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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="relat_1.html#V2">V3</a>  <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="msafree.html#K5" target="_self">DTConMSA</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#K11" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="msafree.html#K13" target="_self">FreeGen</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>let </b><font color="Maroon">c<sub>6</sub></font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="msafree.html#D5" target="_self">MSAFREE:def 5</a><br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="pboole.html#M3">ManySortedFunction</a> of  <a href="msafree.html#K13" target="_self">FreeGen</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>8</sub></font> = 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>;<br/>
<b>set </b><font color="Maroon">c<sub>9</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>10</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>12</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>13</sub></font> =  <a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <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>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> =  <a href="msafree.html#K16" target="_self">pi</a> <font color="Maroon">c<sub>7</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <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>,  <a href="finseq_1.html#NM1">FinSequence</a>,  <a href="finseq_1.html#NM1">FinSequence</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> =  <a href="msafree.html#K18" target="_self">pi</a> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">a<sub>3</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="funct_2.html#NM1">Function</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>, <a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E2:47"/><i><font color="Green">E30</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">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>and </b><br/><a NAME="E3:47"/><i><font color="Green">E31</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> holds <br/><font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>, <a href="dtconstr.html#K1">roots</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</a> <font color="Olive">b<sub>2</sub></font>)
 <b>from </b><i><a class="ref" href="dtconstr.html#S8">DTCONSTR:sch 8</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="relset_1.html#M2">Relation</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>, <a href="card_3.html#K3">Union</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default">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> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:47"/><i><font color="Green">E32</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>15</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="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</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> holds <br/><font color="Maroon">c<sub>15</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>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</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#E5:47"><i><font color="Green">E32</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E7:47"/>
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> <font color="Maroon">c<sub>16</sub></font> holds <br/><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:47_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<a NAME="E2:47_1"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default">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> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E5:47"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:47_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="pboole.html#NM2">ManySortedFunction</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="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means 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> st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="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="Olive">b<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>;<br/>
<a NAME="E9:47"/><i><font color="Green">E33</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="47_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</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="E1:47_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>18</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/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>20</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:47_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>20</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>19</sub></font> &amp; <font color="Maroon">c<sub>18</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>20</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="msafree.html#T7" target="_self">Th7</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <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> : ( <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> &amp; <font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>22</sub></font> =  <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>18</sub></font>;<br/>
<a NAME="E4:47_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E3:47_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E5:47_2"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <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> : ( <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> &amp; <font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="msafree.html#E1:47_2"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E6:47_2"/><i><font color="Green">E38</font></i>: 
( <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <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> : ( <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> &amp; <font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="msafree.html#K12" target="_self">FreeGen</a> <font color="Maroon">c<sub>19</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>19</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>19</sub></font> )
 
