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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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being <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/> for <font color="Olive" title="b2">R</font> being  <a href="msualg_4.html#NM3" title="MSUALG_4:NM.3">ManySortedRelation</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Olive" title="b1">A</font><br/> for <font color="Olive" title="b3">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="b4">a</font>, <font color="Olive" title="b5">b</font> being  <a href="msualg_6.html#NM1" title="MSUALG_6:NM.1">Element</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b3">s</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">a</font>,<font color="Olive" title="b5">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#K8" title="MSUALG_6:func.8">TRS</a> <font color="Olive" title="b2">R</font></span>)</span> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Olive" title="b3">s</font> iff  ex <font color="Olive" title="b6">s'</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font> st <br/>(  <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="Olive" title="b6">s'</font>,<font color="Olive" title="b3">s</font> &amp;  ex <font color="Olive" title="b7">l</font>, <font color="Olive" title="b8">r</font> being  <a href="msualg_6.html#NM1" title="MSUALG_6:NM.1">Element</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b6">s'</font> ex <font color="Olive" title="b9">h</font> being   <a href="msualg_6.html#M1" title="MSUALG_6:mode.1">Endomorphism</a> of <font color="Olive" title="b1">A</font> ex <font color="Olive" title="b10">t</font> being   <a href="msualg_6.html#M2" title="MSUALG_6:mode.2">Translation</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b6">s'</font>,<font color="Olive" title="b3">s</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b7">l</font>,<font color="Olive" title="b8">r</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="Olive" title="b2">R</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Olive" title="b6">s'</font> &amp; <font color="Olive" title="b4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b10">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="Olive" title="b9">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b6">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b7">l</font></span>)</span> &amp; <font color="Olive" title="b5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b10">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="Olive" title="b9">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b6">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b8">r</font></span>)</span> ) ) )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">R</font> being  <a href="msualg_4.html#NM3" title="MSUALG_4:NM.3">ManySortedRelation</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b2">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="b3">a</font>, <font color="Olive" title="b4">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="b2">s</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">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#K8" title="MSUALG_6:func.8">TRS</a> <font color="Olive" title="b1">R</font></span>)</span> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Olive" title="b2">s</font> iff  ex <font color="Olive" title="b5">s'</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font> st <br/>(  <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="Olive" title="b5">s'</font>,<font color="Olive" title="b2">s</font> &amp;  ex <font color="Olive" title="b6">l</font>, <font color="Olive" title="b7">r</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="b5">s'</font> ex <font color="Olive" title="b8">h</font> being   <a href="msualg_6.html#M1" title="MSUALG_6:mode.1">Endomorphism</a> of <font color="Maroon" title="c2">A</font> ex <font color="Olive" title="b9">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="Olive" title="b5">s'</font>,<font color="Olive" title="b2">s</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">l</font>,<font color="Olive" title="b7">r</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="Olive" title="b1">R</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Olive" title="b5">s'</font> &amp; <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b9">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="Olive" title="b8">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b5">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">l</font></span>)</span> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b9">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="Olive" title="b8">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b5">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b7">r</font></span>)</span> ) ) )</span><br/><b>let </b><font color="Maroon" title="c3">R</font> be   <a href="msualg_4.html#NM3" title="MSUALG_4:NM.3">ManySortedRelation</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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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> <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> iff  ex <font color="Olive" title="b4">s'</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font> st <br/>(  <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="Olive" title="b4">s'</font>,<font color="Olive" title="b1">s</font> &amp;  ex <font color="Olive" title="b5">l</font>, <font color="Olive" title="b6">r</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="b4">s'</font> ex <font color="Olive" title="b7">h</font> being   <a href="msualg_6.html#M1" title="MSUALG_6:mode.1">Endomorphism</a> of <font color="Maroon" title="c2">A</font> ex <font color="Olive" title="b8">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="Olive" title="b4">s'</font>,<font color="Olive" title="b1">s</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">l</font>,<font color="Olive" title="b6">r</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="Olive" title="b4">s'</font> &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">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="Olive" title="b7">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b4">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">l</font></span>)</span> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">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="Olive" title="b7">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b4">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">r</font></span>)</span> ) ) )</span><br/><b>let </b><font color="Maroon" title="c4">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">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="Maroon" title="c4">s</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">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#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="c4">s</font> iff  ex <font color="Olive" title="b3">s'</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font> st <br/>(  <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="Olive" title="b3">s'</font>,<font color="Maroon" title="c4">s</font> &amp;  ex <font color="Olive" title="b4">l</font>, <font color="Olive" title="b5">r</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="b3">s'</font> ex <font color="Olive" title="b6">h</font> being   <a href="msualg_6.html#M1" title="MSUALG_6:mode.1">Endomorphism</a> of <font color="Maroon" title="c2">A</font> ex <font color="Olive" title="b7">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="Olive" title="b3">s'</font>,<font color="Maroon" title="c4">s</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">l</font>,<font color="Olive" title="b5">r</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="Olive" title="b3">s'</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">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="Olive" title="b6">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b3">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">l</font></span>)</span> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">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="Olive" title="b6">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b3">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">r</font></span>)</span> ) ) )</span><br/><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <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="c4">s</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">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#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="c4">s</font> iff  ex <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> st <br/>(  <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="Olive" title="b1">s'</font>,<font color="Maroon" title="c4">s</font> &amp;  ex <font color="Olive" title="b2">l</font>, <font color="Olive" title="b3">r</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> ex <font color="Olive" title="b4">h</font> being   <a href="msualg_6.html#M1" title="MSUALG_6:mode.1">Endomorphism</a> of <font color="Maroon" title="c2">A</font> ex <font color="Olive" title="b5">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="Olive" title="b1">s'</font>,<font color="Maroon" title="c4">s</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">l</font>,<font color="Olive" title="b3">r</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="Olive" title="b1">s'</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">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="Olive" title="b4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">l</font></span>)</span> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">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="Olive" title="b4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">r</font></span>)</span> ) ) )</span><br/>









<a NAME="E1:64"/><i><font color="Green" title="E42">A1</font></i>: 
 <a href="msualg_6.html#K6" title="MSUALG_6:func.6">InvCl</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K7" title="MSUALG_6:func.7">StabCl</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_6.html#K8" title="MSUALG_6:func.8">TRS</a> <font color="Maroon" title="c3">R</font>
 
<b>by </b><i><a class="ref" href="msualg_6.html#T37" target="_self" title="MSUALG_6:th.37">Th37</a></i>;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <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> st <br/>(  <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="Olive" title="b1">s'</font>,<font color="Maroon" title="c4">s</font> &amp;  ex <font color="Olive" title="b2">l</font>, <font color="Olive" title="b3">r</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> ex <font color="Olive" title="b4">h</font> being   <a href="msualg_6.html#M1" title="MSUALG_6:mode.1">Endomorphism</a> of <font color="Maroon" title="c2">A</font> ex <font color="Olive" title="b5">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="Olive" title="b1">s'</font>,<font color="Maroon" title="c4">s</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">l</font>,<font color="Olive" title="b3">r</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="Olive" title="b1">s'</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">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="Olive" title="b4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">l</font></span>)</span> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">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="Olive" title="b4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">r</font></span>)</span> ) ) implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">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#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="c4">s</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:64_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">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#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="c4">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <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> st <br/>(  <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="Olive" title="b1">s'</font>,<font color="Maroon" title="c4">s</font> &amp;  ex <font color="Olive" title="b2">u</font>, <font color="Olive" title="b3">v</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> ex <font color="Olive" title="b4">h</font> being   <a href="msualg_6.html#M1" title="MSUALG_6:mode.1">Endomorphism</a> of <font color="Maroon" title="c2">A</font> ex <font color="Olive" title="b5">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="Olive" title="b1">s'</font>,<font color="Maroon" title="c4">s</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">u</font>,<font