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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> be  <a href="relat_1.html#V3">V4</a>  <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="msafree3.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" target="_self">Free</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>









<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></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/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:47_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="msafree3.html#K2" target="_self">variables_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:47_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> 
 <b>by </b><i><a class="ref" href="msafree3.html#D3" target="_self">Def3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E4:47_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><a NAME="E5:47_1"/>
( <font color="Maroon">c<sub>4</sub></font> is   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></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> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="msafree3.html#T9" target="_self">Th9</a></i>;<br/><b>hence </b><a NAME="E6:47_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="msafree3.html#E4:47_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="msafree3.html#T35" target="_self">Th35</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:47"/><i><font color="Green">E35</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:47"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></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">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="msafree3.html#E4:47"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="trees_2.html#K5">|</a> <font color="Maroon">c<sub>8</sub></font> as   <a href="msafree3.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" target="_self">Free</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="ref" href="msafree3.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E7:47"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E8:47"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E9:47"/><b>then </b>
not <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:47"/><b>then </b><i><font color="Green">E37</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <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">c<sub>2</sub></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">c<sub>2</sub></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/>
<a NAME="E11:47"/><i><font color="Green">E38</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>9</sub></font> is   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></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="txt" href="msafree3.html#E4:47"><i><font color="Green">E36</font></i></a>, <a class="ref" href="msafree3.html#T9" target="_self">Th9</a>, <a class="ref" href="qc_lang4.html#T8">QC_LANG4:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></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">c<sub>10</sub></font><b> such that </b><br/><a NAME="E13:47"/><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="txt" href="msafree3.html#E10:47"><i><font color="Green">E37</font></i></a>, <a class="ref" href="msaterm.html#T2">MSATERM:2</a></i>;<br/>
<a NAME="E14:47"/><i><font color="Green">E40</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E15:47"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <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">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="msafree3.html#E11:47"><i><font color="Green">E38</font></i></a>, <a class="txt" href="msafree3.html#E13:47"><i><font color="Green">E39</font></i></a>, <a class="ref" href="msaterm.html#T5">MSATERM:5</a></i>;<br/>
<a NAME="E16:47"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="msafree3.html#K2" target="_self">variables_in</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="msafree3.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E17:47"/>
<font color="Maroon">c<sub>2</sub></font> <a href="msafree3.html#K2" target="_self">variables_in</a> <font color="Maroon">c<sub>9</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="msafree3.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E18:47"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="msafree3.html#K2" target="_self">variables_in</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E19:47"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="msafree3.html#E13:47"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msafree3.html#E16:47"><i><font color="Green">E41</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E20:47"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="msafree3.html#E2:47"><i><font color="Green">E35</font></i></a>, <a class="txt" href="msafree3.html#E14:47"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>hence </b><a NAME="E21:47"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="msafree3.html#K2" target="_self">variables_in</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="msafree3.html#E19:47"><i><font color="Green">E42</font></i></a>, <a class="ref" href="msafree3.html#D3" target="_self">Def3</a></i>;<br/>


</div>
