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

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<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>set </b><font color="Maroon">Y</font> = <font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/>
<b>set </b><font color="Maroon">T</font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" target="_self">Free</a> <font color="Maroon">S</font>,<font color="Maroon">X</font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>  st <font color="Maroon">a<sub>1</sub></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>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> holds <br/><font color="Maroon">a<sub>1</sub></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="msafree3.html#K1" target="_self">Free</a> <font color="Maroon">S</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font>;<br/>
<b>consider </b><font color="Maroon">A</font> being   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E2:35"/><i><font color="Green">E42</font></i>: 
(  <a href="msafree3.html#K1" target="_self">Free</a> <font color="Maroon">S</font>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K15">Reverse</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="equation.html#K1">""</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">TT</font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" target="_self">Free</a> <font color="Maroon">S</font>,<font color="Maroon">X</font></span>)</span> as   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <b>by </b><i>, <a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E4:35"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S</font> <a href="msafree3.html#K5" target="_self">-Terms</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s1</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">v</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>;<br/><b>thus </b><a NAME="E1:35_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K2">root-tree</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">s1</font></span><a href="domain_1.html#K1">]</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E1:35_1_1"/><i><font color="Green">E49</font></i>: 
 <a href="trees_4.html#K2">root-tree</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">s1</font></span><a href="domain_1.html#K1">]</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>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">t</font> =  <a href="trees_4.html#K2">root-tree</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">s1</font></span><a href="domain_1.html#K1">]</a></span> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="msaterm.html#T4">MSATERM:4</a></i>;<br/>
<a NAME="E3:35_1_1"/>
<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>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <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/>
<a NAME="E4:35_1_1"/><b>then </b>
(  <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="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> )
 
<b>by </b><i>, , <a class="ref" href="msaterm.html#T14">MSATERM:14</a></i>;<br/>
<a NAME="E5:35_1_1"/><b>then </b>
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">v</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> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:35_1_1"/>
 <a href="trees_4.html#K2">root-tree</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">s1</font></span><a href="domain_1.html#K1">]</a></span> <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="msafree3.html#K1" target="_self">Free</a> <font color="Maroon">S</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>now </b></a><div class="add"><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>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>;<br/><b>assume </b><a NAME="E1:35_2"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">t</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>  st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t</font>]
 ;<br/><b>thus </b><a NAME="E2:35_2"/>
<font color="Maroon">S<sub>1</sub></font>[<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>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E1:35_2_1"/><i><font color="Green">E64</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#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>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E2:35_2_1"/><i><font color="Green">E75</font></i>: 
 <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <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/>
<a NAME="E3:35_2_1"/>
 <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>,<span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <span class="p4">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></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="E4:35_2_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">s</font> <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>, , </i>;<br/>
<a NAME="E5:35_2_1"/><b>then </b><i><font color="Green">E82</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>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:35_2_1"/><i><font color="Green">E84</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> <font color="Maroon">TT</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_1"><b>proof </b></a><div class="add">

<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="E1:35_2_1_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">x</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">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:35_2_1_1"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <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>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> &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#K5" target="_self">-Terms</a> <font color="Maroon">X</font>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></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/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="msualg_1.html#NM1">SortSymbol</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="E5:35_2_1_1"/>
<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>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></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="Maroon">S</font> <a href="msafree3.html#K5" target="_self">-Terms</a> <font color="Maroon">X</font>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <b>by </b><i>, </i>;<br/>
<a NAME="E7:35_2_1_1"/>
(  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" target="_self">Free</a> <font color="Maroon">S</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <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="msafree3.html#K1" target="_self">Free</a> <font color="Maroon">S</font>,<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="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:35_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">TT</font>
 <b>by </b><i><a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:35_2_1"/>
<font color="Maroon">TT</font> is <a href="msualg_2.html#V3">opers_closed</a>
 
<b>by </b><i>, <a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<b>hence </b><a NAME="E8:35_2_1"/>
<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#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" target="_self">Free</a> <font color="Maroon">S</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>thus </b><a NAME="E7:35"/>
for <font color="Olive">t</font> being  <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t</font>]
 <b>from </b><i><a class="ref" href="msaterm.html#S1">MSATERM:sch 1</a>(, );</i><br/>


</div>
