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

<b>reconsider </b><font color="Maroon">D</font> =  <a href="card_3.html#K3">Union</a> <font color="Maroon">F<sub>3</sub></font>() as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E2:20_1"/><i><font color="Green">E41</font></i>: 
 <a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a>  <a href="lang1.html#G1">DTConstrStr</a>(# <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">F<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">F<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">coprod</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K4">REL</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> #)
 
<b>by </b><i><a class="ref" href="msafree.html#D10">MSAFREE:def 10</a></i>;<br/>
<a NAME="E3:20_1"/><i><font color="Green">E43</font></i>: 
 <a href="dtconstr.html#K6">Terminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K3">coprod</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<b>by </b><i><a class="ref" href="msafree.html#T6">MSAFREE:6</a></i>;<br/>
<i><font color="Green">E44</font></i>:  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> = 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="msafree.html#K8">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="msafree.html#T12">MSAFREE:12</a></i>
<br/>.= 
 <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>
;<br/>
<a NAME="E5:20_1"/><i><font color="Green">E45</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<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> <font color="Maroon">F<sub>3</sub></font>() = <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>);<br/>
<b>consider </b><font color="Maroon">G</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>, <a href="card_3.html#K3">Union</a> <font color="Maroon">F<sub>3</sub></font>()<b> such that </b><br/><a NAME="E7:20_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> holds  <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<font color="Maroon">D</font> ;<br/>
<div><i><font color="Green">E47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">f</font> be    <a href="pboole.html#M3">ManySortedFunction</a> of  <a href="msafree.html#K8">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>();<br/><b>assume </b><a NAME="E1:20_1_2"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K12">FreeGen</a> <font color="Olive">s</font>,<font color="Maroon">F<sub>2</sub></font>() holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">y</font>)
 ;<br/><b>let </b><font color="Maroon">t</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>;<br/><b>assume </b><a NAME="E2:20_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">t</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">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 ;<br/><a NAME="E3:20_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">t</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">t</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, <a class="ref" href="card_3.html#T33">CARD_3:33</a></i>;<br/><b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">t</font> <a href="mcart_1.html#K2">`2</a>  as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i>, , , <a class="ref" href="card_3.html#T33">CARD_3:33</a></i>;<br/><a NAME="E5:20_1_2"/><i><font color="Green">E52</font></i>: 
 <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K12">FreeGen</a> <font color="Maroon">s</font>,<font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="ref" href="msafree1.html#T1" target="_self">Th1</a>, <a class="ref" href="msafree.html#D17">MSAFREE:def 17</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="msafree1.html#K1" target="_self">Flatten</a> <font color="Maroon">f</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">t</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="msafree1.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">t</font></span>)</span>)
<b>by </b><i>, <a class="ref" href="msafree1.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>
<b>by </b><i/>
;<br/><br/></div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>,   <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="finseq_1.html#K13">*</a> ,   <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="finseq_1.html#K13">*</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K3">Union</a> <font color="Maroon">F<sub>3</sub></font>() = <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font>);<br/>
<b>consider </b><font color="Maroon">H</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p4">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="finseq_1.html#K13">*</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_3.html#K3">Union</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="finseq_1.html#K13">*</a> </span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span>, <a href="card_3.html#K3">Union</a> <font color="Maroon">F<sub>3</sub></font>()<b> such that </b><br/><a NAME="E11:20_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">nt</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span><br/> for <font color="Olive">ts</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="finseq_1.html#K13">*</a> <br/> for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="finseq_1.html#K13">*</a>  holds  <font color="Maroon">H</font> <a href="multop_1.html#K2">.</a> <font color="Olive">nt</font>,<font color="Olive">ts</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">nt</font>,<font color="Olive">ts</font>,<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="multop_1.html#S4">MULTOP_1:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">H</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p4">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="finseq_1.html#K13">*</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="finseq_1.html#K13">*</a> </span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span>,<font color="Maroon">D</font> ;<br/>
<div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">f</font> be    <a href="pboole.html#M3">ManySortedFunction</a> of  <a href="msafree.html#K8">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>();<br/><b>assume </b><a NAME="E1:20_1_3"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">ts</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Olive">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span><br/> for <font color="Olive">x</font> being   <a href="msafree1.html#M1" target="_self">FinSequence</a> of <font color="Maroon">D</font>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree1.html#K1" target="_self">Flatten</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">ts</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</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="Olive">o</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="Olive">o</font>,<span class="p3">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">ts</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">o</font>,<font color="Olive">ts</font>,<font color="Olive">x</font>)
 ;<br/><a NAME="E2:20_1_3"/><i><font color="Green">E92</font></i>: 
 <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K10">FreeOper</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> #)
