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

<b>let </b><font color="Maroon">S</font> be  <a href="osalg_4.html#V2">locally_directed</a> <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">X</font> be  <a href="relat_1.html#V2">V5</a> <a href="struct_0.html#NM8">ManySortedSet</a> of <font color="Maroon">S</font>;<br/>



<b>set </b><font color="Maroon">D</font> =  <a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">PTA</font> =  <a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">C</font> =  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p3">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>set </b><font color="Maroon">SPTA</font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">F</font> =  <a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><br/> for <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">z</font>,<font color="Olive">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Olive">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font>;<br/>
<a NAME="E1:53"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">s</font> being  <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>  st <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6">Terminals</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</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">s</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">sy</font> be   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:53_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">sy</font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6">Terminals</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>
 ;<br/>

<i><font color="Green">E37</font></i>: <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</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">sy</font></span>)</span> = 
 <a href="osafree.html#K17" target="_self">@</a> <font color="Maroon">sy</font>
<b>by </b><i><a class="txt" href="osafree.html#E1:53_1"><i><font color="Green">E35</font></i></a>, </i>
<br/>.= 
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">sy</font></span>)</span>,<font color="Olive">s1</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">s1</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> :  ex <font color="Olive">s2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s2</font> &amp; <font color="Maroon">sy</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">s2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">s2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s1</font> ) </span>}</span> 

;<br/>
<b>thus </b><a NAME="E3:53_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">sy</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">y</font> be    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">z</font> be    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:53_1_2"/><i><font color="Green">E39</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</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">sy</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">s2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:53_1_2"/><i><font color="Green">E54</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">sy</font></span>)</span>,<font color="Maroon">s2</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E4:53_1_2"/>
 ex <font color="Olive">s0</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s0</font> &amp; <font color="Maroon">sy</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">s0</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">s0</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s2</font> )
 <b>by </b><i/>;<br/>
<a NAME="E5:53_1_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">sy</font> &amp; <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>then consider </b><font color="Maroon">s3</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E7:53_1_2"/><i><font color="Green">E61</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">sy</font></span>)</span>,<font color="Maroon">s3</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E8:53_1_2"/>
 ex <font color="Olive">s0</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s0</font> &amp; <font color="Maroon">sy</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">s0</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">s0</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s3</font> )
 <b>by </b><i>, <a class="txt" href="osafree.html#E2:53_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>thus </b><a NAME="E9:53_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">sy</font></span>)</span>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="osafree.html#E2:53_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="osafree.html#E1:53_1_2"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E2:53"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">nt</font> being  <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span><br/> for <font color="Olive">ts</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="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>  st <font color="Olive">nt</font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Olive">ts</font> &amp; ( for <font color="Olive">t</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="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>  st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">ts</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Olive">ts</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">nt</font> be   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">ts</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="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:53_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">nt</font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">ts</font>
 <b>and </b><br/><a NAME="E2:53_2"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">t</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="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>  st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t</font>]
 ;<br/>

<b>consider </b><font color="Maroon">o</font> being   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E4:53_2"/><i><font color="Green">E67</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>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">ts</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ts</font> &amp; ( for <font color="Olive">s1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds <br/> ( <font color="Maroon">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts</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="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font> iff  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s1</font> ) ) )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">ts1</font> = <font color="Maroon">ts</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">w</font> =  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="lang1.html#K15">*</a> <font color="Maroon">ts</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p3">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<a NAME="E7:53_2"/><i><font color="Green">E68</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts</font> <a href="tarski.html#R1">c=</a>  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:53_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">ts</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T33">FINSEQ_2:33</a></i>;<br/>
<a NAME="E9:53_2"/><b>then </b><i><font color="Green">E69</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ts</font> &amp;  <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">o</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ts</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a>, <a class="ref" href="msualg_3.html#T6">MSUALG_3:6</a></i>;<br/>
<i><font color="Green">E70</font></i>: <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</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#K12">-tree</a> <font color="Maroon">ts</font></span>)</span> = 
 <a href="osafree.html#K18" target="_self">@</a> <font color="Maroon">nt</font>,<font color="Maroon">x</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o2</font>,<span class="p5">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x2</font></span>)</span>,<font color="Olive">s3</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">o2</font> is   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>, <font color="Olive">x2</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Olive">o2</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span>, <font color="Olive">s3</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : (  ex <font color="Olive">o1</font> being  <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( <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="Olive">o1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">o1</font> <a href="osalg_1.html#R1">~=</a> <font color="Olive">o2</font> &amp;  <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="Olive">o1</font></span>)</span> <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="Olive">o2</font></span>)</span> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Olive">o1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s3</font> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Olive">o2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s3</font> ) &amp;  ex <font color="Olive">w3</font> being   <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a>  st <br/>(  <a href="finseq_1.html#K4">dom</a> <font color="Olive">w3</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) ) ) </span>}</span> 