<b>by </b><i><a class="ref" href="msafree.html#T14" target="_self">Th14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="msafree.html#K13" target="_self">FreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> ;<br/>
<a NAME="E8:47_2"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><a href="msafree.html#K13" target="_self">FreeGen</a> <font color="Maroon">c<sub>2</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> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <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> : ( <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> &amp; <font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="msafree.html#E6:47_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="msafree.html#D18" target="_self">Def18</a></i>;<br/>
<a NAME="E9:47_2"/><i><font color="Green">E40</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K13" target="_self">FreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <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>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E10:47_2"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E5:47_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="msafree.html#E8:47_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E11:47_2"/>
<font color="Maroon">c<sub>17</sub></font> <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>14</sub></font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default">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> <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#E5:47"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>then </b><i><font color="Green">E42</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> = 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msafree.html#E5:47_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="msafree.html#E6:47_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="msafree.html#K16" target="_self">pi</a> <font color="Maroon">c<sub>7</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="msafree.html#E2:47"><i><font color="Green">E30</font></i></a>, <a class="txt" href="msafree.html#E1:47_2"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msafree.html#E1:47_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="msafree.html#E4:47_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="msafree.html#D21" target="_self">Def21</a></i>
;<br/>
<b>let </b><font color="Maroon">c<sub>24</sub></font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E13:47_2"/><i><font color="Green">E43</font></i>: 
 <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>18</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>24</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_2_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>24</sub></font>
 ;<br/><a NAME="E2:47_2_2"/>
 <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</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>19</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>24</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="msafree.html#E5:47_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="msafree.html#E6:47_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="msafree.html#E13:47_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:47_2_2"/><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>19</sub></font> <a href="subset_1.html#R2">meets</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>24</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/><b>hence </b><a NAME="E4:47_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="msafree.html#E1:47_2_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="msafree.html#T13" target="_self">Th13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:47_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <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#E9:47_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="msafree.html#E10:47_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="msafree.html#E12:47_2"><i><font color="Green">E42</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:47"/><i><font color="Green">E34</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="47_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</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>19</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="E1:47_3"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">c<sub>19</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>19</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>20</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>19</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>21</sub></font> =  <a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>22</sub></font> =  <a href="msualg_1.html#K1">the_arity_of</a> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>23</sub></font> =  <a href="msualg_1.html#K2">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>24</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>set </b><font color="Maroon">c<sub>25</sub></font> =  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">c<sub>19</sub></font>;<br/>
<a NAME="E2:47_3"/><i><font color="Green">E36</font></i>: 
(  <a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</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="msualg_1.html#K1">the_arity_of</a> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 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> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>;<br/>
<i><font color="Green">E37</font></i>:  <a href="msualg_1.html#K3">Args</a> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</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> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D9">MSUALG_1:def 9</a></i>
<br/>.= 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</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> <span class="p3">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="msafree.html#E2:47_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="msafree.html#T1" target="_self">Th1</a></i>
;<br/>
<a NAME="E4:47_3"/><i><font color="Green">E38</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</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> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="alg_1.html#T1">ALG_1:1</a></i>;<br/>
<a NAME="E5:47_3"/><i><font color="Green">E39</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>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="ref" href="trees_3.html#D18">TREES_3:def 18</a></i>;<br/>
<a NAME="E6:47_3"/><i><font color="Green">E40</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,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>
 