color="Olive" title="b3">v</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="Olive" title="b1">s'</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">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="Olive" title="b4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">u</font></span>)</span> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">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="Olive" title="b4">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">v</font></span>)</span> ) )</span><br/><b>then consider </b><font color="Maroon" title="c7">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="c8">x</font>, <font color="Maroon" title="c9">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="c7">s'</font>, <font color="Maroon" title="c10">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="c7">s'</font>,<font color="Maroon" title="c4">s</font><b> such that </b><br/><a NAME="E3:64_1"/><i><font color="Green" title="E43">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="c7">s'</font>,<font color="Maroon" title="c4">s</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><a href="msualg_6.html#K7" title="MSUALG_6:func.7">StabCl</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="c7">s'</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">y</font> )
 <b>by </b><i><a class="txt" href="msualg_6.html#E1:64"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="msualg_6.html#T29" target="_self" title="MSUALG_6:th.29">Th29</a></i>;<br/><b>take </b><font color="Maroon" title="c11">s'</font> = <font color="Maroon" title="c7">s'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <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="c11">s'</font>,<font color="Maroon" title="c4">s</font> &amp;  ex <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</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="c11">s'</font> ex <font color="Olive" title="b3">h</font> being   <a href="msualg_6.html#M1" title="MSUALG_6:mode.1">Endomorphism</a> of <font color="Maroon" title="c2">A</font> ex <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="c2">A</font>,<font color="Maroon" title="c11">s'</font>,<font color="Maroon" title="c4">s</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</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="c11">s'</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">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="Olive" title="b3">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font></span>)</span> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">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="Olive" title="b3">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font></span>)</span> ) )</span><br/><b>thus </b><a NAME="E4:64_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="c11">s'</font>,<font color="Maroon" title="c4">s</font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E3:64_1"><i><font color="Green" title="E43">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</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="c11">s'</font> ex <font color="Olive" title="b3">h</font> being   <a href="msualg_6.html#M1" title="MSUALG_6:mode.1">Endomorphism</a> of <font color="Maroon" title="c2">A</font> ex <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="c2">A</font>,<font color="Maroon" title="c11">s'</font>,<font color="Maroon" title="c4">s</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</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="c11">s'</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">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="Olive" title="b3">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font></span>)</span> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">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="Olive" title="b3">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font></span>)</span> )</span><br/><b>consider </b><font color="Maroon" title="c12">u</font>, <font color="Maroon" title="c13">v</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="c11">s'</font>, <font color="Maroon" title="c14">h</font> being    <a href="msualg_6.html#M1" title="MSUALG_6:mode.1">Endomorphism</a> of <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E6:64_1"/><i><font color="Green" title="E44">A3</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">u</font>,<font color="Maroon" title="c13">v</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="c11">s'</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">u</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> )
 <b>by </b><i><a class="txt" href="msualg_6.html#E3:64_1"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="msualg_6.html#T31" target="_self" title="MSUALG_6:th.31">Th31</a></i>;<br/><b>take </b><font color="Maroon" title="c15">u</font> = <font color="Maroon" title="c12">u</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">v</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="c11">s'</font> ex <font color="Olive" title="b2">h</font> being   <a href="msualg_6.html#M1" title="MSUALG_6:mode.1">Endomorphism</a> of <font color="Maroon" title="c2">A</font> ex <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="c2">A</font>,<font color="Maroon" title="c11">s'</font>,<font color="Maroon" title="c4">s</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">u</font>,<font color="Olive" title="b1">v</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="c11">s'</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">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="Olive" title="b2">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">u</font></span>)</span> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">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="Olive" title="b2">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span> )</span><br/><b>take </b><font color="Maroon" title="c16">v</font> = <font color="Maroon" title="c13">v</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">h</font> being   <a href="msualg_6.html#M1" title="MSUALG_6:mode.1">Endomorphism</a> of <font color="Maroon" title="c2">A</font> ex <font color="Olive" title="b2">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="c11">s'</font>,<font color="Maroon" title="c4">s</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">v</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="c11">s'</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">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="Olive" title="b1">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">u</font></span>)</span> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">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="Olive" title="b1">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">v</font></span>)</span> )</span><br/><b>take </b><font color="Maroon" title="c17">h</font> = <font color="Maroon" title="c14">h</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <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="c2">A</font>,<font color="Maroon" title="c11">s'</font>,<font color="Maroon" title="c4">s</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">v</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="c11">s'</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="c17">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">u</font></span>)</span> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="c17">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">v</font></span>)</span> )</span><br/><b>reconsider </b><font color="Maroon" title="c18">t</font> = <font color="Maroon" title="c10">t</font> as    <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="c11">s'</font>,<font color="Maroon" title="c4">s</font> ;<br/><b>take </b><font color="Maroon" title="c19">t</font> = <font color="Maroon" title="c18">t</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">v</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="c11">s'</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">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="c17">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">u</font></span>)</span> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">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="c17">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">v</font></span>)</span> )</span><br/><b>thus </b><a NAME="E8:64_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c16">v</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="c11">s'</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">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="c17">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">u</font></span>)</span> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">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="c17">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">v</font></span>)</span> )
 <b>by </b><i><a class="txt" href="msualg_6.html#E3:64_1"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="msualg_6.html#E6:64_1"><i><font color="Green" title="E44">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c7">s'</font> being   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font><b> such that </b><a NAME="E4:64"/><i><font color="Green" title="E43">A4</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="c7">s'</font>,<font color="Maroon" title="c4">s</font>
 <b>and </b><br/><a NAME="E5:64"/><i><font color="Green" title="E44">A5</font></i>: 
 ex <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">r</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="c7">s'</font> ex <font color="Olive" title="b3">h</font> being   <a href="msualg_6.html#M1" title="MSUALG_6:mode.1">Endomorphism</a> of <font color="Maroon" title="c2">A</font> ex <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="c2">A</font>,<font color="Maroon" title="c7">s'</font>,<font color="Maroon" title="c4">s</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">l</font>,<font color="Olive" title="b2">r</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="c7">s'</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">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="Olive" title="b3">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c7">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">l</font></span>)</span> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">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="Olive" title="b3">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c7">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">r</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">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#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="c4">s</font></span><br/>

<b>consider </b><font color="Maroon" title="c8">l</font>, <font color="Maroon" title="c9">r</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="c7">s'</font>, <font color="Maroon" title="c10">h</font> being    <a href="msualg_6.html#M1" title="MSUALG_6:mode.1">Endomorphism</a> of <font color="Maroon" title="c2">A</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="c7">s'</font>,<font color="Maroon" title="c4">s</font><b> such that </b><br/><a NAME="E7:64"/><i><font color="Green" title="E45">A6</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">l</font>,<font color="Maroon" title="c9">r</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="c7">s'</font> &amp; <font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c7">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">l</font></span>)</span> &amp; <font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c7">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">r</font></span>)</span> )
 <b>by </b><i><a class="txt" href="msualg_6.html#E5:64"><i><font color="Green" title="E44">A5</font></i></a></i>;<br/>
<a NAME="E8:64"/>
<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="c10">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c7">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">l</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c7">s'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">r</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#K7" title="MSUALG_6:func.7">StabCl</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="c7">s'</font>
 
<b>by </b><i><a class="txt" href="msualg_6.html#E7:64"><i><font color="Green" title="E45">A6</font></i></a>, <a class="ref" href="msualg_6.html#T31" target="_self" title="MSUALG_6:th.31">Th31</a></i>;<br/>
<b>hence </b><a NAME="E9:64"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">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#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="c4">s</font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E1:64"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="msualg_6.html#E4:64"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="msualg_6.html#E7:64"><i><font color="Green" title="E45">A6</font></i></a>, <a class="ref" href="msualg_6.html#T29" target="_self" title="MSUALG_6:th.29">Th29</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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