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

<b>let </b><font color="Maroon" title="c1">S</font> be  <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>set </b><font color="Maroon" title="c3">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="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">C</font> =  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p3">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>,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="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c6">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="c7">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="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <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">y</font>, <font color="Olive" title="b3">z</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 <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">s</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"><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> $1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">z</font>,<font color="Olive" title="b1">s</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"><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="Olive" title="b2">y</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">$1,<font color="Olive" title="b1">s</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"><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="Olive" title="b3">z</font>;<br/>
<a NAME="E1:53"/><i><font color="Green" title="E34">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>1</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="53_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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="E1:53_1"/><i><font color="Green" title="E35">A2</font></i>: 
<font color="Maroon" title="c8">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/>

<i><font color="Green" title="E36">A3</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_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="c8">sy</font></span>)</span> = 
 <a href="osafree.html#K17" title="OSAFREE:func.17">@</a> <font color="Maroon" title="c8">sy</font>
<b>by </b><i><a class="txt" href="osafree.html#E1:53_1"><i><font color="Green" title="E35">A2</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"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c8">sy</font></span>)</span>,<font color="Olive" title="b1">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">s1</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="b2">s2</font> being  <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="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b3">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="b2">s2</font> &amp; <font color="Maroon" title="c8">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="Olive" title="b3">x</font>,<font color="Olive" title="b2">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">s2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">s1</font> ) </span>}</span> 

;<br/>
<b>thus </b><a NAME="E3:53_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c8">sy</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">s1</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="c10">y</font>, <font color="Maroon" title="c11">z</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>assume </b><a NAME="E1:53_1_2"/><i><font color="Green" title="E37">A4</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c9">s1</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"><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"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c8">sy</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c9">s1</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"><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="c10">y</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">s2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E3:53_1_2"/><i><font color="Green" title="E38">A5</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c9">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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"><span class="p2">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c8">sy</font></span>)</span>,<font color="Maroon" title="c12">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E4:53_1_2"/>
 ex <font color="Olive" title="b1">s0</font> being  <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="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <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">s0</font> &amp; <font color="Maroon" title="c8">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="Olive" title="b2">x</font>,<font color="Olive" title="b1">s0</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">s0</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">s2</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E2:53_1"><i><font color="Green" title="E36">A3</font></i></a></i>;<br/>
<a NAME="E5:53_1_2"/><i><font color="Green" title="E39">A6</font></i>: 
( <font color="Maroon" title="c10">y</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="c8">sy</font> &amp; <font color="Maroon" title="c9">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s2</font> )
 
<b>by </b><i><a class="txt" href="osafree.html#E3:53_1_2"><i><font color="Green" title="E38">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">s3</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E7:53_1_2"/><i><font color="Green" title="E40">A7</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c9">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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"><span class="p2">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c8">sy</font></span>)</span>,<font color="Maroon" title="c13">s3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E8:53_1_2"/>
 ex <font color="Olive" title="b1">s0</font> being  <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="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <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">s0</font> &amp; <font color="Maroon" title="c8">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="Olive" title="b2">x</font>,<font color="Olive" title="b1">s0</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">s0</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">s3</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E2:53_1"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="osafree.html#E1:53_1_2"><i><font color="Green" title="E37">A4</font></i></a></i>;<br/>
<b>thus </b><a NAME="E9:53_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c8">sy</font></span>)</span>,<font color="Maroon" title="c9">s1</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"><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="c11">z</font>
 <b>by </b><i><a class="txt" href="osafree.html#E1:53_1_2"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="osafree.html#E5:53_1_2"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="osafree.html#E7:53_1_2"><i><font color="Green" title="E40">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E2:53"/><i><font color="Green" title="E35">A8</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">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 <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">ts</font> &amp; ( for <font color="Olive" title="b3">t</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">t</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">ts</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">t</font>] ) holds <br/><font color="Maroon">S<sub>1</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">ts</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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="c9">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><b>that </b><br/><a NAME="E1:53_2"/><i><font color="Green" title="E36">A9</font></i>: 
<font color="Maroon" title="c8">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="c9">ts</font>
 <b>and </b><br/><a NAME="E2:53_2"/><i><font color="Green" title="E37">A10</font></i>: 
for <font color="Olive" title="b1">t</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">t</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="c9">ts</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>]
 ;<br/>

