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

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="msafree.html#K11" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> =  <a href="msafree.html#K12" target="_self">FreeGen</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:39_1_1"/><i><font color="Green">E23</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S4">FUNCT_1:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <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> <b>by </b><i><a class="txt" href="msafree.html#E2:39_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:39_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <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" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_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="pboole.html#D5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E1:39_1_1_1"/>
<font color="Maroon">c<sub>7</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>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:39_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="msafree.html#K12" target="_self">FreeGen</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="msafree.html#K12" target="_self">FreeGen</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="msafree.html#E2:39_1_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:39_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</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" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E6:39_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</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" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:39_1_1_2"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> is   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<b>hence </b><a NAME="E2:39_1_1_2"/><i><font color="Green">E24</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</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="msafree.html#K11" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D13">PBOOLE:def 13</a><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E3:39_1_1_2"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6">Terminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K2">root-tree</a> <font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:39_1_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6">Terminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:39_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K3" target="_self">coprod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msafree.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E3:39_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="msafree.html#K3" target="_self">coprod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:39_1_1_2_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msafree.html#K3" target="_self">coprod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:39_1_1_2_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msafree.html#K3" target="_self">coprod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="msafree.html#K3" target="_self">coprod</a> <font color="Maroon">c<sub>2</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> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="msafree.html#E5:39_1_1_2_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="msafree.html#E7:39_1_1_2_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:39_1_1_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="msafree.html#K2" target="_self">coprod</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E7:39_1_1_2_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="msafree.html#D3" target="_self">Def3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:39_1_1_2_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="msafree.html#E5:39_1_1_2_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="msafree.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E12:39_1_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D18">MSUALG_2:def 18</a></i>;<br/>
<a NAME="E13:39_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <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="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E14:39_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</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="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E15:39_1_1_2_1"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="msafree.html#K12" target="_self">FreeGen</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>2</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="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E2:39_1_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E16:39_1_1_2_1"/><i><font color="Green">E30</font></i>: 
 <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K12" target="_self">FreeGen</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E11:39_1_1_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="msafree.html#D17" target="_self">Def17</a></i>;<br/>
<a NAME="E17:39_1_1_2_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="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E18:39_1_1_2_1"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E19:39_1_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>8</sub></font>]
 <b>by </b><i><a class="txt" href="msafree.html#E15:39_1_1_2_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="msafree.html#E16:39_1_1_2_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:39_1_1_2"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="trees_2.html#NM6">DecoratedTree</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>3</sub></font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:39_1_1_2_2"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">c<sub>9</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="trees_2.html#NM6">DecoratedTree</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = <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>1</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>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p2">(<span class="default"><a href="msafree.html#K3" target="_self">coprod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> as   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> = the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E3:39_1_1_2_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>12</sub></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/>
<a NAME="E4:39_1_1_2_2"/><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><a href="dtconstr.html#K1">roots</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="lang1.html#U1">Rules</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E1:39_1_1_2_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="lang1.html#D1">LANG1:def 1</a></i>;<br/>
<a NAME="E5:39_1_1_2_2"/><i><font color="Green">E30</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><a href="dtconstr.html#K1">roots</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K4" target="_self">REL</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E1:39_1_1_2_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="lang1.html#D1">LANG1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="subset_1.html#M1">Element</a> of <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>1</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>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p2">(<span class="default"><a href="msafree.html#K3" target="_self">coprod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> =  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">c<sub>9</sub></font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><span class="p2"><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>1</sub></font>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p3">(<span class="default"><a href="msafree.html#K3" target="_self">coprod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K13">*</a>  <b>by </b><i><a class="txt" href="msafree.html#E4:39_1_1_2_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E8:39_1_1_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K4" target="_self">REL</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E1:39_1_1_2_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="lang1.html#D1">LANG1:def 1</a></i>;<br/>