;<br/>
<b>thus </b><a NAME="E11:53_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">y</font> be    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">z</font> be    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:53_2_2"/><i><font color="Green">E71</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</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#K12">-tree</a> <font color="Maroon">ts</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">o2</font> being   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>, <font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o2</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">s3</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:53_2_2"/><i><font color="Green">E72</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span>,<font color="Maroon">s3</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E4:53_2_2"/><i><font color="Green">E74</font></i>: 
(  ex <font color="Olive">o1</font> being  <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( <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="Olive">o1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">o1</font> <a href="osalg_1.html#R1">~=</a> <font color="Maroon">o2</font> &amp;  <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="Olive">o1</font></span>)</span> <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">o2</font></span>)</span> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Olive">o1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s3</font> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s3</font> ) &amp;  ex <font color="Olive">w3</font> being   <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a>  st <br/>(  <a href="finseq_1.html#K4">dom</a> <font color="Olive">w3</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) ) )
 <b>by </b><i/>;<br/>
<a NAME="E5:53_2_2"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s3</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>consider </b><font color="Maroon">o1</font> being   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E7:53_2_2"/><i><font color="Green">E76</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">o1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">o1</font> <a href="osalg_1.html#R1">~=</a> <font color="Maroon">o2</font> &amp;  <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">o1</font></span>)</span> <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">o2</font></span>)</span> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s3</font> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s3</font> )
 <b>by </b><i><a class="ref" href="osafree.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E8:53_2_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E9:53_2_2"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">x2</font> is    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o2</font>,<font color="Maroon">X</font> <a href="lang1.html#R1">==&gt;</a>  <a href="trees_3.html#K16">roots</a> <font color="Maroon">x2</font> )
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x3</font> = <font color="Maroon">x2</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="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">o3</font> being   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E12:53_2_2"/><i><font color="Green">E80</font></i>: 
(  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o2</font>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o3</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x3</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o3</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o2</font>,<font color="Maroon">X</font></span>)</span> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">x3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o3</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x3</font> &amp; ( for <font color="Olive">s2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds <br/> ( <span class="p1">(<span class="default"><a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o2</font>,<font color="Maroon">X</font></span>)</span> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">x3</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="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s2</font> iff  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o3</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s2</font> ) ) )
 <b>by </b><i>, </i>;<br/>
<a NAME="E13:53_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o3</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E14:53_2_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o3</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E15:53_2_2"/><i><font color="Green">E86</font></i>: 
 <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">o2</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">o</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>consider </b><font color="Maroon">w3</font> being    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E17:53_2_2"/><i><font color="Green">E105</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w3</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) )
 <b>by </b><i><a class="ref" href="osafree.html#D4" target="_self">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xy</font> = <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="lang1.html#K15">*</a> <font color="Maroon">x3</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p3">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<a NAME="E19:53_2_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">x3</font> <a href="tarski.html#R1">c=</a>  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E20:53_2_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">x3</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E21:53_2_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">xy</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x3</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T33">FINSEQ_2:33</a></i>;<br/>
<a NAME="E22:53_2_2"/><b>then </b><i><font color="Green">E106</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x3</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">xy</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E23:53_2_2"/><i><font color="Green">E107</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="ref" href="msualg_3.html#T6">MSUALG_3:6</a></i>;<br/>
<i><font color="Green">E108</font></i>: <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> = 
 <a href="osafree.html#K18" target="_self">@</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o2</font>,<font color="Maroon">X</font></span>)</span>,<font color="Maroon">xy</font>
<b>by </b><i>, , , , </i>
<br/>.= 
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o4</font>,<span class="p5">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x4</font></span>)</span>,<font color="Olive">s4</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">o4</font> is   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>, <font color="Olive">x4</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Olive">o4</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span>, <font color="Olive">s4</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : (  ex <font color="Olive">o1</font> being  <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>(  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o2</font>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">o1</font> <a href="osalg_1.html#R1">~=</a> <font color="Olive">o4</font> &amp;  <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="Olive">o1</font></span>)</span> <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="Olive">o4</font></span>)</span> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Olive">o1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s4</font> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Olive">o4</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s4</font> ) &amp;  ex <font color="Olive">w4</font> being   <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a>  st <br/>(  <a href="finseq_1.html#K4">dom</a> <font color="Olive">w4</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">xy</font> &amp; ( for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">xy</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x4</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">w4</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">xy</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) ) ) </span>}</span> 

