<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="msualg_1.html#V2">void</a> <a href="catalg_1.html#NM1">Signature</a>;<br/><b>let </b><font color="Maroon">Y</font> be  <a href="relat_1.html#V2">V3</a>  <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/>









<a NAME="E1:30"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msafree3.html#K5" target="_self">-Terms</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">Y</font> : (  <a href="msaterm.html#K7">the_sort_of</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  <a href="msafree3.html#K4" target="_self">variables_in</a> <font color="Olive">t</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">X</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:30_1"/>
 <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> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msafree3.html#K5" target="_self">-Terms</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/><b>then consider </b><font color="Maroon">t</font> being   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:30_1"/><i><font color="Green">E47</font></i>: 
(  <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> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  <a href="msafree3.html#K4" target="_self">variables_in</a> <font color="Maroon">t</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/><a NAME="E4:30_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/><a NAME="E5:30_1"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/><a NAME="E6:30_1"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <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> &amp; not <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/><a NAME="E7:30_1"/><b>then </b>
not <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon">s'</font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>, <font color="Maroon">v</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font><b> such that </b><br/><a NAME="E9:30_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">s'</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="msaterm.html#T2">MSATERM:2</a></i>;<br/><a NAME="E10:30_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s'</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E11:30_1"/>
(  <a href="msafree3.html#K4" target="_self">variables_in</a> <font color="Maroon">t</font> <a href="pboole.html#R6">=</a> <font color="Maroon">S</font> <a href="msafree3.html#K2" target="_self">variables_in</a> <font color="Maroon">t</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msafree3.html#K2" target="_self">variables_in</a> <font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="msafree3.html#K4" target="_self">variables_in</a> <font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 <b>by </b><i>, , <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><b>hence </b><a NAME="E12:30_1"/>
( <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">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:30"/><i><font color="Green">E51</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">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">t</font> =  <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> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="msaterm.html#T4">MSATERM:4</a></i>;<br/>
<a NAME="E5:30"/><i><font color="Green">E64</font></i>: 
 <a href="msafree3.html#K4" target="_self">variables_in</a> <font color="Maroon">t</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E1:30_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:30_2"/>
( ( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s</font> implies <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msafree3.html#K2" target="_self">variables_in</a> <font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msafree3.html#K2" target="_self">variables_in</a> <font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:30_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msafree3.html#K2" target="_self">variables_in</a> <font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>hence </b><a NAME="E4:30_2"/>
<span class="p1">(<span class="default"><a href="msafree3.html#K4" target="_self">variables_in</a> <font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:30"/>
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="msaterm.html#T14">MSATERM:14</a></i>;<br/>
<b>hence </b><a NAME="E7:30"/>
 <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> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msafree3.html#K5" target="_self">-Terms</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, </i>;<br/>


</div>
