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

<b>let </b><font color="Maroon" title="c1">S</font> be  <a href="osalg_1.html#V8" title="OSALG_1:attr.8">monotone</a> <a href="osalg_1.html#V10" title="OSALG_1:attr.10">regular</a> <a href="osalg_4.html#V2" title="OSALG_4:attr.2">locally_directed</a> <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a> <a href="struct_0.html#NM8" title="STRUCT_0:NM.8">ManySortedSet</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">t</font> be    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>





<b>set </b><font color="Maroon" title="c4">PTA</font> =  <a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font>;<br/>
<b>set </b><font color="Maroon" title="c5">D</font> =  <a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font>;<br/>
<b>set </b><font color="Maroon" title="c6">R</font> =  <a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font>;<br/>
<b>set </b><font color="Maroon" title="c7">SPTA</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">M</font> =  <a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font>;<br/>
<b>set </b><font color="Maroon" title="c9">F</font> =  <a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>] <b>means </b><span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,$1;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>] <b>means </b> <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> $1;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>] <b>means </b>for <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>] <b>means </b>for <font color="Olive" title="b1">o</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">ts</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">ts</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b2">ts</font> holds <br/>(  <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Olive" title="b2">ts</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b1">o</font> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Olive" title="b2">ts</font></span>)</span> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Olive" title="b1">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Olive" title="b2">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Olive" title="b2">ts</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Olive" title="b1">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Olive" title="b2">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Olive" title="b2">ts</font></span>)</span> );<br/>
<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>] <b>means </b> ex <font color="Olive" title="b1">t1</font> being   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> st <br/>( <font color="Olive" title="b1">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t1</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">t1</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">t1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">t1</font>] );<br/>
<a NAME="E1:86"/><i><font color="Green" title="E59">A1</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> holds <br/><font color="Maroon">S<sub>5</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b1">s</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">sy</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:86_1"/><i><font color="Green" title="E60">A2</font></i>: 
<font color="Maroon" title="c10">sy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c11">t1</font> =  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c10">sy</font> as    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="txt" href="osafree.html#E1:86_1"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="dtconstr.html#D4" title="DTCONSTR:def.4">DTCONSTR:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c11">t1</font>
;<br/>

<b>thus </b><a NAME="E3:86_1"/>
<font color="Maroon" title="c11">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c10">sy</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c12">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:86_1"/><i><font color="Green" title="E61">A3</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font> &amp; <font color="Maroon" title="c10">sy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c12">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="osafree.html#E1:86_1"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="osafree.html#T4" target="_self" title="OSAFREE:th.4">Th4</a></i>;<br/>
<i><font color="Green" title="E62">A4</font></i>: <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c10">sy</font></span>)</span> = 
 <a href="osafree.html#K14" title="OSAFREE:func.14">pi</a> <font color="Maroon" title="c10">sy</font>
<b>by </b><i><a class="txt" href="osafree.html#E1:86_1"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="osafree.html#D32" target="_self" title="OSAFREE:def.32">Def32</a></i>
<br/>.= 
 <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c10">sy</font>
<b>by </b><i><a class="txt" href="osafree.html#E1:86_1"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="osafree.html#D17" target="_self" title="OSAFREE:def.17">Def17</a></i>
;<br/>
<b>hence </b><a NAME="E7:86_1"/>
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c11">t1</font>
 <b>by </b><i><a class="ref" href="osafree.html#T33" target="_self" title="OSAFREE:th.33">Th33</a></i>;<br/>

<b>thus </b><a NAME="E8:86_1"/>
 <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t1</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c11">t1</font>
 <b>by </b><i><a class="txt" href="osafree.html#E6:86_1"><i><font color="Green" title="E62">A4</font></i></a></i>;<br/>