<b>by </b><i><a class="txt" href="msafree.html#E1:47_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="msafree.html#D22" target="_self">Def22</a></i>;<br/>
<a NAME="E7:47_3"/><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,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>  <a href="msafree.html#K6" target="_self">Sym</a> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E8:47_3"/><i><font color="Green">E42</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></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> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</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="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> <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="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:47_3"/><b>then </b><i><font color="Green">E43</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</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> <span class="p3">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></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> <span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">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> <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> <span class="p3">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></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> <span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E10:47_3"/><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,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>,<span class="p2">(<span class="default"><a href="dtconstr.html#K1">roots</a> <font color="Maroon">c<sub>19</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#E1:47_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="msafree.html#E6:47_3"><i><font color="Green">E40</font></i></a>, <a class="ref" href="lang1.html#D1">LANG1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>26</sub></font> =  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">c<sub>19</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#E10:47_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E12:47_3"/><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>26</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> <span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E10:47_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="msafree.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E13:47_3"/><i><font color="Green">E46</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <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>26</sub></font></span>)</span> &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> <span class="p3">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></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> <span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E14:47_3"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</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> <span class="p3">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E4:47_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="msafree.html#E5:47_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msafree.html#E9:47_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="msafree.html#E10:47_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="msafree.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E15:47_3"/>
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">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</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> <span class="p3">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</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> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</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> <span class="p3">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></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="47_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>27</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:47_3_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> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</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> <span class="p3">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>27</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="msafree.html#E9:47_3"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E3:47_3_2"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E14:47_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="msafree.html#E1:47_3_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="alg_1.html#T1">ALG_1:1</a></i>;<br/>
<a NAME="E4:47_3_2"/><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>19</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="E5:47_3_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E5:47_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msafree.html#E9:47_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="msafree.html#E12:47_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="msafree.html#E13:47_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="msafree.html#E1:47_3_2"><i><font color="Green">E48</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>29</sub></font> = <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</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#E4:47_3_2"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E7:47_3_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</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> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E1:47_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="msafree.html#E6:47_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="msafree.html#E7:47_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="msafree.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E8:47_3_2"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</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#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E2:47_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="msafree.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E9:47_3_2"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</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#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E9:47_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="msafree.html#E1:47_3_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E10:47_3_2"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</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> <span class="p3">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E9:47_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="msafree.html#E1:47_3_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E11:47_3_2"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</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="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E9:47_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="msafree.html#E1:47_3_2"><i><font color="Green">E48</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>30</sub></font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <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#E8:47_3"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E13:47_3_2"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E1:47_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="msafree.html#E5:47_3_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="msafree.html#E10:47_3_2"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E14:47_3_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default">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> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E5:47"><i><font color="Green">E32</font></i></a></i>;<br/>
<i><font color="Green">E55</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> = 
 <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="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <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="ref" href="msafree.html#T12" target="_self">Th12</a></i>
;<br/>
<a NAME="E16:47_3_2"/>
 <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> <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="E17:47_3_2"/><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>30</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="E18:47_3_2"/><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>30</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E15:47_3_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E19:47_3_2"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font></span>)</span> <a href="hidden.html#R1">=</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>30</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E14:47_3_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E20:47_3_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E10:47_3_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="msafree.html#E13:47_3_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="msafree.html#E14:47_3_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E21:47_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</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> <span class="p3">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="msafree.html#E1:47_3_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="msafree.html#E3:47_3_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:47_3"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E3:47_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="msafree.html#E4:47_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="msafree.html#E5:47_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msafree.html#E9:47_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="msafree.html#E12:47_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="msafree.html#E13:47_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E17:47_3"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="msafree.html#K18" target="_self">pi</a> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msafree.html#D23" target="_self">Def23</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>27</sub></font> =  <a href="msafree.html#K18" target="_self">pi</a> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>;<br/>
<a NAME="E18:47_3"/><i><font color="Green">E50</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K4">Result</a> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E19:47_3"/><b>then </b>
