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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">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">R</font> be   <a href="msualg_4.html#NM3">ManySortedRelation</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>;<br/><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>let </b><font color="Maroon">a</font> be   <a href="msualg_6.html#NM1">Element</a> of <font color="Maroon">A</font>,<font color="Maroon">s</font>, <font color="Maroon">b</font> be   <a href="msualg_6.html#NM1">Element</a> of <font color="Maroon">A</font>,<font color="Maroon">s</font>;<br/>









<a NAME="E1:64"/><i><font color="Green">E32</font></i>: 
 <a href="msualg_6.html#K6" target="_self">InvCl</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K7" target="_self">StabCl</a> <font color="Maroon">R</font></span>)</span> <a href="pboole.html#R6">=</a>  <a href="msualg_6.html#K8" target="_self">TRS</a> <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:64_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K8" target="_self">TRS</a> <font color="Maroon">R</font></span>)</span> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/><b>then consider </b><font color="Maroon">s'</font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>, <font color="Maroon">x</font> being   <a href="msualg_6.html#NM1">Element</a> of <font color="Maroon">A</font>,<font color="Maroon">s'</font>, <font color="Maroon">y</font> being   <a href="msualg_6.html#NM1">Element</a> of <font color="Maroon">A</font>,<font color="Maroon">s'</font>, <font color="Maroon">t</font> being    <a href="msualg_6.html#M2" target="_self">Translation</a> of <font color="Maroon">A</font>,<font color="Maroon">s'</font>,<font color="Maroon">s</font><b> such that </b><br/><a NAME="E3:64_1"/><i><font color="Green">E34</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">s'</font>,<font color="Maroon">s</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K7" target="_self">StabCl</a> <font color="Maroon">R</font></span>)</span> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s'</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="msualg_6.html#T2" target="_self">Th2</a>, </i>;<br/><b>take </b><font color="Maroon">s'</font> = <font color="Maroon">s'</font>;<br/><b>thus </b><a NAME="E4:64_1"/>
 <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">s'</font>,<font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E6:64_1"><i><font color="Green">E43</font></i></a></i>;<br/><b>consider </b><font color="Maroon">u</font> being   <a href="msualg_6.html#NM1">Element</a> of <font color="Maroon">A</font>,<font color="Maroon">s'</font>, <font color="Maroon">v</font> being   <a href="msualg_6.html#NM1">Element</a> of <font color="Maroon">A</font>,<font color="Maroon">s'</font>, <font color="Maroon">h</font> being    <a href="msualg_6.html#M1" target="_self">Endomorphism</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E6:64_1"/><i><font color="Green">E43</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s'</font> &amp; <font color="Maroon">x</font> <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">s'</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font> &amp; <font color="Maroon">y</font> <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">s'</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="txt" href="msualg_6.html#E6:64_1"><i><font color="Green">E43</font></i></a>, </i>;<br/><b>take </b><font color="Maroon">u</font> = <font color="Maroon">u</font>;<br/><b>take </b><font color="Maroon">v</font> = <font color="Maroon">v</font>;<br/><b>take </b><font color="Maroon">h</font> = <font color="Maroon">h</font>;<br/><b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t</font> as    <a href="msualg_6.html#M2" target="_self">Translation</a> of <font color="Maroon">A</font>,<font color="Maroon">s'</font>,<font color="Maroon">s</font> ;<br/><b>take </b><font color="Maroon">t</font> = <font color="Maroon">t</font>;<br/><b>thus </b><a NAME="E8:64_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s'</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="funct_2.html#K3">.</a> <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">s'</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="funct_2.html#K3">.</a> <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">s'</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> )
 <b>by </b><i><a class="txt" href="msualg_6.html#E6:64_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="msualg_6.html#D1" target="_self">Def1</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">s'</font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font><b> such that </b><a NAME="E3:64"/><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">s'</font>,<font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E4:64"/><i><font color="Green">E55</font></i>: 
 ex <font color="Olive">l</font>, <font color="Olive">r</font> being  <a href="msualg_6.html#NM1">Element</a> of <font color="Maroon">A</font>,<font color="Maroon">s'</font> ex <font color="Olive">h</font> being   <a href="msualg_6.html#M1" target="_self">Endomorphism</a> of <font color="Maroon">A</font> ex <font color="Olive">t</font> being   <a href="msualg_6.html#M2" target="_self">Translation</a> of <font color="Maroon">A</font>,<font color="Maroon">s'</font>,<font color="Maroon">s</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">l</font>,<font color="Olive">r</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s'</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s'</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">l</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s'</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font></span>)</span> )
 ;<br/>

<b>consider </b><font color="Maroon">l</font> being   <a href="msualg_6.html#NM1">Element</a> of <font color="Maroon">A</font>,<font color="Maroon">s'</font>, <font color="Maroon">r</font> being   <a href="msualg_6.html#NM1">Element</a> of <font color="Maroon">A</font>,<font color="Maroon">s'</font>, <font color="Maroon">h</font> being    <a href="msualg_6.html#M1" target="_self">Endomorphism</a> of <font color="Maroon">A</font>, <font color="Maroon">t</font> being    <a href="msualg_6.html#M2" target="_self">Translation</a> of <font color="Maroon">A</font>,<font color="Maroon">s'</font>,<font color="Maroon">s</font><b> such that </b><br/><a NAME="E6:64"/><i><font color="Green">E56</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">l</font>,<font color="Maroon">r</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s'</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="funct_2.html#K3">.</a> <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">s'</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">l</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="funct_2.html#K3">.</a> <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">s'</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span> )
 <b>by </b><i><a class="ref" href="msualg_6.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E7:64"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K3">.</a> <font color="Maroon">l</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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#K3">.</a> <font color="Maroon">r</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K7" target="_self">StabCl</a> <font color="Maroon">R</font></span>)</span> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s'</font>
 
<b>by </b><i><a class="ref" href="msualg_6.html#T3" target="_self">Th3</a>, </i>;<br/>
<b>hence </b><a NAME="E8:64"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K8" target="_self">TRS</a> <font color="Maroon">R</font></span>)</span> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="msualg_6.html#T2" target="_self">Th2</a>, , <a class="ref" href="msualg_6.html#T3" target="_self">Th3</a>, </i>;<br/>


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