<a NAME="E9:39_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <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>1</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>1</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="msafree.html#D9" target="_self">Def9</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>20</sub></font> being    <a href="subset_1.html#M1">Element</a> of <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>1</sub></font></span><a href="tarski.html#K1">}</a></span><b> such that </b><br/><a NAME="E11:39_1_1_2_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E12:39_1_1_2_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>21</sub></font> =  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>22</sub></font> =  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>19</sub></font>;<br/>
<a NAME="E13:39_1_1_2_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="msualg_2.html#R1">is_closed_on</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E3:39_1_1_2_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a></i>;<br/>
<a NAME="E14:39_1_1_2_2"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>19</sub></font>,<span class="p3">(<span class="default"><a href="msafree.html#K11" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D6">MSUALG_2:def 6</a></i>;<br/>
<a NAME="E15:39_1_1_2_2"/><i><font color="Green">E34</font></i>: 
(  <a href="msafree.html#K6" target="_self">Sym</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i><a class="txt" href="msafree.html#E11:39_1_1_2_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<i><font color="Green">E35</font></i>:  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>19</sub></font>,<span class="p1">(<span class="default"><a href="msafree.html#K11" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="msafree.html#K10" target="_self">FreeOper</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
<br/>.= 
 <a href="msafree.html#K9" target="_self">DenOp</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="msafree.html#D15" target="_self">Def15</a></i>
;<br/>
<a NAME="E17:39_1_1_2_2"/><i><font color="Green">E36</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font> &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>19</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a>, <a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E18:39_1_1_2_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msafree.html#K9" target="_self">DenOp</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E19:39_1_1_2_2"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msafree.html#K9" target="_self">DenOp</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E1:39_1_1_2_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="msafree.html#E15:39_1_1_2_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="msafree.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E20:39_1_1_2_2"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E17:39_1_1_2_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="msafree.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E21:39_1_1_2_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E22:39_1_1_2_2"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E17:39_1_1_2_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E23:39_1_1_2_2"/><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1">roots</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="trees_3.html#D18">TREES_3:def 18</a></i>;<br/>
<a NAME="E24:39_1_1_2_2"/><i><font color="Green">E41</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font> holds <br/>( ( <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <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>1</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>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> implies for <font color="Olive">b<sub>2</sub></font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K3" target="_self">coprod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> implies <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K2" target="_self">coprod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> ) ) ) )
 
<b>by </b><i><a class="txt" href="msafree.html#E5:39_1_1_2_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="msafree.html#E11:39_1_1_2_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="msafree.html#E12:39_1_1_2_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="msafree.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E25:39_1_1_2_2"/><i><font color="Green">E42</font></i>: 
(  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font> &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E26:39_1_1_2_2"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E4:39_1_1_2_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="msafree.html#E11:39_1_1_2_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="msafree.html#E12:39_1_1_2_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="msafree.html#E23:39_1_1_2_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="msafree.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E27:39_1_1_2_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></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">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:39_1_1_2_2_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>23</sub></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">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="msafree.html#E22:39_1_1_2_2"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E3:39_1_1_2_2_2"/><i><font color="Green">E45</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E4:39_1_1_2_2_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E22:39_1_1_2_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msafree.html#E23:39_1_1_2_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="msafree.html#E24:39_1_1_2_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="msafree.html#E25:39_1_1_2_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="msafree.html#E1:39_1_1_2_2_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> as    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="msafree.html#E3:39_1_1_2_2_2"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E6:39_1_1_2_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>25</sub></font>]
 
<b>by </b><i><a class="txt" href="msafree.html#E1:39_1_1_2_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="msafree.html#E4:39_1_1_2_2_2"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>26</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:39_1_1_2_2_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>27</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:39_1_1_2_2_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <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="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> )
 <b>by </b><i><a class="txt" href="msafree.html#E8:39_1_1_2_2_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>27</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="msafree.html#E10:39_1_1_2_2_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<i><font color="Green">E49</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> = 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msafree.html#E1:39_1_1_2_2_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msafree.html#E22:39_1_1_2_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msafree.html#E1:39_1_1_2_2_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>29</sub></font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>30</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : ( ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) or ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font> ) ) </span>}</span> ;<br/>
<i><font color="Green">E50</font></i>: <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : ( ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) or ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font> ) ) </span>}</span>  = 