 <a href="msafree.html#K18" target="_self">pi</a> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</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> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E16:47_3"><i><font color="Green">E48</font></i></a>, <a class="txt" href="msafree.html#E17:47_3"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E20:47_3"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="msafree.html#K18" target="_self">pi</a> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K4">Result</a> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E18:47_3"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E21:47_3"/>
(  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</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="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;  <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> <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>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E22:47_3"/><b>then </b><i><font color="Green">E52</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</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>  <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> &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#K1">dom</a> <span class="p1">(<span class="default">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> <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>  <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> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<i><font color="Green">E53</font></i>:  <a href="msualg_1.html#K4">Result</a> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</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> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D10">MSUALG_1:def 10</a></i>
<br/>.= 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">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> <span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="msafree.html#E22:47_3"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</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> <span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>
;<br/>
<a NAME="E24:47_3"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msafree.html#K18" target="_self">pi</a> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E3:47"><i><font color="Green">E31</font></i></a>, <a class="txt" href="msafree.html#E1:47_3"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E25:47_3"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><a href="msafree.html#K9" target="_self">DenOp</a> <span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>2</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>18</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E1:47_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="msafree.html#E6:47_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="msafree.html#E7:47_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="msafree.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E26:47_3"/><i><font color="Green">E56</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msafree.html#K9" target="_self">DenOp</a> <span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<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> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msafree.html#K9" target="_self">DenOp</a> <span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</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="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> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E27:47_3"/><b>then </b>
<font color="Maroon">c<sub>19</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> <span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E1:47_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="msafree.html#E6:47_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="msafree.html#E7:47_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="msafree.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E28:47_3"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>19</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#K9" target="_self">DenOp</a> <span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E25:47_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E29:47_3"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">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> <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> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E26:47_3"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E30:47_3"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>19</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> <span class="p1">(<span class="default">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> <span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E22:47_3"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E31:47_3"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>19</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> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>;<br/>
<a NAME="E32:47_3"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="msafree.html#K18" target="_self">pi</a> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <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> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <span class="p4">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E24:47_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>28</sub></font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E33:47_3"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>19</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>28</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_3_4"/><i><font color="Green">E60</font></i>: 
 <a href="msualg_1.html#K2">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>28</sub></font>
 ;<br/><a NAME="E2:47_3_4"/>
<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"><a href="msualg_1.html#K2">the_result_sort_of</a> <span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="subset_1.html#R2">meets</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>28</sub></font>
 <b>by </b><i><a class="txt" href="msafree.html#E31:47_3"><i><font color="Green">E57</font></i></a>, <a class="txt" href="msafree.html#E33:47_3"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E3:47_3_4"/>
contradiction
 <b>by </b><i><a class="txt" href="msafree.html#E1:47_3_4"><i><font color="Green">E60</font></i></a>, <a class="ref" href="msafree.html#T13" target="_self">Th13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E35:47_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <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#E5:47"><i><font color="Green">E32</font></i></a>, <a class="txt" href="msafree.html#E20:47_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="msafree.html#E23:47_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="msafree.html#E32:47_3"><i><font color="Green">E58</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:47"/><i><font color="Green">E35</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#E9:47"><i><font color="Green">E33</font></i></a>, <a class="txt" href="msafree.html#E10:47"><i><font color="Green">E34</font></i></a>);<br/></i>
<a NAME="E12:47"/>
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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is   <a href="funct_2.html#NM1">Function</a> of 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="Olive">b<sub>1</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <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="47_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:47_4"/>
<font color="Maroon">c<sub>18</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>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:47_4"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <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>14</sub></font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default">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> <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#E5:47"><i><font color="Green">E32</font></i></a></i>;<br/>
<i><font color="Green">E37</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> = 
 <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="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <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="ref" href="msafree.html#T12" target="_self">Th12</a></i>
;<br/>
<a NAME="E5:47_4"/>
 <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> <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="E6:47_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>19</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="E7:47_4"/><b>then </b><i><font color="Green">E38</font></i>: 
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>19</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E4:47_4"><i><font color="Green">E37</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E8:47_4"/><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>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</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>19</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E3:47_4"><i><font color="Green">E36</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E9:47_4"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>20</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="E1:47_4_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:47_4_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>21</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>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</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#E7:47_4"><i><font color="Green">E38</font></i></a>, <a class="txt" href="msafree.html#E8:47_4"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msafree.html#E3:47_4_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:47_4_2"/>
<font color="Maroon">c<sub>22</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>19</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E3:47_4"><i><font color="Green">E36</font></i></a>, <a class="txt" href="msafree.html#E7:47_4"><i><font color="Green">E38</font></i></a>, <a class="txt" href="msafree.html#E3:47_4_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>hence </b><a NAME="E6:47_4_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <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#E11:47"><i><font color="Green">E35</font></i></a>, <a class="txt" href="msafree.html#E3:47_4_2"><i><font color="Green">E40</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:47_4"/>
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> is   <a href="funct_2.html#NM1">Function</a> of 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>18</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="msafree.html#E8:47_4"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as   <a href="msualg_3.html#NM1">ManySortedFunction</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>,<font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>18</sub></font>
;<br/>