;<br/>
<b>consider </b><font color="Maroon">o5</font> being   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>, <font color="Maroon">x5</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o5</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">s5</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E26:53_2_2"/><i><font color="Green">E109</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o5</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x5</font></span>)</span>,<font color="Maroon">s5</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E27:53_2_2"/><i><font color="Green">E110</font></i>: 
(  ex <font color="Olive">o1</font> being  <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>(  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o2</font>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">o1</font> <a href="osalg_1.html#R1">~=</a> <font color="Maroon">o5</font> &amp;  <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="Olive">o1</font></span>)</span> <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">o5</font></span>)</span> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Olive">o1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s5</font> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o5</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s5</font> ) &amp;  ex <font color="Olive">w3</font> being   <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a>  st <br/>(  <a href="finseq_1.html#K4">dom</a> <font color="Olive">w3</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">xy</font> &amp; ( for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">xy</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x5</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">xy</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) ) )
 <b>by </b><i>, <a class="ref" href="osafree.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E28:53_2_2"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o5</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x5</font> &amp; <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s5</font> )
 
<b>by </b><i><a class="ref" href="osafree.html#T3" target="_self">Th3</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>consider </b><font color="Maroon">o6</font> being   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E30:53_2_2"/><i><font color="Green">E113</font></i>: 
(  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o2</font>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o6</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">o6</font> <a href="osalg_1.html#R1">~=</a> <font color="Maroon">o5</font> &amp;  <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">o6</font></span>)</span> <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">o5</font></span>)</span> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o6</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s5</font> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o5</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s5</font> )
 <b>by </b><i/>;<br/>
<a NAME="E31:53_2_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">o6</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E32:53_2_2"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">x5</font> is    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o5</font>,<font color="Maroon">X</font> <a href="lang1.html#R1">==&gt;</a>  <a href="trees_3.html#K16">roots</a> <font color="Maroon">x5</font> )
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x6</font> = <font color="Maroon">x5</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="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">o7</font> being   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E35:53_2_2"/><i><font color="Green">E116</font></i>: 
(  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o5</font>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o7</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x6</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o7</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o5</font>,<font color="Maroon">X</font></span>)</span> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">x6</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o7</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x6</font> &amp; ( for <font color="Olive">s2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds <br/> ( <span class="p1">(<span class="default"><a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o5</font>,<font color="Maroon">X</font></span>)</span> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">x6</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="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s2</font> iff  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o7</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s2</font> ) ) )
 <b>by </b><i>, </i>;<br/>
<a NAME="E36:53_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o5</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o7</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="osafree.html#E2:53"><i><font color="Green">E62</font></i></a></i>;<br/>
<a NAME="E37:53_2_2"/><b>then </b><i><font color="Green">E176</font></i>: 
<font color="Maroon">o5</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o7</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E38:53_2_2"/><i><font color="Green">E177</font></i>: 
 <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">o5</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">o2</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>consider </b><font color="Maroon">w5</font> being    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E40:53_2_2"/><i><font color="Green">E193</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w5</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">xy</font> &amp; ( for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">xy</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x5</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">w5</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">xy</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">xz</font> = <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="lang1.html#K15">*</a> <font color="Maroon">x6</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p3">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<a NAME="E42:53_2_2"/><i><font color="Green">E194</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">x6</font> <a href="tarski.html#R1">c=</a>  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">x3</font> <a href="tarski.html#R1">c=</a>  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E43:53_2_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">x6</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E44:53_2_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">xz</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x6</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T33">FINSEQ_2:33</a></i>;<br/>
<a NAME="E45:53_2_2"/><b>then </b><i><font color="Green">E195</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x6</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">xz</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<i><font color="Green">E196</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x5</font> = 
 <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">o2</font></span>)</span>
<b>by </b><i><a class="txt" href="osafree.html#E1:53_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="msualg_3.html#T6">MSUALG_3:6</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">xy</font>
<b>by </b><i>, <a class="ref" href="msualg_3.html#T6">MSUALG_3:6</a></i>
;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ts1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">a<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">xz</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E47:53_2_2"/><i><font color="Green">E197</font></i>: 
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="finseq_1.html#K4">dom</a> <font color="Maroon">xz</font> holds <br/> ex <font color="Olive">sy</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">sy</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Olive">sy</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_2_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:53_2_2_3"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">xz</font>
 ;<br/>