<b>thus </b><a NAME="E9:86_1"/>
for <font color="Olive" title="b1">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s1</font> &amp; <font color="Maroon" title="c11">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x1</font>,<font color="Olive" title="b1">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">t1</font>
 <b>by </b><i><a class="txt" href="osafree.html#E6:86_1"><i><font color="Green" title="E62">A4</font></i></a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c14">o</font> be   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c15">ts</font> be    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/><b>assume </b><a NAME="E1:86_1_2"/><i><font color="Green" title="E63">A5</font></i>: 
(  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c14">o</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c15">ts</font> &amp; <font color="Maroon" title="c11">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c14">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c15">ts</font> )
 ;<br/><a NAME="E2:86_1_2"/>
<font color="Maroon" title="c11">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="osafree.html#E1:86_1_2"><i><font color="Green" title="E63">A5</font></i></a>, <a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></i>;<br/><b>hence </b><a NAME="E3:86_1_2"/>
(  <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c15">ts</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c14">o</font> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c14">o</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c15">ts</font></span>)</span> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c14">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c15">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c15">ts</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c14">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c15">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c15">ts</font></span>)</span> )
 <b>by </b><i><a class="txt" href="osafree.html#E5:86_1"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="osafree.html#T10" target="_self" title="OSAFREE:th.10">Th10</a></i>;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E2:86"/><i><font color="Green" title="E60">A6</font></i>: 
for <font color="Olive" title="b1">nt</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span><br/> for <font color="Olive" title="b2">ts1</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">ts1</font> &amp; ( for <font color="Olive" title="b3">dt1</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b3">dt1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">ts1</font> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b3">dt1</font>] ) holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b2">ts1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">nt</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c11">ts1</font> be    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:86_2"/><i><font color="Green" title="E61">A7</font></i>: 
<font color="Maroon" title="c10">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c11">ts1</font>
 <b>and </b><br/><a NAME="E2:86_2"/><i><font color="Green" title="E62">A8</font></i>: 
for <font color="Olive" title="b1">dt1</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">dt1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">ts1</font> holds <br/> ex <font color="Olive" title="b2">t2</font> being   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> st <br/>( <font color="Olive" title="b2">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">dt1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">t2</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">t2</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">t2</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b2">t2</font>] )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c12">t1</font> = <font color="Maroon" title="c10">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c11">ts1</font> as    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="txt" href="osafree.html#E1:86_2"><i><font color="Green" title="E61">A7</font></i></a>, <a class="ref" href="osafree.html#T12" target="_self" title="OSAFREE:th.12">Th12</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">t1</font>
;<br/>

<b>thus </b><a NAME="E4:86_2"/>
<font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c11">ts1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c13">ta1</font> = <font color="Maroon" title="c12">t1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c12">t1</font></span>)</span> <b>by </b><i><a class="ref" href="osafree.html#D12" target="_self" title="OSAFREE:def.12">Def12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">o</font> being   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E7:86_2"/><i><font color="Green" title="E63">A9</font></i>: 
( <font color="Maroon" title="c10">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">ts1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c14">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> &amp; <font color="Maroon" title="c10">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c11">ts1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c14">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">ts1</font> &amp; ( for <font color="Olive" title="b1">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Maroon" title="c10">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c11">ts1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s1</font> iff  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c14">o</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">s1</font> ) ) )
 <b>by </b><i><a class="txt" href="osafree.html#E1:86_2"><i><font color="Green" title="E61">A7</font></i></a>, <a class="ref" href="osafree.html#T12" target="_self" title="OSAFREE:th.12">Th12</a></i>;<br/>
<i><font color="Green" title="E64">A10</font></i>:  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c12">t1</font> = 
 <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c13">ta1</font>
<b>by </b><i><a class="ref" href="osafree.html#D28" target="_self" title="OSAFREE:def.28">Def28</a></i>
<br/>.= 
 <a href="funct_5.html#K1" title="FUNCT_5:func.1">proj1</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font></span>)</span>
<b>by </b><i><a class="ref" href="osafree.html#T26" target="_self" title="OSAFREE:th.26">Th26</a></i>
;<br/>
<b>set </b><font color="Maroon" title="c15">Ms</font> = <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font>;<br/>
<b>set </b><font color="Maroon" title="c16">LSMs</font> =  <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c17">w</font> =  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c14">o</font>;<br/>
<b>set </b><font color="Maroon" title="c18">so</font> =  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c14">o</font>;<br/>
<b>set </b><font color="Maroon" title="c19">Lo</font> =  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c14">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span>;<br/>
<a NAME="E9:86_2"/><i><font color="Green" title="E65">A11</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">ts1</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">ts1</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">ts1</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">ts1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c14">o</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="osafree.html#E7:86_2"><i><font color="Green" title="E63">A9</font></i></a>, <a class="ref" href="osafree.html#D14" target="_self" title="OSAFREE:def.14">Def14</a>, <a class="ref" href="alg_1.html#T1" title="ALG_1:th.1">ALG_1:1</a>, <a class="ref" href="msualg_3.html#T6" title="MSUALG_3:th.6">MSUALG_3:6</a></i>;<br/>
<a NAME="E10:86_2"/><b>then </b><i><font color="Green" title="E66">A12</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">ts1</font>
 