<b>thus </b><a NAME="E14:47"/>
<font color="Maroon">c<sub>18</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msafree.html#K11" target="_self">FreeMSA</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a><br/>

<b>assume </b><a NAME="E1:47_5"/>
 <a href="msualg_1.html#K3">Args</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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>20</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</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>;<br/>

<b>set </b><font color="Maroon">c<sub>21</sub></font> =  <a href="msualg_1.html#K2">the_result_sort_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#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>;<br/>
<b>set </b><font color="Maroon">c<sub>23</sub></font> =  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>24</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>set </b><font color="Maroon">c<sub>25</sub></font> =  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font>;<br/>
<i><font color="Green">E36</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="E3:47_5"/><i><font color="Green">E37</font></i>: 
 <a href="msualg_1.html#K3">Args</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> <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="msualg_1.html#D9">MSUALG_1:def 9</a></i>;<br/>
<a NAME="E4:47_5"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</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>
 
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>20</sub></font> as    <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> <b>by </b><i><a class="txt" href="msafree.html#E3:47_5"><i><font color="Green">E37</font></i></a>, <a class="ref" href="msafree.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E6:47_5"/><i><font color="Green">E39</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="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">c<sub>26</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E3:47_5"><i><font color="Green">E37</font></i></a>, <a class="ref" href="msafree.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E7:47_5"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>19</sub></font>,<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="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>26</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E2:47_5"><i><font color="Green">E36</font></i></a>, <a class="ref" href="msafree.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E8:47_5"/><i><font color="Green">E41</font></i>: 
( <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="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> 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> &amp;  <a href="relset_1.html#K4">dom</a> the <a href="msualg_1.html#U2">Arity</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> )
 
<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></i>;<br/>
<a NAME="E9:47_5"/>
(  <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; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><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></span>)</span></span>)</span>,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>  <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> )
 
<b>by </b><i><a class="txt" href="msafree.html#E6:47_5"><i><font color="Green">E39</font></i></a>, <a class="ref" href="msafree.html#D22" target="_self">Def22</a></i>;<br/>
<a NAME="E10:47_5"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><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></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E11:47_5"/>
(  <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;  <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> <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>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E12:47_5"/><b>then </b><i><font color="Green">E43</font></i>: 
(  <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#K1">dom</a> <span class="p1">(<span class="default">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> <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>  <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> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E13:47_5"/><i><font color="Green">E44</font></i>: 
<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>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>26</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E6:47_5"><i><font color="Green">E39</font></i></a>, <a class="ref" href="msafree.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E14:47_5"/><i><font color="Green">E45</font></i>: 
(  <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> &amp;  <a href="relat_1.html#K2">rng</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="tarski.html#R1">c=</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="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="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E15:47_5"/><b>then </b>
<span class="p1">(<span class="default"><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></span>)</span> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>26</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#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#E3:47_5"><i><font color="Green">E37</font></i></a>, <a class="txt" href="msafree.html#E13:47_5"><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="E16:47_5"/><b>then </b>
<span class="p1">(<span class="default"><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></span>)</span> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">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> <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="txt" href="msafree.html#E14:47_5"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E17:47_5"/><b>then </b>
<span class="p1">(<span class="default"><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></span>)</span> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>26</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> <span class="p1">(<span class="default">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></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E12:47_5"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E18:47_5"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><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></span>)</span> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>26</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> <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>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>;<br/>
<a NAME="E19:47_5"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="msualg_3.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#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default">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> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E5:47"><i><font color="Green">E32</font></i></a></i>;<br/>
<i><font color="Green">E48</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> = 
 <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="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <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="ref" href="msafree.html#T12" target="_self">Th12</a></i>
;<br/>
<a NAME="E21:47_5"/><i><font color="Green">E49</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>26</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>26</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="alg_1.html#T1">ALG_1:1</a></i>;<br/>
<a NAME="E22:47_5"/><i><font color="Green">E50</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>20</sub></font></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> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>20</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="ref" href="msualg_3.html#T6">MSUALG_3:6</a></i>;<br/>
<a NAME="E23:47_5"/>
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>26</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>20</sub></font></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="47_5_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>27</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:47_5_3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>26</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>27</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E3:47_5_3"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E21:47_5"><i><font color="Green">E49</font></i></a>, <a class="txt" href="msafree.html#E1:47_5_3"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E4:47_5_3"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="msualg_3.html#K1">.</a> <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>28</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E1:47_5_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="msualg_3.html#D8">MSUALG_3:def 8</a></i>;<br/>
<a NAME="E5:47_5_3"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="msualg_3.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>28</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default">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> <a href="funct_1.html#K1">.</a> <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>28</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E5:47"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E6:47_5_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</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#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#E4:47_5"><i><font color="Green">E38</font></i></a>, <a class="txt" href="msafree.html#E8:47_5"><i><font color="Green">E41</font></i></a>, <a class="ref" href="msafree.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E7:47_5_3"/>
(  <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> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</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="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> <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="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E8:47_5_3"/><b>then </b><i><font color="Green">E56</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</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> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">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> <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="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>29</sub></font> =  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">c<sub>26</sub></font>;<br/>
<a NAME="E9:47_5_3"/><i><font color="Green">E57</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>26</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>26</sub></font>
 