 <b>by </b><i><a class="ref" href="msafree.html#D16">MSAFREE:def 16</a></i>;<br/><b>let </b><font color="Maroon">nt</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>;<br/><b>let </b><font color="Maroon">ts</font> be    <a href="msafree1.html#M1" target="_self">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>;<br/><b>assume </b><a NAME="E3:20_1_3"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">nt</font> <a href="lang1.html#R1">==&gt;</a>  <a href="trees_3.html#K16">roots</a> <font color="Maroon">ts</font>
 ;<br/><a NAME="E4:20_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">nt</font>,<span class="p2">(<span class="default"><a href="trees_3.html#K16">roots</a> <font color="Maroon">ts</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K4">REL</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i>, <a class="ref" href="lang1.html#D1">LANG1:def 1</a></i>;<br/><a NAME="E5:20_1_3"/><b>then </b>
<font color="Maroon">nt</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">F<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">F<sub>1</sub></font>()</span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">o</font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="tarski.html#K1">}</a></span><b> such that </b><br/><a NAME="E7:20_1_3"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">nt</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/><a NAME="E8:20_1_3"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E9:20_1_3"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">nt</font> <a href="hidden.html#R1">=</a>  <a href="msafree.html#K6">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="msafree1.html#E1:20"><i><font color="Green">E24</font></i></a>, <a class="ref" href="msafree.html#D11">MSAFREE:def 11</a></i>;<br/><a NAME="E10:20_1_3"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">ts</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">FreeSort</a> <font color="Maroon">F<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">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i>, <a class="ref" href="msafree.html#T10">MSAFREE:10</a></i>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="msafree1.html#M1" target="_self">FinSequence</a> of <font color="Maroon">D</font>;<br/><b>assume </b><a NAME="E11:20_1_3"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree1.html#K1" target="_self">Flatten</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">ts</font>
 ;<br/><a NAME="E12:20_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K8">FreeSort</a> <font color="Maroon">F<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">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> <font color="Maroon">F<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">o</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/><a NAME="E13:20_1_3"/><b>then </b><i><font color="Green">E100</font></i>: 
<span class="p1">(<span class="default"><a href="msafree.html#K9">DenOp</a> <font color="Maroon">o</font>,<font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ts</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> <font color="Maroon">F<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">o</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E14:20_1_3"/><i><font color="Green">E101</font></i>: 
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">F<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">FreeSort</a> <font color="Maroon">F<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">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><a href="msafree1.html#K1" target="_self">Flatten</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">ts</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="msafree1.html#K1" target="_self">Flatten</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K9">DenOp</a> <font color="Maroon">o</font>,<font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ts</font></span>)</span>
<b>by </b><i>, <a class="txt" href="msafree1.html#E1:20"><i><font color="Green">E24</font></i></a>, <a class="txt" href="msafree1.html#E2:20"><i><font color="Green">E25</font></i></a>, , <a class="ref" href="msafree.html#D14">MSAFREE:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</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">o</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="msafree.html#K9">DenOp</a> <font color="Maroon">o</font>,<font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ts</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</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">o</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">o</font>,<span class="p3">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ts</font></span>)</span>
<b>by </b><i>, <a class="ref" href="msafree.html#D15">MSAFREE:def 15</a></i>
<br/>.= 
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">o</font>,<font color="Maroon">ts</font>,<font color="Maroon">x</font>)
<b>by </b><i><a class="ref" href="msafree1.html#T1" target="_self">Th1</a>, , <a class="txt" href="msafree1.html#E3:20"><i><font color="Green">E28</font></i></a>, </i>
<br/>.= 
<font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">nt</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<font color="Maroon">ts</font>,<font color="Maroon">x</font>)
<b>by </b><i><a class="txt" href="msafree1.html#E1:20"><i><font color="Green">E24</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/><b>hence </b><a NAME="E16:20_1_3"/>
<span class="p1">(<span class="default"><a href="msafree1.html#K1" target="_self">Flatten</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">nt</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">ts</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="multop_1.html#K2">.</a> <font color="Maroon">nt</font>,<font color="Maroon">ts</font>,<font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="msafree1.html#E1:20"><i><font color="Green">E24</font></i></a>, <a class="txt" href="msafree1.html#E2:20"><i><font color="Green">E25</font></i></a></i>;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">f1</font> =  <a href="msafree1.html#K1" target="_self">Flatten</a> <font color="Maroon">F<sub>6</sub></font>(), <font color="Maroon">f2</font> =  <a href="msafree1.html#K1" target="_self">Flatten</a> <font color="Maroon">F<sub>7</sub></font>() as   <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">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<font color="Maroon">D</font> <b>by </b><i/>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> = <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>,   <a href="msafree1.html#M1" target="_self">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>,   <a href="msafree1.html#M1" target="_self">FinSequence</a> of <font color="Maroon">D</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> = <font color="Maroon">H</font> <a href="multop_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font>;<br/>
<a NAME="E15:20_1"/><i><font color="Green">E104</font></i>: 
for <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>  st <font color="Olive">t</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">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> holds <br/><font color="Maroon">f1</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">t</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">t</font>)
 