<b>by </b><i><a class="ref" href="osafree.html#D14" target="_self" title="OSAFREE:def.14">Def14</a></i>;<br/>
<a NAME="E11:86_2"/><i><font color="Green" title="E67">A13</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">ts1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E12:86_2"/>
 <a href="osafree.html#K13" title="OSAFREE:func.13">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c10">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">o</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E1:86_2"><i><font color="Green" title="E61">A7</font></i></a>, <a class="txt" href="osafree.html#E7:86_2"><i><font color="Green" title="E63">A9</font></i></a>, <a class="ref" href="osafree.html#D16" target="_self" title="OSAFREE:def.16">Def16</a></i>;<br/>
<a NAME="E13:86_2"/><b>then </b><i><font color="Green" title="E68">A14</font></i>: 
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osafree.html#K12" title="OSAFREE:func.12">pi</a> <font color="Maroon" title="c14">o</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="osafree.html#E1:86_2"><i><font color="Green" title="E61">A7</font></i></a>, <a class="ref" href="osafree.html#D32" target="_self" title="OSAFREE:def.32">Def32</a></i>;<br/>
<a NAME="E14:86_2"/><i><font color="Green" title="E69">A15</font></i>: 
<font color="Maroon" title="c10">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c14">o</font>,<font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E7:86_2"><i><font color="Green" title="E63">A9</font></i></a></i>;<br/>
<a NAME="E15:86_2"/><i><font color="Green" title="E70">A16</font></i>: 
 <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c14">o</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E7:86_2"><i><font color="Green" title="E63">A9</font></i></a>, <a class="ref" href="osafree.html#T18" target="_self" title="OSAFREE:th.18">Th18</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E17:86_2"/><i><font color="Green" title="E71">A17</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">ts1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c20">l</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/>
<i><font color="Green" title="E72">A18</font></i>: <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> = 
 <a href="osafree.html#K18" title="OSAFREE:func.18">@</a> <font color="Maroon" title="c10">nt</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span>
<b>by </b><i><a class="txt" href="osafree.html#E1:86_2"><i><font color="Green" title="E61">A7</font></i></a>, <a class="ref" href="osafree.html#D22" target="_self" title="OSAFREE:def.22">Def22</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b1">o2</font>,<span class="p5">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x2</font></span>)</span>,<font color="Olive" title="b3">s3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">o2</font> is   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>, <font color="Olive" title="b2">x2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Olive" title="b1">o2</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>, <font color="Olive" title="b3">s3</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> : (  ex <font color="Olive" title="b4">o1</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Maroon" title="c10">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">o1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b4">o1</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Olive" title="b1">o2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b4">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b1">o2</font></span>)</span> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b4">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">s3</font> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b1">o2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">s3</font> ) &amp;  ex <font color="Olive" title="b4">w3</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b4">w3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> &amp; ( for <font color="Olive" title="b5">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b5">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">y</font> ) ) ) </span>}</span> 

;<br/>
<a NAME="E19:86_2"/><i><font color="Green" title="E73">A19</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c12">t1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">o2</font> be   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c22">ts</font> be    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:86_2_3"/><i><font color="Green" title="E74">A20</font></i>: 
(  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c21">o2</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c22">ts</font> &amp; <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c21">o2</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c22">ts</font> )
 ;<br/>

<a NAME="E2:86_2_3"/><i><font color="Green" title="E75">A21</font></i>: 
(  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c21">o2</font>,<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">nt</font> &amp; <font color="Maroon" title="c22">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">ts1</font> )
 
<b>by </b><i><a class="txt" href="osafree.html#E1:86_2_3"><i><font color="Green" title="E74">A20</font></i></a>, <a class="ref" href="trees_4.html#T15" title="TREES_4:th.15">TREES_4:15</a></i>;<br/>
<a NAME="E3:86_2_3"/><b>then </b><i><font color="Green" title="E76">A22</font></i>: 
<font color="Maroon" title="c21">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">o</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E7:86_2"><i><font color="Green" title="E63">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>set </b><font color="Maroon" title="c23">Ms1</font> = <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c22">ts</font>;<br/>
<b>set </b><font color="Maroon" title="c24">LSMs1</font> =  <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c22">ts</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c25">Lo2</font> =  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c21">o2</font>,<span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c22">ts</font></span>)</span></span>)</span>;<br/>
<a NAME="E4:86_2_3"/><i><font color="Green" title="E77">A23</font></i>: 
 <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c14">o</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2_3_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:86_2_3_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c14">o</font></span>)</span>
 <b>by </b><i><a class="txt" href="osafree.html#E9:86_2"><i><font color="Green" title="E65">A11</font></i></a>, <a class="txt" href="osafree.html#E10:86_2"><i><font color="Green" title="E66">A12</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_1.html#D7" title="OSALG_1:def.7">OSALG_1:def 7</a><br/>

