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

<b>let </b><font color="Maroon">S</font> be  <a href="osalg_1.html#V8">monotone</a> <a href="osalg_1.html#V10">regular</a> <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>let </b><font color="Maroon">t</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>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">D</font> =  <a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">R</font> =  <a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font>;<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">M</font> =  <a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font>;<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="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>means </b>for <font color="Olive">t1</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 <font color="Olive">t1</font> <a href="hidden.html#R2">in</a>  <a href="osafree.html#K25" target="_self">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">a<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">t1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <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>] <b>means </b> ex <font color="Olive">t1</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 <br/>( <font color="Olive">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t1</font>] );<br/>
<a NAME="E1:87"/><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>2</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="87_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="E1:87_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/>

<b>reconsider </b><font color="Maroon">t1</font> =  <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">sy</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>, <a class="ref" href="dtconstr.html#D4">DTCONSTR:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon">t1</font>
;<br/>

<b>thus </b><a NAME="E3:87_1"/>
<font color="Maroon">t1</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">sy</font>
 ;<br/>

<b>consider </b><font color="Maroon">s</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:87_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</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="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, </i>;<br/>
<b>let </b><font color="Maroon">t2</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="E6:87_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">t2</font> <a href="hidden.html#R2">in</a>  <a href="osafree.html#K25" target="_self">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">t1</font>
 ;<br/>

<b>thus </b><a NAME="E7:87_1"/>
<span class="p1">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t1</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:87"/><i><font color="Green">E54</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">ts1</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">ts1</font> &amp; ( for <font color="Olive">dt1</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">dt1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">ts1</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">dt1</font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Olive">ts1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_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">ts1</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="E1:87_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">nt</font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">ts1</font>
 <b>and </b><br/><a NAME="E2:87_2"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">dt1</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">dt1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts1</font> holds <br/> ex <font color="Olive">t2</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 <br/>( <font color="Olive">t2</font> <a href="hidden.html#R1">=</a> <font color="Olive">dt1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t2</font>] )
 ;<br/>

<b>reconsider </b><font color="Maroon">t1</font> = <font color="Maroon">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts1</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><a class="txt" href="osafree.html#E7:87_2"><i><font color="Green">E62</font></i></a>, </i>;<br/>
<b>take </b>
<font color="Maroon">t1</font>
;<br/>

<b>thus </b><a NAME="E4:87_2"/>
<font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">ta1</font> = <font color="Maroon">t1</font> as    <a href="subset_1.html#M1">Element</a> of 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"><a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Maroon">t1</font></span>)</span> <b>by </b><i/>;<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="E7:87_2"/><i><font color="Green">E62</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">ts1</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">ts1</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">ts1</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">ts1</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><a class="txt" href="osafree.html#E7:87_2"><i><font color="Green">E62</font></i></a>, </i>;<br/>
<i><font color="Green">E64</font></i>:  <a href="osafree.html#K25" target="_self">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">t1</font> = 
 <a href="osalg_4.html#K8">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">ta1</font>
<b>by </b><i/>
<br/>.= 
 <a href="funct_5.html#K1">proj1</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span>
<b>by </b><i/>
;<br/>
<b>set </b><font color="Maroon">Ms</font> = <span class="p1">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="lang1.html#K15">*</a> <font color="Maroon">ts1</font>;<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/>
<a NAME="E9:87_2"/><i><font color="Green">E65</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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">ts1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ts1</font> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="lang1.html#K15">*</a> <font color="Maroon">ts1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ts1</font> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="osafree.html#K11" target="_self">LeastSorts</a> <font color="Maroon">ts1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ts1</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ts1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="osafree.html#E8:87_2"><i><font color="Green">E64</font></i></a>, , <a class="ref" href="alg_1.html#T1">ALG_1:1</a>, <a class="ref" href="msualg_3.html#T6">MSUALG_3:6</a></i>;<br/>
<a NAME="E10:87_2"/><i><font color="Green">E67</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts1</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="E11:87_2"/>
( <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font></span>)</span> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts1</font> &amp;  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">ts1</font> )
 
<b>by </b><i><a class="txt" href="osafree.html#E7:87_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="osafree.html#E8:87_2"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E12:87_2"/><b>then </b><i><font color="Green">E68</font></i>: 
( <span class="p1">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" target="_self">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3">LBound</a> <font color="Maroon">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" target="_self">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="lang1.html#K15">*</a> <font color="Maroon">ts1</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="trees_4.html#K4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="lang1.html#K15">*</a> <font color="Maroon">ts1</font></span>)</span> &amp;  <a href="osafree.html#K11" target="_self">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="lang1.html#K15">*</a> <font color="Maroon">ts1</font></span>)</span> <a href="osalg_1.html#R2">&lt;=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font> )
 
