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

<b>let </b><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a> <a href="catalg_1.html#NM1" title="CATALG_1:NM.1">Signature</a>;<br/><b>let </b><font color="Maroon" title="c3">X</font> be  <a href="relat_1.html#V3" title="RELAT_1:attr.3">V4</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c4">t</font> be   <a href="msafree3.html#NM1" title="MSAFREE3:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">X</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c5">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S</font>;<br/>









<b>set </b><font color="Maroon" title="c6">Y</font> = <font color="Maroon" title="c3">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:47_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="msafree3.html#K2" title="MSAFREE3:func.2">variables_in</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font>
 ;<br/><a NAME="E2:47_1"/><b>then </b>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">t</font> : <font color="Olive" title="b1">a</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="msafree3.html#D3" target="_self" title="MSAFREE3:def.3">Def3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">t</font><b> such that </b><br/><a NAME="E4:47_1"/><i><font color="Green" title="E34">A1</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c7">a</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> )
 ;<br/><a NAME="E5:47_1"/>
( <font color="Maroon" title="c4">t</font> is   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c2">S</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">t</font> )
 <b>by </b><i><a class="ref" href="msafree3.html#T9" target="_self" title="MSAFREE3:th.9">Th9</a></i>;<br/><b>hence </b><a NAME="E6:47_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">t</font>
 <b>by </b><i><a class="txt" href="msafree3.html#E4:47_1"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="msafree3.html#T35" target="_self" title="MSAFREE3:th.35">Th35</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:47"/><i><font color="Green" title="E34">A2</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">t</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:47"/><i><font color="Green" title="E35">A3</font></i>: 
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">t</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">z</font> = <font color="Maroon" title="c7">z</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">t</font> <b>by </b><i><a class="txt" href="msafree3.html#E4:47"><i><font color="Green" title="E35">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c4">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c8">z</font> as   <a href="msafree3.html#NM1" title="MSAFREE3:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">X</font></span>)</span> <b>by </b><i><a class="ref" href="msafree3.html#T34" target="_self" title="MSAFREE3:th.34">Th34</a></i>;<br/>
<a NAME="E7:47"/>
<font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 
;<br/>
<a NAME="E8:47"/><b>then </b>
<font color="Maroon" title="c5">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 
;<br/>
<a NAME="E9:47"/><b>then </b>
not <font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:47"/><b>then </b><i><font color="Green" title="E36">A4</font></i>: 
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E11:47"/><i><font color="Green" title="E37">A5</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c9">q</font> is   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c2">S</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="msafree3.html#E4:47"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="msafree3.html#T9" target="_self" title="MSAFREE3:th.9">Th9</a>, <a class="ref" href="qc_lang4.html#T8" title="QC_LANG4:th.8">QC_LANG4:8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">s'</font> being   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S</font>, <font color="Maroon" title="c11">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s'</font><b> such that </b><br/><a NAME="E13:47"/><i><font color="Green" title="E38">A6</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">v</font>,<font color="Maroon" title="c10">s'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="msafree3.html#E10:47"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="msaterm.html#T2" title="MSATERM:th.2">MSATERM:2</a></i>;<br/>
<a NAME="E14:47"/><i><font color="Green" title="E39">A7</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E15:47"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">v</font>,<font color="Maroon" title="c10">s'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="msafree3.html#E11:47"><i><font color="Green" title="E37">A5</font></i></a>, <a class="txt" href="msafree3.html#E13:47"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="msaterm.html#T5" title="MSATERM:th.5">MSATERM:5</a></i>;<br/>
<a NAME="E16:47"/><b>then </b><i><font color="Green" title="E40">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="msafree3.html#K2" title="MSAFREE3:func.2">variables_in</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="msafree3.html#T11" target="_self" title="MSAFREE3:th.11">Th11</a></i>;<br/>
<a NAME="E17:47"/>
<font color="Maroon" title="c2">S</font> <a href="msafree3.html#K2" title="MSAFREE3:func.2">variables_in</a> <font color="Maroon" title="c9">q</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="ref" href="msafree3.html#T28" target="_self" title="MSAFREE3:th.28">Th28</a></i>;<br/>
<a NAME="E18:47"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="msafree3.html#K2" title="MSAFREE3:func.2">variables_in</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s'</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E19:47"/><b>then </b><i><font color="Green" title="E41">A9</font></i>: 
( <font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s'</font> &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">v</font> &amp; <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s'</font> )
 
<b>by </b><i><a class="txt" href="msafree3.html#E13:47"><i><font color="Green" title="E38">A6</font></i></a>, <a class="txt" href="msafree3.html#E16:47"><i><font color="Green" title="E40">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E20:47"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">t</font> : <font color="Olive" title="b1">a</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="msafree3.html#E2:47"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="msafree3.html#E14:47"><i><font color="Green" title="E39">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E21:47"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="msafree3.html#K2" title="MSAFREE3:func.2">variables_in</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> )
 <b>by </b><i><a class="txt" href="msafree3.html#E19:47"><i><font color="Green" title="E41">A9</font></i></a>, <a class="ref" href="msafree3.html#D3" target="_self" title="MSAFREE3:def.3">Def3</a></i>;<br/>


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