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

<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">o</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">Y</font>;<br/>









<b>set </b><font color="Maroon">s</font> =  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font>;<br/>
<a NAME="E1:31"/><i><font color="Green">E42</font></i>: 
 <a href="relat_1.html#K1">dom</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="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="E2:31"/><i><font color="Green">E47</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> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span> <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>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</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/>
<a NAME="E3:31"/><i><font color="Green">E48</font></i>: 
 <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="msafree.html#D11">MSAFREE:def 11</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_1"/>
<span class="p1">(<span class="default"><a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">Y</font></span>)</span> <a href="msaterm.html#K6">-tree</a> <font color="Maroon">p</font> <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> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 ;<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="E3:31_1"/><i><font color="Green">E49</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font> <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>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</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>, </i>;<br/><b>thus </b><a NAME="E4:31_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K3">Union</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:31_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i>, <a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> 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#T22">MSATERM:22</a></i>;<br/>
<b>set </b><font color="Maroon">sq</font> =  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">q</font>;<br/>
<a NAME="E6:31_1_1"/><i><font color="Green">E64</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> <span class="p1">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">q</font></span>)</span> <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>  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">q</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/>
<a NAME="E7:31_1_1"/>
 <a href="msafree3.html#K4" target="_self">variables_in</a> <font color="Maroon">q</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:31_1_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<b>let </b><font color="Maroon">a</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="E2:31_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree3.html#K4" target="_self">variables_in</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 ;<br/>

<a NAME="E3:31_1_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <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">z</font> &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">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 
<b>by </b><i>, , , , , <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:31_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:31_1_1"/><b>then </b>
<font color="Maroon">q</font> <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> <span class="p1">(<span class="default"><a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:31_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</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>
 <b>by </b><i>, , <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>


</div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E5:31"/><i><font color="Green">E76</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K3">Union</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>
 ;<br/>

<b>set </b><font color="Maroon">t</font> = <span class="p1">(<span class="default"><a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">Y</font></span>)</span> <a href="msaterm.html#K6">-tree</a> <font color="Maroon">p</font>;<br/>
<a NAME="E6:31"/><i><font color="Green">E82</font></i>: 
 <a href="msaterm.html#K7">the_sort_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">Y</font></span>)</span> <a href="msaterm.html#K6">-tree</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="msaterm.html#T20">MSATERM:20</a></i>;<br/>
<a NAME="E7:31"/>
 <a href="msafree3.html#K4" target="_self">variables_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">Y</font></span>)</span> <a href="msaterm.html#K6">-tree</a> <font color="Maroon">p</font></span>)</span> <a href="pboole.html#R2">c=</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:31_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> 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="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="E2:31_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree3.html#K4" target="_self">variables_in</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">Y</font></span>)</span> <a href="msaterm.html#K6">-tree</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 ;<br/>

<b>then consider </b><font color="Maroon">q</font> being   <a href="trees_2.html#NM4">DecoratedTree</a><b> such that </b><br/><a NAME="E4:31_2"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="msafree3.html#K2" target="_self">variables_in</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:31_2"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">q</font> <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">y</font> )
 <b>by </b><i>, , , <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>
<a NAME="E7:31_2"/>
<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">y</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">y</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>, </i>;<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="E9:31_2"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">q</font> <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">y</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="E10:31_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">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> &amp;  <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">q</font> )
 
<b>by </b><i>, , <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:31_2"/>
<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">z</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:31"/>
<span class="p1">(<span class="default"><a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">Y</font></span>)</span> <a href="msaterm.html#K6">-tree</a> <font color="Maroon">p</font> <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> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div>
