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<b>set </b><font color="Maroon">FG</font> =  <a href="osafree.html#K22" target="_self">PTVars</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>consider </b><font color="Maroon">s</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:78_1_1"/><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> <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> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E3:78_1_1"/>
<span class="p1">(<span class="default"><a href="osafree.html#K22" target="_self">PTVars</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="osafree.html#K21" target="_self">PTVars</a> <font color="Maroon">s</font>,<font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>then </b><i><font color="Green">E37</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> = 
 <a href="osafree.html#K21" target="_self">PTVars</a> <font color="Maroon">s</font>,<font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Olive">tt</font></span>)</span> where <font color="Olive">tt</font> is   <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> : ( <font color="Olive">tt</font> <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> &amp; <font color="Olive">tt</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> ) </span>}</span> 
<b>by </b><i/>
;<br/>
<a NAME="E5:78_1_1"/>
<font color="Maroon">t</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E8:78_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E6:78_1_1"/><b>then </b>
 <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="osafree.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E7:78_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</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">t</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:78_1_1"/><i><font color="Green">E54</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E9:78_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E10:78_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E11:78_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</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">t</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="osafree.html#T3" target="_self">Th3</a>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">eu</font> = <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</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">t</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K3">Union</a> <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon">eu</font>
;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E13:78_1_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="mcart_1.html#K2">`2</a> </span>)</span>
 ;<br/>

<b>hence </b><a NAME="E14:78_1_1"/>
<font color="Maroon">eu</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <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">t</font></span>)</span>
 <b>by </b><i><a class="txt" href="osafree.html#E8:78_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>


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