<b>let </b><font color="Maroon" title="c26">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E2:86_2_3_1"/><i><font color="Green" title="E78">A24</font></i>: 
<font color="Maroon" title="c26">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c27">k</font> = <font color="Maroon" title="c26">i</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="osafree.html#E2:86_2_3_1"><i><font color="Green" title="E78">A24</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E4:86_2_3_1"/><i><font color="Green" title="E79">A25</font></i>: 
<font color="Maroon" title="c11">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">ts1</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E10:86_2"><i><font color="Green" title="E66">A12</font></i></a>, <a class="txt" href="osafree.html#E2:86_2_3_1"><i><font color="Green" title="E78">A24</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c28">tr</font> = <font color="Maroon" title="c11">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k</font> as    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="txt" href="osafree.html#E11:86_2"><i><font color="Green" title="E67">A13</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c29">tr1</font> being    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E7:86_2_3_1"/><i><font color="Green" title="E80">A26</font></i>: 
( <font color="Maroon" title="c29">tr1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">tr</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c29">tr1</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c29">tr1</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c29">tr1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c29">tr1</font>] )
 <b>by </b><i><a class="txt" href="osafree.html#E2:86_2"><i><font color="Green" title="E62">A8</font></i></a>, <a class="txt" href="osafree.html#E4:86_2_3_1"><i><font color="Green" title="E79">A25</font></i></a></i>;<br/>
<b>let </b><font color="Maroon" title="c30">s1</font>, <font color="Maroon" title="c31">s2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/>

<b>assume </b><a NAME="E8:86_2_3_1"/><i><font color="Green" title="E81">A27</font></i>: 
( <font color="Maroon" title="c30">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font> &amp; <font color="Maroon" title="c31">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c14">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c32">t3</font> being    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E10:86_2_3_1"/><i><font color="Green" title="E82">A28</font></i>: 
( <font color="Maroon" title="c32">t3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k</font> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c32">t3</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E9:86_2"><i><font color="Green" title="E65">A11</font></i></a>, <a class="txt" href="osafree.html#E10:86_2"><i><font color="Green" title="E66">A12</font></i></a>, <a class="txt" href="osafree.html#E2:86_2_3_1"><i><font color="Green" title="E78">A24</font></i></a>, <a class="ref" href="osafree.html#D14" target="_self" title="OSAFREE:def.14">Def14</a></i>;<br/>
<a NAME="E11:86_2_3_1"/><i><font color="Green" title="E83">A29</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">tr</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E9:86_2"><i><font color="Green" title="E65">A11</font></i></a>, <a class="txt" href="osafree.html#E10:86_2"><i><font color="Green" title="E66">A12</font></i></a>, <a class="txt" href="osafree.html#E2:86_2_3_1"><i><font color="Green" title="E78">A24</font></i></a>, <a class="ref" href="alg_1.html#T1" title="ALG_1:th.1">ALG_1:1</a></i>;<br/>
<a NAME="E12:86_2_3_1"/>
<font color="Maroon" title="c11">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c14">o</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c27">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="osafree.html#E7:86_2"><i><font color="Green" title="E63">A9</font></i></a>, <a class="txt" href="osafree.html#E9:86_2"><i><font color="Green" title="E65">A11</font></i></a>, <a class="txt" href="osafree.html#E10:86_2"><i><font color="Green" title="E66">A12</font></i></a>, <a class="txt" href="osafree.html#E2:86_2_3_1"><i><font color="Green" title="E78">A24</font></i></a>, <a class="ref" href="msualg_6.html#T2" title="MSUALG_6:th.2">MSUALG_6:2</a></i>;<br/>
<a NAME="E13:86_2_3_1"/><b>then </b><i><font color="Green" title="E84">A30</font></i>: 
 <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c28">tr</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c14">o</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c27">k</font>
 
<b>by </b><i><a class="ref" href="osafree.html#D12" target="_self" title="OSAFREE:def.12">Def12</a></i>;<br/>
<a NAME="E14:86_2_3_1"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c14">o</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c27">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c14">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E9:86_2"><i><font color="Green" title="E65">A11</font></i></a>, <a class="txt" href="osafree.html#E10:86_2"><i><font color="Green" title="E66">A12</font></i></a>, <a class="txt" href="osafree.html#E2:86_2_3_1"><i><font color="Green" title="E78">A24</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E15:86_2_3_1"/>
<font color="Maroon" title="c30">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c31">s2</font>
 <b>by </b><i><a class="txt" href="osafree.html#E7:86_2_3_1"><i><font color="Green" title="E80">A26</font></i></a>, <a class="txt" href="osafree.html#E8:86_2_3_1"><i><font color="Green" title="E81">A27</font></i></a>, <a class="txt" href="osafree.html#E10:86_2_3_1"><i><font color="Green" title="E82">A28</font></i></a>, <a class="txt" href="osafree.html#E11:86_2_3_1"><i><font color="Green" title="E83">A29</font></i></a>, <a class="txt" href="osafree.html#E13:86_2_3_1"><i><font color="Green" title="E84">A30</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:86_2_3"/>
 <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c22">ts</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o2</font>
 <b>by </b><i><a class="txt" href="osafree.html#E7:86_2"><i><font color="Green" title="E63">A9</font></i></a>, <a class="txt" href="osafree.html#E2:86_2_3"><i><font color="Green" title="E75">A21</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>

