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<div class="add">

<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="E1:37"/><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>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="funct_1.html#NM1">Function</a>,  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>,  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>] <b>means </b> ex <font color="Olive">T</font> being   <a href="msualg_6.html#M2" target="_self">Translation</a> of <font color="Maroon">A2</font>,<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font> st <font color="Olive">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">a<sub>2</sub></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">a<sub>3</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:37"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font></span>)</span>,<font color="Olive">s</font>,<font color="Olive">s</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/>

<a NAME="E1:37_1"/>
(  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default">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">s</font></span>)</span> is    <a href="msualg_6.html#M2" target="_self">Translation</a> of <font color="Maroon">A2</font>,<font color="Maroon">s</font>,<font color="Maroon">s</font> &amp;  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span> is    <a href="msualg_6.html#M2" target="_self">Translation</a> of <font color="Maroon">A1</font>,<font color="Maroon">s</font>,<font color="Maroon">s</font> &amp; <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default">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">s</font></span>)</span></span>)</span> <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">s</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font> &amp; <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="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font> )
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>;<br/>
<b>hence </b><a NAME="E2:37_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span>,<font color="Maroon">s</font>,<font color="Maroon">s</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:37"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">s1</font>, <font color="Olive">s2</font>, <font color="Olive">s3</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>  st  <a href="msualg_6.html#K3" target="_self">TranslationRel</a> <font color="Maroon">S</font> <a href="rewrite1.html#R1">reduces</a> <font color="Olive">s1</font>,<font color="Olive">s2</font> holds <br/>for <font color="Olive">t</font> being   <a href="msualg_6.html#M2" target="_self">Translation</a> of <font color="Maroon">A1</font>,<font color="Olive">s1</font>,<font color="Olive">s2</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t</font>,<font color="Olive">s1</font>,<font color="Olive">s2</font>] holds <br/>for <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">f</font> <a href="msualg_6.html#R1" target="_self">is_e.translation_of</a> <font color="Maroon">A1</font>,<font color="Olive">s2</font>,<font color="Olive">s3</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">t</font>,<font color="Olive">s1</font>,<font color="Olive">s3</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<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>, <font color="Maroon">s3</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E1:37_2"/><i><font color="Green">E46</font></i>: 
 <a href="msualg_6.html#K3" target="_self">TranslationRel</a> <font color="Maroon">S</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">s1</font>,<font color="Maroon">s2</font>
 ;<br/>

<b>let </b><font color="Maroon">t</font> be    <a href="msualg_6.html#M2" target="_self">Translation</a> of <font color="Maroon">A1</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>;<br/>

<b>given </b><font color="Maroon">T</font> being    <a href="msualg_6.html#M2" target="_self">Translation</a> of <font color="Maroon">A2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font><b> such that </b><a NAME="E2:37_2"/><i><font color="Green">E55</font></i>: 
<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="funct_2.html#R2">=</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="partfun1.html#K1">*</a> <font color="Maroon">t</font>
 ;<br/>

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

<b>assume </b><a NAME="E3:37_2"/>
<font color="Maroon">f</font> <a href="msualg_6.html#R1" target="_self">is_e.translation_of</a> <font color="Maroon">A1</font>,<font color="Maroon">s2</font>,<font color="Maroon">s3</font>
 ;<br/>

<b>then consider </b><font color="Maroon">T1</font> being   <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">s2</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">s3</font><b> such that </b><br/><a NAME="E5:37_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">T1</font> <a href="msualg_6.html#R1" target="_self">is_e.translation_of</a> <font color="Maroon">A2</font>,<font color="Maroon">s2</font>,<font color="Maroon">s3</font> &amp; <font color="Maroon">T1</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">s2</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">s3</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font> )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">T2</font> = <font color="Maroon">T1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">T</font> as    <a href="msualg_6.html#M2" target="_self">Translation</a> of <font color="Maroon">A2</font>,<font color="Maroon">s1</font>,<font color="Maroon">s3</font> <b>by </b><i>, <a class="txt" href="msualg_6.html#E1:37"><i><font color="Green">E32</font></i></a>, </i>;<br/>
<b>take </b>
<font color="Maroon">T2</font>
;<br/>

<b>thus </b><font color="Maroon">T2</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> = 
<font color="Maroon">T1</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s3</font></span>)</span> <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> <font color="Maroon">t</font>
<b>by </b><i>, <a class="txt" href="msualg_6.html#E1:37"><i><font color="Green">E32</font></i></a>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s3</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">t</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:37"/>
for <font color="Olive">s1</font>, <font color="Olive">s2</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>  st  <a href="msualg_6.html#K3" target="_self">TranslationRel</a> <font color="Maroon">S</font> <a href="rewrite1.html#R1">reduces</a> <font color="Olive">s1</font>,<font color="Olive">s2</font> holds <br/>for <font color="Olive">t</font> being   <a href="msualg_6.html#M2" target="_self">Translation</a> of <font color="Maroon">A1</font>,<font color="Olive">s1</font>,<font color="Olive">s2</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t</font>,<font color="Olive">s1</font>,<font color="Olive">s2</font>]
 <b>from </b><i><a class="ref" href="msualg_6.html#S1" target="_self">MSUALG_6:sch 1</a>(, );</i><br/>


</div>