<b>by </b><i><a class="ref" href="trees_3.html#D18">TREES_3:def 18</a></i>;<br/>
<a NAME="E10:47_5_3"/><i><font color="Green">E58</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><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>,<span class="p2">(<span class="default"><a href="dtconstr.html#K1">roots</a> <font color="Maroon">c<sub>26</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#E6:47_5"><i><font color="Green">E39</font></i></a>, <a class="ref" href="lang1.html#D1">LANG1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>30</sub></font> =  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">c<sub>26</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#E10:47_5_3"><i><font color="Green">E58</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E12:47_5_3"/><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>30</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>
 
<b>by </b><i><a class="txt" href="msafree.html#E10:47_5_3"><i><font color="Green">E58</font></i></a>, <a class="ref" href="msafree.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E13:47_5_3"/><i><font color="Green">E60</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>30</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>30</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="E14:47_5_3"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</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#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>28</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E1:47_5_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="msafree.html#E6:47_5_3"><i><font color="Green">E55</font></i></a>, <a class="txt" href="msafree.html#E8:47_5_3"><i><font color="Green">E56</font></i></a>, <a class="txt" href="msafree.html#E9:47_5_3"><i><font color="Green">E57</font></i></a>, <a class="txt" href="msafree.html#E12:47_5_3"><i><font color="Green">E59</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<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#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>28</sub></font> = 
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> <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>28</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msafree.html#E1:47_5_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="msafree.html#E8:47_5_3"><i><font color="Green">E56</font></i></a>, <a class="txt" href="msafree.html#E9:47_5_3"><i><font color="Green">E57</font></i></a>, <a class="txt" href="msafree.html#E12:47_5_3"><i><font color="Green">E59</font></i></a>, <a class="txt" href="msafree.html#E13:47_5_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
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> <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>28</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msafree.html#E1:47_5_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="msafree.html#E9:47_5_3"><i><font color="Green">E57</font></i></a>, <a class="txt" href="msafree.html#E12:47_5_3"><i><font color="Green">E59</font></i></a>, <a class="txt" href="msafree.html#E13:47_5_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E16:47_5_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>26</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="msafree.html#E3:47_5_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="msafree.html#E4:47_5_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="msafree.html#E5:47_5_3"><i><font color="Green">E54</font></i></a>, <a class="txt" href="msafree.html#E14:47_5_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E24:47_5"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E21:47_5"><i><font color="Green">E49</font></i></a>, <a class="txt" href="msafree.html#E22:47_5"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E25:47_5"/>
 <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> <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="E26:47_5"/><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> <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> 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="E27:47_5"/><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> <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="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E20:47_5"><i><font color="Green">E48</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E28:47_5"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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> <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>
 
