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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">A1</font>, <font color="Maroon" title="c3">A2</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="c4">h</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c3">A2</font>;<br/>





<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c4">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c3">A2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">s1</font>, <font color="Maroon" title="c6">s2</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="c7">t</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>



<b>assume </b><a NAME="E2:35"/>
<font color="Maroon" title="c7">t</font> <a href="msualg_6.html#R1" title="MSUALG_6:pred.1">is_e.translation_of</a> <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">o</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E4:35"/><i><font color="Green" title="E27">A2</font></i>: 
 <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c8">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s2</font>
 <b>and </b><br/><a NAME="E5:35"/><i><font color="Green" title="E28">A3</font></i>: 
 ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c8">o</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s1</font> &amp;  ex <font color="Olive" title="b2">a</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c2">A1</font> &amp; <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_6.html#K4" title="MSUALG_6:func.4">transl</a> <font color="Maroon" title="c8">o</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">a</font>,<font color="Maroon" title="c2">A1</font> ) )
 <b>by </b><i><a class="ref" href="msualg_6.html#D5" target="_self" title="MSUALG_6:def.5">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c10">a</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E7:35"/><i><font color="Green" title="E29">A4</font></i>: 
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c8">o</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s1</font> )
 <b>and </b><br/><a NAME="E8:35"/><i><font color="Green" title="E30">A5</font></i>: 
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c2">A1</font> &amp; <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_6.html#K4" title="MSUALG_6:func.4">transl</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c2">A1</font> )
 <b>by </b><i><a class="txt" href="msualg_6.html#E5:35"><i><font color="Green" title="E28">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">a</font> = <font color="Maroon" title="c10">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c2">A1</font> <b>by </b><i><a class="txt" href="msualg_6.html#E8:35"><i><font color="Green" title="E30">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">T</font> =  <a href="msualg_6.html#K4" title="MSUALG_6:func.4">transl</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c9">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">a</font></span>)</span>,<font color="Maroon" title="c3">A2</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s2</font> <b>by </b><i><a class="txt" href="msualg_6.html#E4:35"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="msualg_6.html#E7:35"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="msualg_6.html#T10" target="_self" title="MSUALG_6:th.10">Th10</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">T</font>
;<br/>

<b>thus </b><a NAME="E11:35"/>
<font color="Maroon" title="c12">T</font> <a href="msualg_6.html#R1" title="MSUALG_6:pred.1">is_e.translation_of</a> <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E4:35"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="msualg_6.html#E7:35"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="msualg_6.html#D5" target="_self" title="MSUALG_6:def.5">Def5</a></i>;<br/>

<b>thus </b><a NAME="E12:35"/>
<font color="Maroon" title="c12">T</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s2</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">t</font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E1:35"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="msualg_6.html#E4:35"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="msualg_6.html#E7:35"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="msualg_6.html#E8:35"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="msualg_6.html#T21" target="_self" title="MSUALG_6:th.21">Th21</a></i>;<br/>


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