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<b>let </b><font color="Maroon">S</font> be  <a href="osalg_4.html#V2">locally_directed</a> <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">X</font> be  <a href="relat_1.html#V2">V5</a> <a href="struct_0.html#NM8">ManySortedSet</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">t</font> be    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>;<br/>







<b>set </b><font color="Maroon">PTA</font> =  <a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">D</font> =  <a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">R</font> =  <a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">t</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Maroon">t</font></span>)</span> <b>by </b><i/>;<br/>
<i><font color="Green">E34</font></i>:  <a href="osafree.html#K25" target="_self">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">t</font> = 
 <a href="osalg_4.html#K8">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">y</font>
<b>by </b><i/>
<br/>.= 
 <a href="funct_5.html#K1">proj1</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="txt" href="osafree.html#E4:74"><i><font color="Green">E37</font></i></a></i>
;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E3:74"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/>

<a NAME="E4:74"/><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6">Terminals</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">sy</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> as   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> ;<br/>
<i><font color="Green">E39</font></i>: <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">sy</font></span>)</span> = 
 <a href="osafree.html#K17" target="_self">@</a> <font color="Maroon">sy</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">sy</font></span>)</span>,<font color="Olive">s1</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">s1</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> :  ex <font color="Olive">s2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ex <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s2</font> &amp; <font color="Maroon">sy</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x2</font>,<font color="Olive">s2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">s2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s1</font> ) </span>}</span> 

;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:74_3"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="osafree.html#K25" target="_self">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">t</font>
 ;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:74_3"/><i><font color="Green">E54</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="funct_5.html#D1">FUNCT_5:def 1</a></i>;<br/><b>consider </b><font color="Maroon">s1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E5:74_3"/><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">sy</font></span>)</span>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E6:74_3"/>
 ex <font color="Olive">s2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ex <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s2</font> &amp; <font color="Maroon">sy</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x2</font>,<font color="Olive">s2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">s2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s1</font> )
 <b>by </b><i>, , <a class="txt" href="osafree.html#E3:74_3"><i><font color="Green">E54</font></i></a></i>;<br/><b>thus </b><a NAME="E7:74_3"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><i>, <a class="ref" href="osafree.html#T2" target="_self">Th2</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E8:74"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 ;<br/>

<a NAME="E9:74"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E10:74"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="osafree.html#K25" target="_self">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">t</font>
 <b>by </b><i>, <a class="ref" href="funct_5.html#D1">FUNCT_5:def 1</a></i>;<br/>


</div>
