<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">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="c3">A</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">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S'</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c5">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>







<b>assume </b><a NAME="E1:47"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c2">S'</font>,<font color="Maroon" title="c1">S</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">B</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="c2">S'</font>;<br/>

<b>assume </b><a NAME="E2:47"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <font color="Maroon" title="c2">S'</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S'</font>,  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S'</font>] <b>means </b>(  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon" title="c1">S</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $2,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $3 &amp; $1 is    <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $2,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $3 );<br/>
<a NAME="E3:47"/><i><font color="Green" title="E29">A3</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S'</font> holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c6">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font></span>)</span>,<font color="Olive" title="b1">s</font>,<font color="Olive" title="b1">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" title="c7">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S'</font>;<br/>

<b>thus </b><a NAME="E1:47_1"/>
 <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon" title="c1">S</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T13" title="REWRITE1:th.13">REWRITE1:13</a></i>;<br/>

<a NAME="E2:47_1"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c6">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="instalg1.html#E1:47"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="instalg1.html#E2:47"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="instalg1.html#D3" target="_self" title="INSTALG1:def.3">Def3</a></i>;<br/>
<a NAME="E3:47_1"/><b>then </b>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c6">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/>
<b>hence </b><a NAME="E4:47_1"/>
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c6">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s</font></span>)</span> is    <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font>
 <b>by </b><i><a class="ref" href="msualg_6.html#T16" title="MSUALG_6:th.16">MSUALG_6:16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:47"/><i><font color="Green" title="E30">A4</font></i>: 
for <font color="Olive" title="b1">s1</font>, <font color="Olive" title="b2">s2</font>, <font color="Olive" title="b3">s3</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S'</font>  st  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon" title="c2">S'</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Olive" title="b1">s1</font>,<font color="Olive" title="b2">s2</font> holds <br/>for <font color="Olive" title="b4">t</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon" title="c6">B</font>,<font color="Olive" title="b1">s1</font>,<font color="Olive" title="b2">s2</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b4">t</font>,<font color="Olive" title="b1">s1</font>,<font color="Olive" title="b2">s2</font>] holds <br/>for <font color="Olive" title="b5">h</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b5">h</font> <a href="msualg_6.html#R1" title="MSUALG_6:pred.1">is_e.translation_of</a> <font color="Maroon" title="c6">B</font>,<font color="Olive" title="b2">s2</font>,<font color="Olive" title="b3">s3</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b5">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b4">t</font>,<font color="Olive" title="b1">s1</font>,<font color="Olive" title="b3">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" title="c7">s1</font>, <font color="Maroon" title="c8">s2</font>, <font color="Maroon" title="c9">s3</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S'</font>;<br/>

<b>assume </b><a NAME="E1:47_2"/>
 <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon" title="c2">S'</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c7">s1</font>,<font color="Maroon" title="c8">s2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c10">t</font> be    <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c7">s1</font>,<font color="Maroon" title="c8">s2</font>;<br/>

<b>assume </b><a NAME="E2:47_2"/><i><font color="Green" title="E31">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">t</font>,<font color="Maroon" title="c7">s1</font>,<font color="Maroon" title="c8">s2</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c11">h</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>

<b>assume </b><a NAME="E3:47_2"/><i><font color="Green" title="E32">A6</font></i>: 
<font color="Maroon" title="c11">h</font> <a href="msualg_6.html#R1" title="MSUALG_6:pred.1">is_e.translation_of</a> <font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c8">s2</font>,<font color="Maroon" title="c9">s3</font>
 ;<br/>

<a NAME="E4:47_2"/><b>then </b><i><font color="Green" title="E33">A7</font></i>: 
<font color="Maroon" title="c11">h</font> <a href="msualg_6.html#R1" title="MSUALG_6:pred.1">is_e.translation_of</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s2</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">s3</font>
 
<b>by </b><i><a class="txt" href="instalg1.html#E1:47"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="instalg1.html#E2:47"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="instalg1.html#T37" target="_self" title="INSTALG1:th.37">Th37</a></i>;<br/>
<a NAME="E5:47_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">s3</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="txt" href="instalg1.html#E1:47"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="instalg1.html#E2:47"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="instalg1.html#E3:47_2"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="instalg1.html#T37" target="_self" title="INSTALG1:th.37">Th37</a>, <a class="ref" href="msualg_6.html#T12" title="MSUALG_6:th.12">MSUALG_6:12</a></i>;<br/>
<a NAME="E6:47_2"/><b>then </b>
 <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon" title="c1">S</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s2</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">s3</font>
 
<b>by </b><i><a class="ref" href="rewrite1.html#T16" title="REWRITE1:th.16">REWRITE1:16</a></i>;<br/>
<b>hence </b><a NAME="E7:47_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">t</font>,<font color="Maroon" title="c7">s1</font>,<font color="Maroon" title="c9">s3</font>]
 <b>by </b><i><a class="txt" href="instalg1.html#E2:47_2"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="instalg1.html#E4:47_2"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="msualg_6.html#T19" title="MSUALG_6:th.19">MSUALG_6:19</a>, <a class="ref" href="rewrite1.html#T17" title="REWRITE1:th.17">REWRITE1:17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:47"/><i><font color="Green" title="E31">A8</font></i>: 
for <font color="Olive" title="b1">s1</font>, <font color="Olive" title="b2">s2</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S'</font>  st  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon" title="c2">S'</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Olive" title="b1">s1</font>,<font color="Olive" title="b2">s2</font> holds <br/>for <font color="Olive" title="b3">t</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon" title="c6">B</font>,<font color="Olive" title="b1">s1</font>,<font color="Olive" title="b2">s2</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">t</font>,<font color="Olive" title="b1">s1</font>,<font color="Olive" title="b2">s2</font>]
 
<b>from </b><i><a class="ref" href="msualg_6.html#S1" title="MSUALG_6:sch.1">MSUALG_6:sch 1</a>(<a class="txt" href="instalg1.html#E3:47"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="instalg1.html#E4:47"><i><font color="Green" title="E30">A4</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c7">s1</font>, <font color="Maroon" title="c8">s2</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S'</font>;<br/>

<b>assume </b><a NAME="E6:47"/>
 <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon" title="c2">S'</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c7">s1</font>,<font color="Maroon" title="c8">s2</font>
 ;<br/>

<b>hence </b><a NAME="E7:47"/>
(  <a href="msualg_6.html#K3" title="MSUALG_6:func.3">TranslationRel</a> <font color="Maroon" title="c1">S</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s1</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s2</font> &amp; ( for <font color="Olive" title="b1">t</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c7">s1</font>,<font color="Maroon" title="c8">s2</font> holds  <font color="Olive" title="b1">t</font> is    <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s1</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s2</font> ) )
 <b>by </b><i><a class="txt" href="instalg1.html#E5:47"><i><font color="Green" title="E31">A8</font></i></a></i>;<br/>


</div>