<a NAME="E2:53_2_2_3"/><i><font color="Green">E199</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ts1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x5</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, , , <a class="txt" href="osafree.html#E4:53_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="osafree.html#E7:53_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="osafree.html#E10:53_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="msualg_3.html#T6">MSUALG_3:6</a></i>;<br/>
<a NAME="E3:53_2_2_3"/><i><font color="Green">E200</font></i>: 
( <font color="Maroon">ts1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts1</font> &amp; <font color="Maroon">x5</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">x6</font> &amp; <font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">x3</font> )
 
<b>by </b><i>, , , <a class="txt" href="osafree.html#E4:53_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="osafree.html#E7:53_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="osafree.html#E10:53_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">t1</font> = <font color="Maroon">ts1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>, <font color="Maroon">t2</font> = <font color="Maroon">x3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>, <font color="Maroon">t3</font> = <font color="Maroon">x5</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</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="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i>, </i>;<br/>
<a NAME="E5:53_2_2_3"/><i><font color="Green">E217</font></i>: 
<font color="Maroon">y</font> is   <a href="nat_1.html#NM1">Nat</a>
 
<b>by </b><i><a class="txt" href="osafree.html#E10:53_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E6:53_2_2_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x5</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">w5</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">xy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E2:53_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="osafree.html#E4:53_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="osafree.html#E7:53_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="osafree.html#E10:53_2"><i><font color="Green">E70</font></i></a></i>;<br/>
<a NAME="E7:53_2_2_3"/><b>then </b><i><font color="Green">E218</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t3</font>,<span class="p2">(<span class="default"><font color="Maroon">w5</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t2</font>
 
<b>by </b><i>, <a class="txt" href="osafree.html#E4:53_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="osafree.html#E7:53_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="osafree.html#E10:53_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E8:53_2_2_3"/><b>then </b><i><font color="Green">E219</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t2</font>,<span class="p2">(<span class="default"><font color="Maroon">w5</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t2</font>,<span class="p2">(<span class="default"><font color="Maroon">w5</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t3</font> )
 
<b>by </b><i>, <a class="ref" href="osafree.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E9:53_2_2_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t3</font>,<span class="p2">(<span class="default"><font color="Maroon">w5</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t3</font>
 
<b>by </b><i><a class="ref" href="osafree.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E10:53_2_2_3"/><b>then </b><i><font color="Green">E220</font></i>: 
( <font color="Maroon">t2</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="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w5</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span> &amp; <font color="Maroon">t3</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="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w5</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="osafree.html#E5:53_2_2"><i><font color="Green">E75</font></i></a>, </i>;<br/>
<a NAME="E11:53_2_2_3"/><b>then </b><i><font color="Green">E221</font></i>: 
 <a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Maroon">t2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">w5</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:53_2_2_3"/><i><font color="Green">E222</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , , , <a class="txt" href="osafree.html#E4:53_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="osafree.html#E7:53_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="osafree.html#E10:53_2"><i><font color="Green">E70</font></i></a></i>;<br/>
<a NAME="E13:53_2_2_3"/><b>then </b><i><font color="Green">E223</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ts1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i>, , , <a class="txt" href="osafree.html#E4:53_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="osafree.html#E7:53_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="osafree.html#E10:53_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E14:53_2_2_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t2</font>,<span class="p2">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t1</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E1:53_2_2"><i><font color="Green">E71</font></i></a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E15:53_2_2_3"/><b>then </b><i><font color="Green">E224</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t1</font>,<span class="p2">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t1</font>,<span class="p2">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t1</font> )
 
<b>by </b><i>, <a class="ref" href="osafree.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E16:53_2_2_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t2</font>,<span class="p2">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t2</font>
 
<b>by </b><i><a class="ref" href="osafree.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E17:53_2_2_3"/><b>then </b><i><font color="Green">E225</font></i>: 
( <font color="Maroon">t2</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="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span> &amp; <font color="Maroon">t1</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="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="osafree.html#E12:53_2_2"><i><font color="Green">E80</font></i></a>, </i>;<br/>
<a NAME="E18:53_2_2_3"/><b>then </b>
 <a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Maroon">t2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">s7</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E20:53_2_2_3"/><i><font color="Green">E227</font></i>: 
( <font color="Maroon">w5</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s7</font> &amp; <font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s7</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E8:53_2_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="osalg_4.html#T12">OSALG_4:12</a></i>;<br/>
<a NAME="E21:53_2_2_3"/><i><font color="Green">E228</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t2</font>,<font color="Maroon">s7</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t1</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E9:53_2_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="osafree.html#E14:53_2_2"><i><font color="Green">E81</font></i></a>, , </i>;<br/>
<a NAME="E22:53_2_2_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t3</font>,<font color="Maroon">s7</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t2</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E4:53_2_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="osafree.html#E7:53_2_2"><i><font color="Green">E76</font></i></a>, , </i>;<br/>
<a NAME="E23:53_2_2_3"/><b>then </b><i><font color="Green">E229</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t1</font>,<font color="Maroon">s7</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t3</font>
 