<b>by </b><i/>;<br/>
<i><font color="Green">E69</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">t1</font> = 
 <a href="osafree.html#K18" target="_self">@</a> <font color="Maroon">nt</font>,<span class="p1">(<span class="default"><span class="p2">(<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">ts1</font></span>)</span>
<b>by </b><i><a class="txt" href="osafree.html#E7:87_2"><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">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> <span class="p1">(<span class="default"><span class="p2">(<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">ts1</font></span>)</span> &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> <span class="p1">(<span class="default"><span class="p2">(<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">ts1</font></span>)</span> 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> <span class="p1">(<span class="default"><span class="p2">(<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">ts1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) ) ) </span>}</span> 

;<br/>
<b>thus </b><a NAME="E14:87_2"/>
for <font color="Olive">t3</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 <font color="Olive">t3</font> <a href="hidden.html#R2">in</a>  <a href="osafree.html#K25" target="_self">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">t1</font> holds <br/><span class="p1">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">t3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t3</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="E1:87_2_3"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">t3</font> <a href="hidden.html#R2">in</a>  <a href="osafree.html#K25" target="_self">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">t1</font>
 ;<br/>

<b>consider </b><font color="Maroon">s4</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:87_2_3"/><i><font color="Green">E71</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t3</font>,<font color="Maroon">s4</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#E9:87_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="osafree.html#E1:87_2_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="funct_5.html#D1">FUNCT_5:def 1</a></i>;<br/>
<b>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="E5:87_2_3"/><i><font color="Green">E72</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t3</font>,<font color="Maroon">s4</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="E6:87_2_3"/><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> )
 <b>and </b><br/><a NAME="E7:87_2_3"/><i><font color="Green">E75</font></i>: 
 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> <span class="p1">(<span class="default"><span class="p2">(<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">ts1</font></span>)</span> &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> <span class="p1">(<span class="default"><span class="p2">(<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">ts1</font></span>)</span> 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> <span class="p1">(<span class="default"><span class="p2">(<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">ts1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) )
 <b>by </b><i><a class="txt" href="osafree.html#E13:87_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="osafree.html#E3:87_2_3"><i><font color="Green">E71</font></i></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="E9:87_2_3"/><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/>;<br/>
<a NAME="E10:87_2_3"/><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="txt" href="osafree.html#E8:87_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="osafree.html#E7:87_2_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E11:87_2_3"/><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">ts3</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/>
<a NAME="E13:87_2_3"/>
(  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o2</font>,<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K7">NonTerminals</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</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">ts3</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> &amp;  ex <font color="Olive">o3</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">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">ts3</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Olive">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">ts3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">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">ts3</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/> ( <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">ts3</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="Olive">o3</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s1</font> ) ) ) )
 
<b>by </b><i><a class="txt" href="osafree.html#E10:87_2_3"><i><font color="Green">E77</font></i></a>, </i>;<br/>
<b>then 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="E15:87_2_3"/><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">ts3</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">ts3</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">ts3</font> )
 ;<br/>
<a NAME="E16:87_2_3"/>
<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="E17:87_2_3"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">t3</font> <a href="hidden.html#R1">=</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> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts3</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E5:87_2_3"><i><font color="Green">E72</font></i></a>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E18:87_2_3"/><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">o</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> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ts3</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="lang1.html#K15">*</a> <font color="Maroon">ts3</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ts3</font> )
 
<b>by </b><i><a class="txt" href="osafree.html#E7:87_2_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="osafree.html#E9:87_2_3"><i><font color="Green">E76</font></i></a>, <a class="ref" href="alg_1.html#T1">ALG_1:1</a>, <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/>
<a NAME="E19:87_2_3"/><i><font color="Green">E105</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts3</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="E20:87_2_3"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="lang1.html#K15">*</a> <font color="Maroon">ts3</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="lang1.html#K15">*</a> <font color="Maroon">ts3</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="lang1.html#K15">*</a> <font color="Maroon">ts1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:87_2_3_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="lang1.html#K15">*</a> <font color="Maroon">ts3</font></span>)</span>
 ;<br/>

<a NAME="E2:87_2_3_1"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">ts1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</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">ts3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts3</font> )
 
