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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">A1</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">A2</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">h</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">A1</font>,<font color="Maroon">A2</font>;<br/>





<b>assume </b><a NAME="E1:35"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">h</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">A1</font>,<font color="Maroon">A2</font>
 ;<br/>

<b>let </b><font color="Maroon">s1</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>, <font color="Maroon">s2</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">t</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><a NAME="E2:35"/>
<font color="Maroon">t</font> <a href="msualg_6.html#R1" target="_self">is_e.translation_of</a> <font color="Maroon">A1</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">o</font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E4:35"/><i><font color="Green">E34</font></i>: 
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font>
 <b>and </b><br/><a NAME="E5:35"/><i><font color="Green">E43</font></i>: 
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp;  ex <font color="Olive">a</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">A1</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="msualg_6.html#K4" target="_self">transl</a> <font color="Maroon">o</font>,<font color="Olive">i</font>,<font color="Olive">a</font>,<font color="Maroon">A1</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">a</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:35"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> )
 <b>and </b><br/><a NAME="E8:35"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">A1</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="msualg_6.html#K4" target="_self">transl</a> <font color="Maroon">o</font>,<font color="Maroon">i</font>,<font color="Maroon">a</font>,<font color="Maroon">A1</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <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">A1</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">T</font> =  <a href="msualg_6.html#K4" target="_self">transl</a> <font color="Maroon">o</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">A2</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> <b>by </b><i>, , </i>;<br/>
<b>take </b>
<font color="Maroon">T</font>
;<br/>

<b>thus </b><a NAME="E11:35"/>
<font color="Maroon">T</font> <a href="msualg_6.html#R1" target="_self">is_e.translation_of</a> <font color="Maroon">A2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>
 <b>by </b><i>, , </i>;<br/>

<b>thus </b><a NAME="E12:35"/>
<font color="Maroon">T</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s2</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">t</font>
 <b>by </b><i>, , , , </i>;<br/>


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