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<b>let </b><font color="Maroon">S1</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> , <font color="Maroon">S2</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">f</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><a NAME="E1:35"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font>
 ;<br/>

<b>let </b><font color="Maroon">A</font> be    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S2</font>;<br/><b>let </b><font color="Maroon">o1</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S1</font>;<br/><b>let </b><font color="Maroon">o2</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S2</font>;<br/>





<b>assume </b><a NAME="E2:35"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font>
 ;<br/>

<b>thus </b> <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o2</font>,<font color="Maroon">A</font> = 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#T10">PRALG_2:10</a></i>
<br/>.= 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="pua2mss1.html#D13">PUA2MSS1:def 13</a></i>
<br/>.= 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S1</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S1</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#T10">PRALG_2:10</a></i>
;<br/>

<a NAME="E4:35"/><i><font color="Green">E52</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> )
 
<b>by </b><i>, <a class="ref" href="pua2mss1.html#D13">PUA2MSS1:def 13</a></i>;<br/>
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o2</font> = 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font>
<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">f</font> <a href="funct_1.html#NK3" target="_self">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font>
<b>by </b><i>, <a class="ref" href="pua2mss1.html#D13">PUA2MSS1:def 13</a></i>
<br/>.= 
<font color="Maroon">f</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">o1</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 href="msualg_1.html#K4">Result</a> <font color="Maroon">o2</font>,<font color="Maroon">A</font> = 
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"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#T10">PRALG_2:10</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <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">o1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S1</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <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">o1</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o1</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S1</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#T10">PRALG_2:10</a></i>
;<br/><br/>


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