<b>by </b><i>, <a class="txt" href="osafree.html#E15:87_2_3"><i><font color="Green">E80</font></i></a>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">tk1</font> = <font color="Maroon">ts1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>, <font color="Maroon">tk3</font> = <font color="Maroon">ts3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</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><a class="txt" href="osafree.html#E10:87_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="osafree.html#E17:87_2_3"><i><font color="Green">E81</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">tk2</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><b> such that </b><br/><a NAME="E5:87_2_3_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">tk2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">tk1</font>
 <b>and </b><br/><a NAME="E6:87_2_3_1"/><i><font color="Green">E109</font></i>: 
for <font color="Olive">t4</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 <font color="Olive">t4</font> <a href="hidden.html#R2">in</a>  <a href="osafree.html#K25" target="_self">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">tk2</font> holds <br/><span class="p1">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">t4</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">tk2</font>
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">tak</font> = <font color="Maroon">tk1</font> as    <a href="subset_1.html#M1">Element</a> of 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"><a href="osafree.html#K10" target="_self">LeastSort</a> <font color="Maroon">tk1</font></span>)</span> <b>by </b><i/>;<br/>
<i><font color="Green">E110</font></i>:  <a href="osafree.html#K25" target="_self">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">tk1</font> = 
 <a href="osalg_4.html#K8">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">tak</font>
<b>by </b><i/>
<br/>.= 
 <a href="funct_5.html#K1">proj1</a> <span class="p1">(<span class="default"><span class="p2">(<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">tk1</font></span>)</span>
<b>by </b><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="E10:87_2_3_1"/>
 <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> <span class="p1">(<span class="default"><span class="p2">(<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">ts1</font></span>)</span>
 <b>and </b><br/><a NAME="E11:87_2_3_1"/><i><font color="Green">E112</font></i>: 
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> <span class="p1">(<span class="default"><span class="p2">(<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">ts1</font></span>)</span> 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> <span class="p1">(<span class="default"><span class="p2">(<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">ts1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font>
 <b>by </b><i><a class="txt" href="osafree.html#E6:87_2_3"><i><font color="Green">E74</font></i></a></i>;<br/>
<a NAME="E12:87_2_3_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">tk3</font>,<span class="p2">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<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">ts1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="txt" href="osafree.html#E15:87_2_3"><i><font color="Green">E80</font></i></a>, , <a class="ref" href="osafree.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E13:87_2_3_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">tk3</font>,<span class="p2">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</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">tk1</font>
 
<b>by </b><i>, <a class="txt" href="osafree.html#E15:87_2_3"><i><font color="Green">E80</font></i></a>, , <a class="ref" href="alg_1.html#T1">ALG_1:1</a></i>;<br/>
<a NAME="E14:87_2_3_1"/><b>then </b>
<font color="Maroon">tk3</font> <a href="hidden.html#R2">in</a>  <a href="funct_5.html#K1">proj1</a> <span class="p1">(<span class="default"><span class="p2">(<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">tk1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_5.html#T4">FUNCT_5:4</a></i>;<br/>
<a NAME="E15:87_2_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">tk3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">tk1</font>
 
<b>by </b><i>, , <a class="txt" href="osafree.html#E18:87_2_3"><i><font color="Green">E86</font></i></a></i>;<br/>
<a NAME="E16:87_2_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">tk3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="lang1.html#K15">*</a> <font color="Maroon">ts1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="txt" href="osafree.html#E15:87_2_3"><i><font color="Green">E80</font></i></a>, , <a class="ref" href="alg_1.html#T1">ALG_1:1</a></i>;<br/>
<b>hence </b><a NAME="E17:87_2_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="lang1.html#K15">*</a> <font color="Maroon">ts3</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="lang1.html#K15">*</a> <font color="Maroon">ts1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="alg_1.html#T1">ALG_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:87_2_3"/><b>then </b><i><font color="Green">E113</font></i>: 
<span class="p1">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="lang1.html#K15">*</a> <font color="Maroon">ts3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="lang1.html#K15">*</a> <font color="Maroon">ts1</font>
 
<b>by </b><i>, <a class="txt" href="osafree.html#E15:87_2_3"><i><font color="Green">E80</font></i></a>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>
<a NAME="E22:87_2_3"/>
( <span class="p1">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" target="_self">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3">LBound</a> <font color="Maroon">o2</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" target="_self">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="lang1.html#K15">*</a> <font color="Maroon">ts3</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="trees_4.html#K4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="lang1.html#K15">*</a> <font color="Maroon">ts3</font></span>)</span> &amp;  <a href="osafree.html#K11" target="_self">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="lang1.html#K15">*</a> <font color="Maroon">ts3</font></span>)</span> <a href="osalg_1.html#R2">&lt;=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font> )
 
<b>by </b><i><a class="txt" href="osafree.html#E10:87_2_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="osafree.html#E11:87_2_3"><i><font color="Green">E79</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E23:87_2_3"/>
<span class="p1">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K29" target="_self">PTMin</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#E12:87_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="osafree.html#E7:87_2_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="osafree.html#E9:87_2_3"><i><font color="Green">E76</font></i></a>, , <a class="ref" href="osalg_1.html#T34">OSALG_1:34</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E3:87"/>
for <font color="Olive">dt</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">dt</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>2</sub></font>[<font color="Olive">dt</font>]
 
<b>from </b><i><a class="ref" href="dtconstr.html#S7">DTCONSTR:sch 7</a>(, );<br/></i>
<b>then consider </b><font color="Maroon">t1</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><b> such that </b><br/><a NAME="E5:87"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">t1</font>] )
 ;<br/>
<b>thus </b><a NAME="E6:87"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">t</font>]
 <b>by </b><i/>;<br/>


</div>
