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<div class="add">

<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">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="c3">R</font>, <font color="Maroon" title="c4">E</font> be   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">ManySortedRelation</a> of <font color="Maroon" title="c2">A</font>;<br/>





<b>assume </b><a NAME="E1:79"/><i><font color="Green" title="E54">A1</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="c1">S</font><br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being  <a href="msualg_6.html#NM1" title="MSUALG_6:NM.1">Element</a> of <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">s</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Olive" title="b1">s</font> iff <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K8" title="MSUALG_6:func.8">TRS</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Olive" title="b1">s</font> )
 ;<br/>

<a NAME="E2:79"/><b>then </b><i><font color="Green" title="E55">A2</font></i>: 
<font color="Maroon" title="c4">E</font> is <a href="msualg_4.html#V2" title="MSUALG_4:attr.2">MSEquivalence_Relation-like</a>
 
<b>by </b><i><a class="ref" href="msualg_6.html#T49" target="_self" title="MSUALG_6:th.49">Th49</a></i>;<br/>
<a NAME="E3:79"/><i><font color="Green" title="E56">A3</font></i>: 
<font color="Maroon" title="c4">E</font> is <a href="msualg_6.html#V4" title="MSUALG_6:attr.4">stable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">h</font> be    <a href="msualg_6.html#M1" title="MSUALG_6:mode.1">Endomorphism</a> of <font color="Maroon" title="c2">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_6.html#D9" target="_self" title="MSUALG_6:def.9">MSUALG_6:def 9</a><br/><b>let </b><font color="Maroon" title="c6">s</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">a</font>, <font color="Maroon" title="c8">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><a NAME="E1:79_1"/><i><font color="Green" title="E57">A4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c6">s</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c8">b</font> as   <a href="msualg_6.html#NM1" title="MSUALG_6:NM.1">Element</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c6">s</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">a'</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c12">b'</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> as   <a href="msualg_6.html#NM1" title="MSUALG_6:NM.1">Element</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c6">s</font> ;<br/>
<a NAME="E4:79_1"/>
<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K8" title="MSUALG_6:func.8">TRS</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c6">s</font>
 
<b>by </b><i><a class="txt" href="msualg_6.html#E1:79"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="msualg_6.html#E1:79_1"><i><font color="Green" title="E57">A4</font></i></a></i>;<br/>
<a NAME="E5:79_1"/><b>then </b>
<font color="Maroon" title="c11">a'</font>,<font color="Maroon" title="c12">b'</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K8" title="MSUALG_6:func.8">TRS</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c6">s</font>
 
<b>by </b><i><a class="ref" href="msualg_6.html#T45" target="_self" title="MSUALG_6:th.45">Th45</a></i>;<br/>
<b>hence </b><a NAME="E6:79_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">b</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> <font color="Maroon" title="c4">E</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c6">s</font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E1:79"><i><font color="Green" title="E54">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:79"/>
<font color="Maroon" title="c4">E</font> is <a href="msualg_6.html#V3" title="MSUALG_6:attr.3">invariant</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><b>proof </b></a><div class="add">

<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>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_6.html#D8" target="_self" title="MSUALG_6:def.8">MSUALG_6:def 8</a><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="E1:79_2"/><i><font color="Green" title="E57">A5</font></i>: 
<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">A</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c6">s2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c7">t</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="c2">A</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="c2">A</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="ref" href="msualg_6.html#T11" target="_self" title="MSUALG_6:th.11">Th11</a></i>;<br/>
<b>let </b><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E3:79_2"/><i><font color="Green" title="E58">A6</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c5">s1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c11">x</font> = <font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c12">y</font> = <font color="Maroon" title="c10">b</font> as   <a href="msualg_6.html#NM1" title="MSUALG_6:NM.1">Element</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">s1</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:79_2"/>
<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K8" title="MSUALG_6:func.8">TRS</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c5">s1</font>
 
<b>by </b><i><a class="txt" href="msualg_6.html#E1:79"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="msualg_6.html#E3:79_2"><i><font color="Green" title="E58">A6</font></i></a></i>;<br/>
<a NAME="E6:79_2"/><b>then </b>
( <font color="Maroon" title="c7">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c7">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">y</font> &amp; <font color="Maroon" title="c7">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c7">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K8" title="MSUALG_6:func.8">TRS</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c6">s2</font> )
 
<b>by </b><i><a class="txt" href="msualg_6.html#E1:79_2"><i><font color="Green" title="E57">A5</font></i></a>, <a class="ref" href="msualg_6.html#T47" target="_self" title="MSUALG_6:th.47">Th47</a></i>;<br/>
<b>hence </b><a NAME="E7:79_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="c7">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">b</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> <font color="Maroon" title="c4">E</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c6">s2</font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E1:79"><i><font color="Green" title="E54">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:79"/>
<font color="Maroon" title="c4">E</font> is   <a href="msualg_6.html#NM4" title="MSUALG_6:NM.4">EquationalTheory</a> of <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E2:79"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="msualg_6.html#E3:79"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="msualg_4.html#D5" title="MSUALG_4:def.5">MSUALG_4:def 5</a></i>;<br/>


</div>