 <a href="msafree.html#K7" target="_self">FreeSort</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="msafree.html#D13" target="_self">Def13</a></i>
;<br/>
<b>consider </b><font color="Maroon">c<sub>31</sub></font> being   <a href="trees_2.html#NM4">DecoratedTree</a><b> such that </b><br/><a NAME="E15:39_1_1_2_2_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="msafree.html#E22:39_1_1_2_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msafree.html#E23:39_1_1_2_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="msafree.html#E24:39_1_1_2_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="msafree.html#E25:39_1_1_2_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="msafree.html#E1:39_1_1_2_2_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="trees_3.html#D18">TREES_3:def 18</a></i>;<br/>
<a NAME="E16:39_1_1_2_2_2"/><i><font color="Green">E52</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</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>1</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>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p2">(<span class="default"><a href="msafree.html#K3" target="_self">coprod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E17:39_1_1_2_2_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E22:39_1_1_2_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msafree.html#E24:39_1_1_2_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="msafree.html#E25:39_1_1_2_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="msafree.html#E1:39_1_1_2_2_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_2_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_2_2_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_2_2_2_3_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <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>1</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>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> or <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K3" target="_self">coprod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="msafree.html#E16:39_1_1_2_2_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="msafree.html#E17:39_1_1_2_2_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_2_2_2_3_1_1"><b>suppose </b></a><a NAME="E1:39_1_1_2_2_2_3_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <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>1</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>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>32</sub></font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>33</sub></font> being    <a href="subset_1.html#M1">Element</a> of <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>1</sub></font></span><a href="tarski.html#K1">}</a></span><b> such that </b><br/><a NAME="E3:39_1_1_2_2_2_3_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>33</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/><a NAME="E4:39_1_1_2_2_2_3_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b> <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>32</sub></font> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="msafree.html#E4:39_1_1_2_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="msafree.html#E11:39_1_1_2_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="msafree.html#E12:39_1_1_2_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="msafree.html#E22:39_1_1_2_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msafree.html#E23:39_1_1_2_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="msafree.html#E26:39_1_1_2_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="msafree.html#E1:39_1_1_2_2_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="msafree.html#E1:39_1_1_2_2_2_3_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="msafree.html#E3:39_1_1_2_2_2_3_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="msafree.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="msafree.html#E22:39_1_1_2_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msafree.html#E1:39_1_1_2_2_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E6:39_1_1_2_2_2_3_1_1"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="msafree.html#E13:39_1_1_2_2_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="msafree.html#E15:39_1_1_2_2_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="msafree.html#E3:39_1_1_2_2_2_3_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="msafree.html#E4:39_1_1_2_2_2_3_1_1"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_2_2_2_3_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_2_2_2_3_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K3" target="_self">coprod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:39_1_1_2_2_2_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K2" target="_self">coprod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="msafree.html#E4:39_1_1_2_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="msafree.html#E11:39_1_1_2_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="msafree.html#E12:39_1_1_2_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="msafree.html#E22:39_1_1_2_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msafree.html#E23:39_1_1_2_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="msafree.html#E26:39_1_1_2_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="msafree.html#E1:39_1_1_2_2_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="msafree.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E3:39_1_1_2_2_2_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K2" target="_self">coprod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="msafree.html#E22:39_1_1_2_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msafree.html#E1:39_1_1_2_2_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>32</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:39_1_1_2_2_2_3_1_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="msafree.html#D2" target="_self">Def2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>33</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> as   <a href="dtconstr.html#NM2">Terminal</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="msafree.html#E1:39_1_1_2_2_2_3_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="msafree.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E7:39_1_1_2_2_2_3_1_2"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K8">root-tree</a> <font color="Maroon">c<sub>33</sub></font>
 <b>by </b><i><a class="txt" href="msafree.html#E15:39_1_1_2_2_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="dtconstr.html#T9">DTCONSTR:9</a></i>;<br/><b>hence </b><a NAME="E8:39_1_1_2_2_2_3_1_2"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="msafree.html#E13:39_1_1_2_2_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="msafree.html#E5:39_1_1_2_2_2_3_1_2"><i><font color="Green">E56</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_2_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1_1_2_2_2_4"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>28</sub></font>
 ;<br/><a NAME="E2:39_1_1_2_2_2_4"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="subset_1.html#R1">misses</a> <span class="p1">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="msafree.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E3:39_1_1_2_2_2_4"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</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" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="msafree.html#E2:39_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><b>hence </b><a NAME="E4:39_1_1_2_2_2_4"/>
contradiction