<b>by </b><i>, </i>;<br/>
<a NAME="E16:20_1"/><i><font color="Green">E105</font></i>: 
for <font color="Olive">nt</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span><br/> for <font color="Olive">ts</font> being   <a href="msafree1.html#M1" target="_self">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>  st <font color="Olive">nt</font> <a href="lang1.html#R1">==&gt;</a>  <a href="trees_3.html#K16">roots</a> <font color="Olive">ts</font> holds <br/>for <font color="Olive">x</font> being   <a href="msafree1.html#M1" target="_self">FinSequence</a> of <font color="Maroon">D</font>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="lang1.html#K15">*</a> <font color="Olive">ts</font> holds <br/><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">nt</font> <a href="trees_4.html#K4">-tree</a> <font color="Olive">ts</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">nt</font>,<font color="Olive">ts</font>,<font color="Olive">x</font>)
 
<b>by </b><i>, </i>;<br/>
<a NAME="E17:20_1"/><i><font color="Green">E106</font></i>: 
for <font color="Olive">t</font> being  <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>  st <font color="Olive">t</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">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> holds <br/><font color="Maroon">f2</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">t</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">t</font>)
 
<b>by </b><i>, </i>;<br/>
<a NAME="E18:20_1"/><i><font color="Green">E107</font></i>: 
for <font color="Olive">nt</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span><br/> for <font color="Olive">ts</font> being   <a href="msafree1.html#M1" target="_self">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>  st <font color="Olive">nt</font> <a href="lang1.html#R1">==&gt;</a>  <a href="trees_3.html#K16">roots</a> <font color="Olive">ts</font> holds <br/>for <font color="Olive">x</font> being   <a href="msafree1.html#M1" target="_self">FinSequence</a> of <font color="Maroon">D</font>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="lang1.html#K15">*</a> <font color="Olive">ts</font> holds <br/><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">nt</font> <a href="trees_4.html#K4">-tree</a> <font color="Olive">ts</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">nt</font>,<font color="Olive">ts</font>,<font color="Olive">x</font>)
 
<b>by </b><i>, </i>;<br/>
<a NAME="E19:20_1"/>
<font color="Maroon">f1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f2</font>
 
<b>from </b><i><a class="ref" href="msafree1.html#S1" target="_self">MSAFREE1:sch 1</a>(<a class="ref" href="msafree1.html#T1" target="_self">Th1</a>, , , <a class="txt" href="msafree1.html#E1:20"><i><font color="Green">E24</font></i></a>);<br/></i>
<b>hence </b><a NAME="E20:20_1"/>
<font color="Maroon">F<sub>6</sub></font>() <a href="pboole.html#R6">=</a> <font color="Maroon">F<sub>7</sub></font>()
 <b>by </b><i/>;<br/>


</div>
