<?xml version="1.0"?>
<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> , <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>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S'</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>







<b>assume </b><a NAME="E1:47"/><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">S'</font>,<font color="Maroon">S</font>
 ;<br/>

<b>let </b><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>;<br/>

<b>assume </b><a NAME="E2:47"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S'</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</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>(  <a href="msualg_6.html#K3">TranslationRel</a> <font color="Maroon">S</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>3</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> is    <a href="msualg_6.html#M2">Translation</a> of <font color="Maroon">A</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>3</sub></font> );<br/>
<a NAME="E3:47"/><i><font color="Green">E52</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">B</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="47_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/>

<b>thus </b><a NAME="E1:47_1"/>
 <a href="msualg_6.html#K3">TranslationRel</a> <font color="Maroon">S</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T13">REWRITE1:13</a></i>;<br/>

<a NAME="E2:47_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E3:47_1"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 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_2.html#K3">.</a> <font color="Maroon">s</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="E4:47_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">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span> is    <a href="msualg_6.html#M2">Translation</a> of <font color="Maroon">A</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="msualg_6.html#T16">MSUALG_6:16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:47"/><i><font color="Green">E53</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">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">Translation</a> of <font color="Maroon">B</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">h</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">h</font> <a href="msualg_6.html#R1">is_e.translation_of</a> <font color="Maroon">B</font>,<font color="Olive">s2</font>,<font color="Olive">s3</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">h</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="47_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="E1:47_2"/>
 <a href="msualg_6.html#K3">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">Translation</a> of <font color="Maroon">B</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>;<br/>

<b>assume </b><a NAME="E2:47_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">t</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font>]
 ;<br/>

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

<b>assume </b><a NAME="E3:47_2"/>
<font color="Maroon">h</font> <a href="msualg_6.html#R1">is_e.translation_of</a> <font color="Maroon">B</font>,<font color="Maroon">s2</font>,<font color="Maroon">s3</font>
 ;<br/>

<a NAME="E4:47_2"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">h</font> <a href="msualg_6.html#R1">is_e.translation_of</a> <font color="Maroon">A</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s2</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s3</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:47_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s3</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_6.html#K3">TranslationRel</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="msualg_6.html#T12">MSUALG_6:12</a></i>;<br/>
<a NAME="E6:47_2"/><b>then </b>
 <a href="msualg_6.html#K3">TranslationRel</a> <font color="Maroon">S</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s2</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s3</font>
 
<b>by </b><i><a class="ref" href="rewrite1.html#T16">REWRITE1:16</a></i>;<br/>
<b>hence </b><a NAME="E7:47_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">h</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">t</font>,<font color="Maroon">s1</font>,<font color="Maroon">s3</font>]
 <b>by </b><i>, <a class="ref" href="instalg1.html#T2" target="_self">Th2</a>, <a class="ref" href="msualg_6.html#T19">MSUALG_6:19</a>, <a class="ref" href="rewrite1.html#T17">REWRITE1:17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:47"/><i><font color="Green">E81</font></i>: 
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">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">Translation</a> of <font color="Maroon">B</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">MSUALG_6:sch 1</a>(, );<br/></i>
<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>assume </b><a NAME="E6:47"/>
 <a href="msualg_6.html#K3">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>hence </b><a NAME="E7:47"/>
(  <a href="msualg_6.html#K3">TranslationRel</a> <font color="Maroon">S</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s1</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s2</font> &amp; ( for <font color="Olive">t</font> being   <a href="msualg_6.html#M2">Translation</a> of <font color="Maroon">B</font>,<font color="Maroon">s1</font>,<font color="Maroon">s2</font> holds  <font color="Olive">t</font> is    <a href="msualg_6.html#M2">Translation</a> of <font color="Maroon">A</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s1</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s2</font> ) )
 <b>by </b><i/>;<br/>


</div>
