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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">R1</font> be   <a href="osalg_4.html#NM4">OSCongruence</a> of  <a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font>, <font color="Maroon">R2</font> be   <a href="osalg_4.html#NM4">OSCongruence</a> of  <a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</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>assume </b><a NAME="E1:76"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">R1</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">R2</font>
 ;<br/>

<b>set </b><font color="Maroon">s</font> =  <a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Maroon">t</font>;<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">SPTA</font> = 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>;<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>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:76"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="osafree.html#K25" target="_self">OSClass</a> <font color="Maroon">R1</font>,<font color="Maroon">t</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">xa</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">E37</font></i>:  <a href="osafree.html#K25" target="_self">OSClass</a> <font color="Maroon">R1</font>,<font color="Maroon">t</font> = 
 <a href="osalg_4.html#K8">OSClass</a> <font color="Maroon">R1</font>,<font color="Maroon">xa</font>
<b>by </b><i/>
<br/>.= 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K5">CompClass</a> <font color="Maroon">R1</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K3">CComp</a> <span class="p3">(<span class="default"><a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">xa</font>

;<br/>
<b>set </b><font color="Maroon">CC1</font> =  <a href="osalg_4.html#K3">CComp</a> <span class="p1">(<span class="default"><a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Maroon">t</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">CR1</font> =  <a href="osalg_4.html#K5">CompClass</a> <font color="Maroon">R1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K3">CComp</a> <span class="p2">(<span class="default"><a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Maroon">t</font></span>)</span></span>)</span>;<br/>
<a NAME="E5:76"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">xa</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="osalg_4.html#K5">CompClass</a> <font color="Maroon">R1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K3">CComp</a> <span class="p2">(<span class="default"><a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Maroon">t</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="osafree.html#E8:76"><i><font color="Green">E54</font></i></a>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<b>then 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="E7:76"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a>  <a href="osalg_4.html#K3">CComp</a> <span class="p1">(<span class="default"><a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Maroon">t</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">xa</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s1</font> )
 <b>by </b><i><a class="ref" href="osalg_4.html#D11">OSALG_4:def 11</a></i>;<br/>
<a NAME="E8:76"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> 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> <font color="Maroon">s1</font> &amp; <font color="Maroon">xa</font> <a href="hidden.html#R2">in</a> 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> <font color="Maroon">s1</font> )
 
<b>by </b><i><a class="ref" href="osafree.html#T2" target="_self">Th2</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xa</font> = <font color="Maroon">t</font>, <font color="Maroon">xb</font> = <font color="Maroon">x</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> <font color="Maroon">s1</font> <b>by </b><i><a class="ref" href="osafree.html#T2" target="_self">Th2</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t1</font> = <font color="Maroon">x</font> as    <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> <b>by </b><i><a class="ref" href="osafree.html#T3" target="_self">Th3</a>, </i>;<br/>
<a NAME="E11:76"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">R1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R2</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s1</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
 <a href="osafree.html#K25" target="_self">OSClass</a> <font color="Maroon">R2</font>,<font color="Maroon">t1</font> = 
 <a href="osalg_4.html#K8">OSClass</a> <font color="Maroon">R2</font>,<font color="Maroon">xb</font>
<b>by </b><i/>
<br/>.= 
 <a href="osalg_4.html#K8">OSClass</a> <font color="Maroon">R2</font>,<font color="Maroon">xa</font>
<b>by </b><i><a class="ref" href="osafree.html#T2" target="_self">Th2</a>, , <a class="ref" href="osalg_4.html#T13">OSALG_4:13</a></i>
<br/>.= 
 <a href="osafree.html#K25" target="_self">OSClass</a> <font color="Maroon">R2</font>,<font color="Maroon">t</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E13:76"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="osafree.html#K25" target="_self">OSClass</a> <font color="Maroon">R2</font>,<font color="Maroon">t</font>
 <b>by </b><i/>;<br/>


</div>
