R2">is_epimorphism</a> <font color="Maroon">U0</font>, <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">U0</font>,<font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="msualg_4.html#T3">MSUALG_4:3</a></i>;<br/>
<b>then consider </b><font color="Maroon">h0</font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="equation.html#K3" target="_self">TermAlg</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">U0</font><b> such that </b><br/><a NAME="E5:51"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">h0</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="equation.html#K3" target="_self">TermAlg</a> <font color="Maroon">S</font>,<font color="Maroon">U0</font>
 <b>and </b><br/><a NAME="E6:51"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">h</font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">U0</font>,<font color="Maroon">R</font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">h0</font>
 <b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">n</font> = <span class="p1">(<span class="default"><a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">U0</font>,<font color="Maroon">R</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font>;<br/>
<a NAME="E7:51"/><i><font color="Green">E41</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="equation.html#K3" target="_self">TermAlg</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:51"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">e</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E9:51"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">e</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="equation.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">h</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"><font color="Maroon">e</font> <a href="mcart_1.html#K1">`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">MSNat_Hom</a> <font color="Maroon">U0</font>,<font color="Maroon">R</font></span>)</span> <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"><font color="Maroon">h0</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K1">`1</a> </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"><span class="p2">(<span class="default"><a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">U0</font>,<font color="Maroon">R</font></span>)</span> <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"><font color="Maroon">h0</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</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">U0</font>,<font color="Maroon">R</font></span>)</span> <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"><font color="Maroon">h0</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span>
<b>by </b><i>, , </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">MSNat_Hom</a> <font color="Maroon">U0</font>,<font color="Maroon">R</font></span>)</span> <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"><font color="Maroon">h0</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K2">`2</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h</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"><font color="Maroon">e</font> <a href="mcart_1.html#K2">`2</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
;<br/>


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