 <b>by </b><i><a class="txt" href="msafree.html#E8:39_1_1_2_2_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="msafree.html#E10:39_1_1_2_2_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="msafree.html#E18:39_1_1_2_2_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="msafree.html#E2:39_1_1_2_2_2_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E20:39_1_1_2_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="msafree.html#E8:39_1_1_2_2_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="msafree.html#E10:39_1_1_2_2_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="msafree.html#E12:39_1_1_2_2_2"><i><font color="Green">E49</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:39_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E20:39_1_1_2_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="msafree.html#E22:39_1_1_2_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msafree.html#E23:39_1_1_2_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="msafree.html#E24:39_1_1_2_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="msafree.html#E25:39_1_1_2_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E29:39_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="msafree.html#K9" target="_self">DenOp</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E19:39_1_1_2_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E30:39_1_1_2_2"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K9" target="_self">DenOp</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K9" target="_self">DenOp</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default"><a href="msafree.html#K9" target="_self">DenOp</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="msafree.html#E1:39_1_1_2_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="msafree.html#E15:39_1_1_2_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="msafree.html#D14" target="_self">Def14</a></i>
;<br/>
<a NAME="E32:39_1_1_2_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>19</sub></font>,<span class="p3">(<span class="default"><a href="msafree.html#K11" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E16:39_1_1_2_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="msafree.html#E30:39_1_1_2_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E33:39_1_1_2_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E34:39_1_1_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <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">c<sub>19</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <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">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="msafree.html#E17:39_1_1_2_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E35:39_1_1_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>9</sub></font>]
 <b>by </b><i><a class="txt" href="msafree.html#E14:39_1_1_2_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="msafree.html#E32:39_1_1_2_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:39_1_1_2"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="trees_2.html#NM6">DecoratedTree</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="dtconstr.html#S7">DTCONSTR:sch 7</a>(<a class="txt" href="msafree.html#E3:39_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="msafree.html#E4:39_1_1_2"><i><font color="Green">E26</font></i></a>);<br/></i>
<a NAME="E6:39_1_1_2"/><i><font color="Green">E28</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K11" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</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="E1:39_1_1_2_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K11" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1_2_3"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:39_1_1_2_3"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">c<sub>2</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> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="msafree.html#E3:39_1_1_2_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="msafree.html#E5:39_1_1_2_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<font color="Maroon">c<sub>9</sub></font> = 
 <a href="msafree.html#K7" target="_self">FreeSort</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="msafree.html#E5:39_1_1_2_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="msafree.html#D13" target="_self">Def13</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : ( ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> ) or ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> ) ) </span>}</span> 

;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" target="_self">DTConMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E9:39_1_1_2_3"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E10:39_1_1_2_3"/>
( ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> ) or ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> ) )
 <b>by </b><i><a class="txt" href="msafree.html#E3:39_1_1_2_3"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>thus </b><a NAME="E11:39_1_1_2_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="msafree.html#E5:39_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="msafree.html#E9:39_1_1_2_3"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>let </b><font color="Maroon">c<sub>8</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="E7:39_1_1_2"/>
<font color="Maroon">c<sub>8</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>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E9:39_1_1_2"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</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="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</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="E1:39_1_1_2_4"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>10</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" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<a NAME="E2:39_1_1_2_4"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <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="msafree.html#K11" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:39_1_1_2_4"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:39_1_1_2_4"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K11" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E1:39_1_1_2_4"><i><font color="Green">E29</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:39_1_1_2_4"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="msafree.html#E6:39_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:39_1_1_2_4"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <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="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="msafree.html#E6:39_1_1_2_4"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="msafree.html#E8:39_1_1_2_4"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1_1_2_4_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:39_1_1_2_4_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#R1">misses</a> <span class="p1">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="msafree.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E3:39_1_1_2_4_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K8" target="_self">FreeSort</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E4:39_1_1_2_4_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</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" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="msafree.html#E2:39_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><b>hence </b><a NAME="E5:39_1_1_2_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="msafree.html#E1:39_1_1_2_4"><i><font color="Green">E29</font></i></a>, <a class="txt" href="msafree.html#E6:39_1_1_2_4"><i><font color="Green">E30</font></i></a>, <a class="txt" href="msafree.html#E8:39_1_1_2_4"><i><font color="Green">E31</font></i></a>, <a class="txt" href="msafree.html#E3:39_1_1_2_4_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:39_1_1_2_4"/>
<font color="Maroon">c<sub>10</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="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="msafree.html#E6:39_1_1_2_4"><i><font color="Green">E30</font></i></a>, <a class="txt" href="msafree.html#E8:39_1_1_2_4"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:39_1_1_2"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</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="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="msafree.html#M1" target="_self">GeneratorSet</a> of  <a href="msafree.html#K11" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msafree.html#D4" target="_self">Def4</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E8:39_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="msafree.html#K12" target="_self">FreeGen</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="msafree.html#E2:39_1_1"><i><font color="Green">E23</font></i></a></i>;<br/>


</div>