<a NAME="E6:86_2_3"/>
 <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c21">o2</font>,<span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c22">ts</font></span>)</span></span>)</span> <a href="osalg_1.html#R5" title="OSALG_1:pred.5">has_least_rank_for</a> <font color="Maroon" title="c21">o2</font>, <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c22">ts</font></span>)</span>
 
<b>by </b><i><a class="txt" href="osafree.html#E2:86_2_3"><i><font color="Green" title="E75">A21</font></i></a>, <a class="txt" href="osafree.html#E3:86_2_3"><i><font color="Green" title="E76">A22</font></i></a>, <a class="txt" href="osafree.html#E4:86_2_3"><i><font color="Green" title="E77">A23</font></i></a>, <a class="ref" href="osalg_1.html#T14" title="OSALG_1:th.14">OSALG_1:14</a></i>;<br/>
<a NAME="E7:86_2_3"/><b>then </b>
(  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c21">o2</font>,<span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c22">ts</font></span>)</span></span>)</span> <a href="osalg_1.html#R3" title="OSALG_1:pred.3">has_least_args_for</a> <font color="Maroon" title="c21">o2</font>, <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c22">ts</font></span>)</span> &amp;  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c21">o2</font>,<span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c22">ts</font></span>)</span></span>)</span> <a href="osalg_1.html#R4" title="OSALG_1:pred.4">has_least_sort_for</a> <font color="Maroon" title="c21">o2</font>, <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c22">ts</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="osalg_1.html#D12" title="OSALG_1:def.12">OSALG_1:def 12</a></i>;<br/>
<a NAME="E8:86_2_3"/><b>then </b>
( <font color="Maroon" title="c21">o2</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a>  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c21">o2</font>,<span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c22">ts</font></span>)</span></span>)</span> &amp;  <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c22">ts</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c21">o2</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c22">ts</font></span>)</span></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">o3</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>  st <font color="Maroon" title="c21">o2</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Olive" title="b1">o3</font> &amp;  <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c22">ts</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b1">o3</font> holds <br/> <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c21">o2</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c22">ts</font></span>)</span></span>)</span></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b1">o3</font> ) )
 
<b>by </b><i><a class="ref" href="osalg_1.html#D10" title="OSALG_1:def.10">OSALG_1:def 10</a></i>;<br/>
<a NAME="E9:86_2_3"/><b>then </b>
( <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c22">ts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c21">o2</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c22">ts</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c22">ts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c21">o2</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="osafree.html#E2:86_2_3"><i><font color="Green" title="E75">A21</font></i></a>, <a class="txt" href="osafree.html#E3:86_2_3"><i><font color="Green" title="E76">A22</font></i></a>, <a class="txt" href="osafree.html#E4:86_2_3"><i><font color="Green" title="E77">A23</font></i></a>, <a class="ref" href="osafree.html#T19" target="_self" title="OSAFREE:th.19">Th19</a></i>;<br/>
<b>hence </b><a NAME="E10:86_2_3"/>
(  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c21">o2</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c22">ts</font></span>)</span> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c21">o2</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c22">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c22">ts</font></span>)</span> )
 <b>by </b><i><a class="ref" href="osafree.html#T13" target="_self" title="OSAFREE:th.13">Th13</a></i>;<br/>

