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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>, <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>, <font color="Maroon">C</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">F1</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">F2</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">C</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:28"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">F1</font> <a href="msualg_3.html#R2">is_epimorphism</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E2:28"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">F2</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">A</font>,<font color="Maroon">C</font>
 ;<br/>

<b>let </b><font color="Maroon">G</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E3:28"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">G</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">F1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">F2</font>
 ;<br/>

<b>let </b><font color="Maroon">o</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a><br/>

<b>assume </b><a NAME="E4:28"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">B</font>;<br/>

<b>set </b><font color="Maroon">r</font> =  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font>;<br/>
<a NAME="E5:28"/>
<font color="Maroon">F1</font> is <a href="msualg_3.html#V2">"onto"</a>
 
<b>by </b><i>, <a class="ref" href="msualg_3.html#D10">MSUALG_3:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">A</font><b> such that </b><br/><a NAME="E7:28"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">F1</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>
<a NAME="E8:28"/>
<font color="Maroon">F1</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="msualg_3.html#D10">MSUALG_3:def 10</a></i>;<br/>
<a NAME="E9:28"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a></i>;<br/>
<a NAME="E10:28"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> is    <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> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E51</font></i>: <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">F1</font></span>)</span> <a href="msualg_3.html#K6">#</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="msualg_3.html#T8">MSUALG_3:8</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="msualg_3.html#K1">.</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="msualg_3.html#K1">.</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</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"><font color="Maroon">G</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">B</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/>
<b>hence </b><a NAME="E13:28"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i/>;<br/>


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