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

<b>let </b><font color="Maroon">c<sub>1</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>2</sub></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">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</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>1</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/>
<a NAME="E1:43"/><i><font color="Green">E32</font></i>: 
 <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#T6">MSUALG_2:6</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></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">c<sub>1</sub></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="E3:43"/><i><font color="Green">E33</font></i>: 
(  <a href="msafree3.html#K1" target="_self">Free</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></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">c<sub>1</sub></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">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="msafree3.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E4:43"/><i><font color="Green">E34</font></i>: 
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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="msafree3.html#K5" target="_self">-Terms</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</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>1</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="ref" href="msafree3.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E5:43"/><i><font color="Green">E35</font></i>: 
 <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K10">FreeOper</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msafree.html#D16">MSAFREE:def 16</a></i>;<br/>
<a NAME="E6:43"/>
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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a>,<a class="ref" href="pboole.html#D13">PBOOLE:def 13</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:43_1_1"/><i><font color="Green">E36</font></i>: 
<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>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>thus </b><a NAME="E3:43_1_1"/>
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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></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="E1:43_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>7</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</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>1</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#E4:43"><i><font color="Green">E34</font></i></a>, <a class="txt" href="msafree3.html#E1:43_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="msafree3.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E3:43_1_1_1"/>
 <a href="msafree3.html#K4" target="_self">variables_in</a> <font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="msafree3.html#E4:43"><i><font color="Green">E34</font></i></a>, <a class="txt" href="msafree3.html#E1:43_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="msafree3.html#E1:43_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="msafree3.html#T18" target="_self">Th18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msafree3.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E5:43_1_1_1"/>
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="msafree3.html#E4:43"><i><font color="Green">E34</font></i></a>, <a class="txt" href="msafree3.html#E1:43_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="msafree3.html#E1:43_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="msafree3.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E6:43_1_1_1"/><b>then </b>
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="msafree3.html#T30" target="_self">Th30</a></i>;<br/>
<a NAME="E7:43_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K7">FreeSort</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="msaterm.html#D5">MSATERM:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:43_1_1_1"/>
<font color="Maroon">c<sub>7</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="msafree.html#K11">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#E5:43"><i><font color="Green">E35</font></i></a>, <a class="ref" href="msafree.html#D13">MSAFREE:def 13</a></i>;<br/>


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

<b>thus </b><a NAME="E2:43_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></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="msafree3.html#K1" target="_self">Free</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></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="E1:43_1_2"/>
<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>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>let </b><font color="Maroon">c<sub>7</sub></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="E3:43_1_2"/>
<font color="Maroon">c<sub>7</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="msafree.html#K11">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E4:43_1_2"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K7">FreeSort</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="msafree.html#K7">FreeSort</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="msaterm.html#K1">-Terms</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="msafree3.html#E5:43"><i><font color="Green">E35</font></i></a>, <a class="ref" href="msafree.html#D13">MSAFREE:def 13</a>, <a class="ref" href="msaterm.html#T12">MSATERM:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E6:43_1_2"/><i><font color="Green">E37</font></i>: 
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="msafree3.html#E4:43_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="msaterm.html#D5">MSATERM:def 5</a></i>;<br/>
<a NAME="E7:43_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>2</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>1</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>
 
<b>by </b><i><a class="ref" href="pboole.html#T16">PBOOLE:16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</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>1</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="ref" href="msaterm.html#T26">MSATERM:26</a></i>;<br/>
<a NAME="E9:43_1_2"/>
(  <a href="msafree3.html#K4" target="_self">variables_in</a> <font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="msafree3.html#K2" target="_self">variables_in</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="msafree3.html#K4" target="_self">variables_in</a> <font color="Maroon">c<sub>9</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="msafree3.html#K2" target="_self">variables_in</a> <font color="Maroon">c<sub>8</sub></font> )
 
;<br/>
<a NAME="E10:43_1_2"/><b>then </b>
(  <a href="msafree3.html#K4" 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>2</sub></font> &amp;  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="msafree3.html#E6:43_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="msafree3.html#T29" target="_self">Th29</a>, <a class="ref" href="msafree3.html#T30" target="_self">Th30</a></i>;<br/>
<a NAME="E11:43_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</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>1</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="msaterm.html#K7">the_sort_of</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="msafree3.html#K4" target="_self">variables_in</a> <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E12:43_1_2"/>
<font color="Maroon">c<sub>7</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <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#E4:43"><i><font color="Green">E34</font></i></a>, <a class="ref" href="msafree3.html#D6" target="_self">Def6</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:43"/>
 <a href="msafree3.html#K1" target="_self">Free</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="msafree3.html#E1:43"><i><font color="Green">E32</font></i></a>, <a class="txt" href="msafree3.html#E3:43"><i><font color="Green">E33</font></i></a>, <a class="ref" href="msafree3.html#T27" target="_self">Th27</a></i>;<br/>


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