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<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> be  <a href="msualg_6.html#V4" title="MSUALG_6:attr.4">stable</a> <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">ManySortedRelation</a> of <font color="Maroon" title="c2">A</font>;<br/><b>let </b><font color="Maroon" title="c4">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="c5">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="c6">a</font>, <font color="Maroon" title="c7">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>











<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E37">A1</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">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> <span class="p1">(<span class="default"><a href="msualg_6.html#K6" title="MSUALG_6:func.6">InvCl</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">s</font>
 ;<br/>

<a NAME="E2:53"/><b>then </b>
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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">s</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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">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>then consider </b><font color="Maroon" title="c8">s'</font> being   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</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="Maroon" title="c8">s'</font>, <font color="Maroon" title="c11">t</font> being    <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c8">s'</font>,<font color="Maroon" title="c5">s</font><b> such that </b><br/><a NAME="E4:53"/><i><font color="Green" title="E38">A2</font></i>: 
(  <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="c8">s'</font>,<font color="Maroon" title="c5">s</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</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="c3">R</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c8">s'</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> )
 <b>by </b><i><a class="txt" href="msualg_6.html#E1:53"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="msualg_6.html#T29" target="_self" title="MSUALG_6:th.29">Th29</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">T</font> being    <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c8">s'</font>,<font color="Maroon" title="c5">s</font><b> such that </b><br/><a NAME="E6:53"/><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon" title="c12">T</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c8">s'</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">t</font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E4:53"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="msualg_6.html#T26" target="_self" title="MSUALG_6:th.26">Th26</a></i>;<br/>
<a NAME="E7:53"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c12">T</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c8">s'</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c8">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">T</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c8">s'</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c8">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</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/>
<a NAME="E8:53"/><b>then </b><i><font color="Green" title="E40">A4</font></i>: 
( <font color="Maroon" title="c12">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c8">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c12">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c8">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font> )
 
<b>by </b><i><a class="txt" href="msualg_6.html#E4:53"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="msualg_6.html#E6:53"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/>
<a NAME="E9:53"/>
<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="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c8">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c8">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</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="c3">R</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c8">s'</font>
 
<b>by </b><i><a class="txt" href="msualg_6.html#E4:53"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="msualg_6.html#D9" target="_self" title="MSUALG_6:def.9">Def9</a></i>;<br/>
<b>hence </b><a NAME="E10:53"/>
<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="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">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> <span class="p1">(<span class="default"><a href="msualg_6.html#K6" title="MSUALG_6:func.6">InvCl</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">s</font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E4:53"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="msualg_6.html#E8:53"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="msualg_6.html#T29" target="_self" title="MSUALG_6:th.29">Th29</a></i>;<br/>


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