<b>hence </b><a NAME="E11:86_2_3"/>
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c21">o2</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c22">ts</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c22">ts</font></span>)</span>
 <b>by </b><i><a class="txt" href="osafree.html#E13:86_2"><i><font color="Green" title="E68">A14</font></i></a>, <a class="txt" href="osafree.html#E2:86_2_3"><i><font color="Green" title="E75">A21</font></i></a>, <a class="txt" href="osafree.html#E3:86_2_3"><i><font color="Green" title="E76">A22</font></i></a>, <a class="ref" href="osafree.html#D15" target="_self" title="OSAFREE:def.15">Def15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:86_2"/><b>then </b><i><font color="Green" title="E74">A31</font></i>: 
(  <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c14">o</font> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c14">o</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c14">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c14">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="osafree.html#E1:86_2"><i><font color="Green" title="E61">A7</font></i></a>, <a class="txt" href="osafree.html#E14:86_2"><i><font color="Green" title="E69">A15</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">o3</font> being   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E22:86_2"/><i><font color="Green" title="E75">A32</font></i>: 
(  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c14">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">o3</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c21">o3</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c14">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c21">o3</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> &amp; ( for <font color="Olive" title="b1">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c14">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s1</font> iff  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c21">o3</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">s1</font> ) ) )
 <b>by </b><i><a class="ref" href="osafree.html#T12" target="_self" title="OSAFREE:th.12">Th12</a></i>;<br/>
<a NAME="E23:86_2"/><i><font color="Green" title="E76">A33</font></i>: 
 <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c14">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">o3</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E22:86_2"><i><font color="Green" title="E75">A32</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">Msr</font> = <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c14">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="txt" href="osafree.html#E22:86_2"><i><font color="Green" title="E75">A32</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E25:86_2"/><i><font color="Green" title="E77">A34</font></i>: 
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c14">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">Msr</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E1:86_2"><i><font color="Green" title="E61">A7</font></i></a>, <a class="txt" href="osafree.html#E14:86_2"><i><font color="Green" title="E69">A15</font></i></a>, <a class="txt" href="osafree.html#E19:86_2"><i><font color="Green" title="E73">A19</font></i></a>, <a class="txt" href="osafree.html#E22:86_2"><i><font color="Green" title="E75">A32</font></i></a>, <a class="txt" href="osafree.html#E23:86_2"><i><font color="Green" title="E76">A33</font></i></a></i>;<br/>
<a NAME="E26:86_2"/><i><font color="Green" title="E78">A35</font></i>: 
 <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c14">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="osafree.html#E20:86_2"><i><font color="Green" title="E74">A31</font></i></a>, <a class="txt" href="osafree.html#E22:86_2"><i><font color="Green" title="E75">A32</font></i></a>, <a class="txt" href="osafree.html#E23:86_2"><i><font color="Green" title="E76">A33</font></i></a>, <a class="ref" href="osafree.html#T18" target="_self" title="OSAFREE:th.18">Th18</a></i>;<br/>
<a NAME="E27:86_2"/>
 <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c14">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span> <a href="osalg_1.html#R5" title="OSALG_1:pred.5">has_least_rank_for</a> <font color="Maroon" title="c14">o</font>, <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="osafree.html#E20:86_2"><i><font color="Green" title="E74">A31</font></i></a>, <a class="ref" href="osalg_1.html#T14" title="OSALG_1:th.14">OSALG_1:14</a></i>;<br/>
<a NAME="E28:86_2"/><b>then </b><i><font color="Green" title="E79">A36</font></i>: 
(  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c14">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span> <a href="osalg_1.html#R3" title="OSALG_1:pred.3">has_least_args_for</a> <font color="Maroon" title="c14">o</font>, <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> &amp;  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c14">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span> <a href="osalg_1.html#R4" title="OSALG_1:pred.4">has_least_sort_for</a> <font color="Maroon" title="c14">o</font>, <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="osalg_1.html#D12" title="OSALG_1:def.12">OSALG_1:def 12</a></i>;<br/>
<a NAME="E29:86_2"/><b>then </b><i><font color="Green" title="E80">A37</font></i>: 
( <font color="Maroon" title="c14">o</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a>  <a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c14">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span> &amp;  <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c14">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">o3</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>  st <font color="Maroon" title="c14">o</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Olive" title="b1">o3</font> &amp;  <a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b1">o3</font> holds <br/> <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c14">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b1">o3</font> ) )
 
<b>by </b><i><a class="ref" href="osalg_1.html#D11" title="OSALG_1:def.11">OSALG_1:def 11</a></i>;<br/>
<a NAME="E30:86_2"/><i><font color="Green" title="E81">A38</font></i>: 
(  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c14">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c14">o</font> &amp;  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c14">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span> <a href="osalg_1.html#R2" title="OSALG_1:pred.2">&lt;=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c14">o</font> )
 
<b>by </b><i><a class="txt" href="osafree.html#E20:86_2"><i><font color="Green" title="E74">A31</font></i></a>, <a class="txt" href="osafree.html#E28:86_2"><i><font color="Green" title="E79">A36</font></i></a>, <a class="ref" href="osalg_1.html#D10" title="OSALG_1:def.10">OSALG_1:def 10</a>, <a class="ref" href="osalg_1.html#D11" title="OSALG_1:def.11">OSALG_1:def 11</a></i>;<br/>
<a NAME="E31:86_2"/><b>then </b><i><font color="Green" title="E82">A39</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c14">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c14">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="osalg_1.html#D7" title="OSALG_1:def.7">OSALG_1:def 7</a></i>;<br/>
<a NAME="E32:86_2"/><i><font color="Green" title="E83">A40</font></i>: 
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c12">t1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2_4"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>6</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>,$2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<a NAME="E1:86_2_4"/><i><font color="Green" title="E84">A41</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">s4</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">s4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">s4</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:86_2_4_1"/><i><font color="Green" title="E85">A42</font></i>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span>
 ;<br/>