<b>by </b><i>, <a class="txt" href="osafree.html#E3:53_2_2"><i><font color="Green">E72</font></i></a>, </i>;<br/>
<b>take </b>
<font color="Maroon">s7</font>
;<br/>

<b>thus </b><a NAME="E24:53_2_2_3"/>
<font color="Maroon">s7</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<b>thus </b><a NAME="E25:53_2_2_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ts1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">s7</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">xz</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="osafree.html#E10:53_2"><i><font color="Green">E70</font></i></a>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">xz</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E49:53_2_2"/><i><font color="Green">E230</font></i>: 
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="finseq_1.html#K4">dom</a> <font color="Maroon">xz</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="osafree.html#E9:53_2"><i><font color="Green">E69</font></i></a>);<br/></i>
<a NAME="E50:53_2_2"/><i><font color="Green">E231</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">xz</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E51:53_2_2"/><b>then </b>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Olive">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E53:53_2_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E54:53_2_2"/><b>then </b>
<font color="Maroon">f1</font> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E56:53_2_2"/><i><font color="Green">E232</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">xz</font> &amp; ( for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">xz</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ts1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">xz</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_2_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:53_2_2_4"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">xz</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>

<b>let </b><font color="Maroon">y</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E2:53_2_2_4"/><i><font color="Green">E233</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">xz</font>
 ;<br/>

<a NAME="E3:53_2_2_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ts1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">xz</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E10:53_2"><i><font color="Green">E70</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E4:53_2_2_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ts1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">xz</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="osafree.html#E1:53_2_2"><i><font color="Green">E71</font></i></a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E234</font></i>: <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> = 
 <a href="osafree.html#K18" target="_self">@</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o5</font>,<font color="Maroon">X</font></span>)</span>,<font color="Maroon">xz</font>
<b>by </b><i>, , <a class="txt" href="osafree.html#E2:53"><i><font color="Green">E62</font></i></a>, , </i>
<br/>.= 
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o4</font>,<span class="p5">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x4</font></span>)</span>,<font color="Olive">s4</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">o4</font> is   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>, <font color="Olive">x4</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Olive">o4</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span>, <font color="Olive">s4</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : (  ex <font color="Olive">o1</font> being  <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>(  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o5</font>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">o1</font> <a href="osalg_1.html#R1">~=</a> <font color="Olive">o4</font> &amp;  <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="Olive">o1</font></span>)</span> <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="Olive">o4</font></span>)</span> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Olive">o1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s4</font> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Olive">o4</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s4</font> ) &amp;  ex <font color="Olive">w4</font> being   <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a>  st <br/>(  <a href="finseq_1.html#K4">dom</a> <font color="Olive">w4</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">xz</font> &amp; ( for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">xz</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x4</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">w4</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">xz</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) ) ) </span>}</span> 

;<br/>
<a NAME="E58:53_2_2"/>
(  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o5</font>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o5</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">o5</font> <a href="osalg_1.html#R1">~=</a> <font color="Maroon">o</font> &amp;  <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">o5</font></span>)</span> <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">o</font></span>)</span> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o5</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s1</font> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s1</font> )
 
<b>by </b><i>, , , <a class="ref" href="osafree.html#T3" target="_self">Th3</a>, , , <a class="ref" href="osalg_1.html#T2">OSALG_1:2</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E59:53_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts</font></span>)</span>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="txt" href="osafree.html#E3:53_2_2"><i><font color="Green">E72</font></i></a>, </i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E3:53"/>
for <font color="Olive">t</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="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>  st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t</font>]
 
<b>from </b><i><a class="ref" href="dtconstr.html#S7">DTCONSTR:sch 7</a>(<a class="txt" href="osafree.html#E1:53"><i><font color="Green">E34</font></i></a>, );<br/></i>
<b>hence </b><a NAME="E4:53"/>
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span><br/> for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">z</font>,<font color="Olive">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font>
 ;<br/>


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