<b>by </b><i><a class="txt" href="msafree.html#E19:47_5"><i><font color="Green">E47</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></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#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="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> = 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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></span>)</span> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msafree.html#E7:47_5"><i><font color="Green">E40</font></i></a>, <a class="txt" href="msafree.html#E18:47_5"><i><font color="Green">E46</font></i></a>, <a class="txt" href="msafree.html#E19:47_5"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
 <a href="msafree.html#K18" target="_self">pi</a> <span class="p1">(<span class="default"><a href="msafree.html#K17" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><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></span>)</span></span>)</span>,<font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msafree.html#E3:47"><i><font color="Green">E31</font></i></a>, <a class="txt" href="msafree.html#E6:47_5"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msafree.html#E10:47_5"><i><font color="Green">E42</font></i></a>, <a class="txt" href="msafree.html#E24:47_5"><i><font color="Green">E51</font></i></a>, <a class="ref" href="msafree.html#D23" target="_self">Def23</a></i>
;<br/><br/>


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

<a NAME="E15:47"/>
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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="msafree.html#K1" target="_self">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K13" target="_self">FreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <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="47_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:47_6"/>
<font color="Maroon">c<sub>19</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>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>set </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="msafree.html#K1" target="_self">||</a> <span class="p1">(<span class="default"><a href="msafree.html#K13" target="_self">FreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<a NAME="E3:47_6"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="msafree.html#K1" target="_self">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K13" target="_self">FreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K13" target="_self">FreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msafree.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E4:47_6"/><i><font color="Green">E37</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</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>20</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:47_6"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="msafree.html#K13" target="_self">FreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="msafree.html#K12" target="_self">FreeGen</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="msafree.html#D18" target="_self">Def18</a></i>;<br/>
<a NAME="E6:47_6"/><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="msafree.html#K1" target="_self">||</a> <span class="p3">(<span class="default"><a href="msafree.html#K13" target="_self">FreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K13" target="_self">FreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:47_6"/><i><font color="Green">E40</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K13" target="_self">FreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:47_6"/>
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> <span class="p1">(<span class="default"><a href="msafree.html#K13" target="_self">FreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="msafree.html#K1" target="_self">||</a> <span class="p3">(<span class="default"><a href="msafree.html#K13" target="_self">FreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></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="47_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>22</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:47_6_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K13" target="_self">FreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/>

<a NAME="E2:47_6_1"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <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> : ( <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> &amp; <font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="msafree.html#E5:47_6"><i><font color="Green">E38</font></i></a>, <a class="ref" href="msafree.html#T14" target="_self">Th14</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>23</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><b> such that </b><br/><a NAME="E4:47_6_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>23</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> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> )
 ;<br/>
<a NAME="E5:47_6_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default">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> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msafree.html#E5:47"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="msafree.html#K1" target="_self">||</a> <span class="p3">(<span class="default"><a href="msafree.html#K13" target="_self">FreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="msafree.html#E3:47_6"><i><font color="Green">E36</font></i></a>, <a class="txt" href="msafree.html#E6:47_6"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msafree.html#E1:47_6_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="msafree.html#E4:47_6"><i><font color="Green">E37</font></i></a>, <a class="txt" href="msafree.html#E5:47_6"><i><font color="Green">E38</font></i></a>, <a class="txt" href="msafree.html#E1:47_6_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="msafree.html#E5:47_6_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
 <a href="msafree.html#K16" target="_self">pi</a> <font color="Maroon">c<sub>7</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="txt" href="msafree.html#E2:47"><i><font color="Green">E30</font></i></a>, <a class="txt" href="msafree.html#E4:47_6_1"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="msafree.html#E4:47_6_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="msafree.html#D21" target="_self">Def21</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:47_6"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="msafree.html#K1" target="_self">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K13" target="_self">FreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></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>7</sub></font> <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#E6:47_6"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msafree.html#E7:47_6"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:47"/>
<font color="Maroon">c<sub>18</sub></font> <a href="msafree.html#K1" target="_self">||</a> <span class="p1">(<span class="default"><a href="msafree.html#K13" target="_self">FreeGen</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div>