<a NAME="E2:86_2_4_1"/><i><font color="Green" title="E86">A43</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span> &amp; <font color="Maroon" title="c11">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">ts1</font> )
 
<b>by </b><i><a class="txt" href="osafree.html#E9:86_2"><i><font color="Green" title="E65">A11</font></i></a>, <a class="txt" href="osafree.html#E1:86_2_4_1"><i><font color="Green" title="E85">A42</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c24">td1</font> = <font color="Maroon" title="c11">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> as    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="txt" href="osafree.html#E11:86_2"><i><font color="Green" title="E67">A13</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">tda</font> = <font color="Maroon" title="c24">td1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c24">td1</font></span>)</span> <b>by </b><i><a class="ref" href="osafree.html#D12" target="_self" title="OSAFREE:def.12">Def12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c26">td2</font> being    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E6:86_2_4_1"/><i><font color="Green" title="E87">A44</font></i>: 
( <font color="Maroon" title="c26">td2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">td1</font> &amp; <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">td2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c26">td2</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c26">td2</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c26">td2</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c26">td2</font>] )
 <b>by </b><i><a class="txt" href="osafree.html#E2:86_2"><i><font color="Green" title="E62">A8</font></i></a>, <a class="txt" href="osafree.html#E2:86_2_4_1"><i><font color="Green" title="E86">A43</font></i></a></i>;<br/>
 <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c24">td1</font> = 
 <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c25">tda</font>
<b>by </b><i><a class="ref" href="osafree.html#D28" target="_self" title="OSAFREE:def.28">Def28</a></i>
<br/>.= 
 <a href="funct_5.html#K1" title="FUNCT_5:func.1">proj1</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span>
<b>by </b><i><a class="txt" href="osafree.html#E2:86_2_4_1"><i><font color="Green" title="E86">A43</font></i></a>, <a class="ref" href="osafree.html#T26" target="_self" title="OSAFREE:th.26">Th26</a></i>
;<br/>
<b>then consider </b><font color="Maroon" title="c27">s4</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:86_2_4_1"/><i><font color="Green" title="E88">A45</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span>,<font color="Maroon" title="c27">s4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
 <b>by </b><i><a class="txt" href="osafree.html#E2:86_2_4_1"><i><font color="Green" title="E86">A43</font></i></a>, <a class="txt" href="osafree.html#E6:86_2_4_1"><i><font color="Green" title="E87">A44</font></i></a>, <a class="ref" href="funct_5.html#D1" title="FUNCT_5:def.1">FUNCT_5:def 1</a></i>;<br/>
<a NAME="E10:86_2_4_1"/>
<font color="Maroon" title="c27">s4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E2:86_2_4_1"><i><font color="Green" title="E86">A43</font></i></a>, <a class="txt" href="osafree.html#E9:86_2_4_1"><i><font color="Green" title="E88">A45</font></i></a>, <a class="ref" href="osafree.html#T24" target="_self" title="OSAFREE:th.24">Th24</a></i>;<br/>
<b>hence </b><a NAME="E11:86_2_4_1"/>
 ex <font color="Olive" title="b1">s4</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">s4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c23">i</font>,<font color="Olive" title="b1">s4</font>] )
 <b>by </b><i><a class="txt" href="osafree.html#E9:86_2_4_1"><i><font color="Green" title="E88">A45</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c23">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:86_2_4"/><i><font color="Green" title="E85">A46</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c23">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp; ( for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> holds <br/><font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c23">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font>] ) )
 <b>from </b><i><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="osafree.html#E1:86_2_4"><i><font color="Green" title="E84">A41</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c24">wa</font> = <font color="Maroon" title="c23">f</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="txt" href="osafree.html#E9:86_2"><i><font color="Green" title="E65">A11</font></i></a>, <a class="txt" href="osafree.html#E17:86_2"><i><font color="Green" title="E71">A17</font></i></a>, <a class="txt" href="osafree.html#E3:86_2_4"><i><font color="Green" title="E85">A46</font></i></a>, <a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">wa</font> = <font color="Maroon" title="c24">wa</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="osafree.html#E3:86_2_4"><i><font color="Green" title="E85">A46</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c26">wa</font> = <font color="Maroon" title="c25">wa</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E7:86_2_4"/>
for <font color="Olive" title="b1">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">Msr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">wa</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c27">y</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:86_2_4_2"/><i><font color="Green" title="E86">A47</font></i>: 
<font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span>
 ;<br/>

<a NAME="E2:86_2_4_2"/>
<font color="Maroon" title="c26">wa</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c27">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">wa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E3:86_2_4"><i><font color="Green" title="E85">A46</font></i></a>, <a class="txt" href="osafree.html#E1:86_2_4_2"><i><font color="Green" title="E86">A47</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:86_2_4_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">Msr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">wa</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c27">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font>
 <b>by </b><i><a class="txt" href="osafree.html#E3:86_2_4"><i><font color="Green" title="E85">A46</font></i></a>, <a class="txt" href="osafree.html#E1:86_2_4_2"><i><font color="Green" title="E86">A47</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:86_2_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p4">(<span class="default"><a href="osalg_1.html#K3" title="OSALG_1:func.3">LBound</a> <font color="Maroon" title="c14">o</font>,<span class="p5">(<span class="default"><a href="osafree.html#K11" title="OSAFREE:func.11">LeastSorts</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c11">ts1</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">Msr</font></span>)</span>,<span class="p2">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c14">o</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E7:86_2"><i><font color="Green" title="E63">A9</font></i></a>, <a class="txt" href="osafree.html#E18:86_2"><i><font color="Green" title="E72">A18</font></i></a>, <a class="txt" href="osafree.html#E29:86_2"><i><font color="Green" title="E80">A37</font></i></a>, <a class="txt" href="osafree.html#E30:86_2"><i><font color="Green" title="E81">A38</font></i></a>, <a class="txt" href="osafree.html#E31:86_2"><i><font color="Green" title="E82">A39</font></i></a>, <a class="txt" href="osafree.html#E3:86_2_4"><i><font color="Green" title="E85">A46</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:86_2_4"/>
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c12">t1</font>
 <b>by </b><i><a class="txt" href="osafree.html#E8:86_2"><i><font color="Green" title="E64">A10</font></i></a>, <a class="txt" href="osafree.html#E25:86_2"><i><font color="Green" title="E77">A34</font></i></a>, <a class="ref" href="funct_5.html#T4" title="FUNCT_5:th.4">FUNCT_5:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E33:86_2"/>
for <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c24">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:86_2_5"/><i><font color="Green" title="E84">A48</font></i>: 
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">s</font> &amp; <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">x</font>,<font color="Maroon" title="c23">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ;<br/>

<a NAME="E2:86_2_5"/>
<font color="Maroon" title="c12">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="osafree.html#E7:86_2"><i><font color="Green" title="E63">A9</font></i></a>, <a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:86_2_5"/>
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t1</font>
 <b>by </b><i><a class="txt" href="osafree.html#E1:86_2_5"><i><font color="Green" title="E84">A48</font></i></a>, <a class="ref" href="osafree.html#T10" target="_self" title="OSAFREE:th.10">Th10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E34:86_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">t1</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">t1</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c12">t1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c12">t1</font>] )
 <b>by </b><i><a class="txt" href="osafree.html#E15:86_2"><i><font color="Green" title="E70">A16</font></i></a>, <a class="txt" href="osafree.html#E19:86_2"><i><font color="Green" title="E73">A19</font></i></a>, <a class="txt" href="osafree.html#E20:86_2"><i><font color="Green" title="E74">A31</font></i></a>, <a class="txt" href="osafree.html#E26:86_2"><i><font color="Green" title="E78">A35</font></i></a>, <a class="txt" href="osafree.html#E28:86_2"><i><font color="Green" title="E79">A36</font></i></a>, <a class="txt" href="osafree.html#E32:86_2"><i><font color="Green" title="E83">A40</font></i></a>, <a class="ref" href="osalg_1.html#D11" title="OSALG_1:def.11">OSALG_1:def 11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:86"/>
for <font color="Olive" title="b1">dt</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">dt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">dt</font>]
 
<b>from </b><i><a class="ref" href="dtconstr.html#S7" title="DTCONSTR:sch.7">DTCONSTR:sch 7</a>(<a class="txt" href="osafree.html#E1:86"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="osafree.html#E2:86"><i><font color="Green" title="E60">A6</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c10">t1</font> being    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E5:86"/><i><font color="Green" title="E61">A49</font></i>: 
( <font color="Maroon" title="c10">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">t1</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">t1</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">t1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c10">t1</font>] )
 ;<br/>
<b>thus </b><a NAME="E6:86"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">t</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">t</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c3">t</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c3">t</font>] )
 <b>by </b><i><a class="txt" href="osafree.html#E5:86"><i><font color="Green" title="E61">A49</font></i></a></i>;<br/>


</div>