<b>consider </b><font color="Maroon" title="c10">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="E4:53_2"/><i><font color="Green" title="E38">A11</font></i>: 
( <font color="Maroon" title="c8">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="c10">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="c9">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="c10">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="c8">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c9">ts</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="c10">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="c9">ts</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="c8">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c9">ts</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="c10">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#E1:53_2"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="osafree.html#T12" target="_self" title="OSAFREE:th.12">Th12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">ts1</font> = <font color="Maroon" title="c9">ts</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> <font color="Maroon" title="c10">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> <b>by </b><i><a class="txt" href="osafree.html#E4:53_2"><i><font color="Green" title="E38">A11</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">w</font> =  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">x</font> = <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="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c9">ts</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p3">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>,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="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ;<br/>
<a NAME="E7:53_2"/><i><font color="Green" title="E39">A12</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">ts</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> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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="hidden.html#R1" title="HIDDEN:pred.1">=</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>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:53_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">ts</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T33" title="FINSEQ_2:th.33">FINSEQ_2:33</a></i>;<br/>
<a NAME="E9:53_2"/><b>then </b><i><font color="Green" title="E40">A13</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">x</font> <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="c9">ts</font> &amp;  <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="c10">o</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="c9">ts</font> )
 
<b>by </b><i><a class="txt" href="osafree.html#E4:53_2"><i><font color="Green" title="E38">A11</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a>, <a class="ref" href="msualg_3.html#T6" title="MSUALG_3:th.6">MSUALG_3:6</a></i>;<br/>
<i><font color="Green" title="E41">A14</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_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c9">ts</font></span>)</span> = 
 <a href="osafree.html#K18" title="OSAFREE:func.18">@</a> <font color="Maroon" title="c8">nt</font>,<font color="Maroon" title="c13">x</font>
<b>by </b><i><a class="txt" href="osafree.html#E1:53_2"><i><font color="Green" title="E36">A9</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="c8">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> <font color="Maroon" title="c13">x</font> &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> <font color="Maroon" title="c13">x</font> 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> <font color="Maroon" title="c13">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">y</font> ) ) ) </span>}</span> 

;<br/>
<b>thus </b><a NAME="E11:53_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c9">ts</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">s1</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="c15">y</font>, <font color="Maroon" title="c16">z</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>assume </b><a NAME="E1:53_2_2"/><i><font color="Green" title="E42">A15</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c14">s1</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"><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="c8">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c9">ts</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">z</font>,<font color="Maroon" title="c14">s1</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"><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="c15">y</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c17">o2</font> being   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c18">x2</font> being    <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="Maroon" title="c17">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="Maroon" title="c19">s3</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E3:53_2_2"/><i><font color="Green" title="E43">A16</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c14">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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"><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> <font color="Maroon" title="c17">o2</font>,<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="c18">x2</font></span>)</span>,<font color="Maroon" title="c19">s3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E4:53_2_2"/><i><font color="Green" title="E44">A17</font></i>: 
(  ex <font color="Olive" title="b1">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="c8">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="b1">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="b1">o1</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c17">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="b1">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="Maroon" title="c17">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="b1">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c19">s3</font> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c17">o2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c19">s3</font> ) &amp;  ex <font color="Olive" title="b1">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="b1">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> <font color="Maroon" title="c13">x</font> &amp; ( for <font color="Olive" title="b2">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">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> <font color="Maroon" title="c13">x</font> 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="c18">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">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> <font color="Maroon" title="c13">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> ) ) )
 <b>by </b><i><a class="txt" href="osafree.html#E10:53_2"><i><font color="Green" title="E41">A14</font></i></a></i>;<br/>
<a NAME="E5:53_2_2"/><i><font color="Green" title="E45">A18</font></i>: 
( <font color="Maroon" title="c15">y</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="c17">o2</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_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">x2</font> &amp; <font color="Maroon" title="c14">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">s3</font> )
 
<b>by </b><i><a class="txt" href="osafree.html#E3:53_2_2"><i><font color="Green" title="E43">A16</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">o1</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="E7:53_2_2"/><i><font color="Green" title="E46">A19</font></i>: 
( <font color="Maroon" title="c8">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="c20">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="Maroon" title="c20">o1</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c17">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="Maroon" title="c20">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="Maroon" title="c17">o2</font></span>)</span> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c20">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c19">s3</font> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c17">o2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c19">s3</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E4:53_2_2"><i><font color="Green" title="E44">A17</font></i></a></i>;<br/>
<a NAME="E8:53_2_2"/><i><font color="Green" title="E47">A20</font></i>: 
<font color="Maroon" title="c20">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">o</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E4:53_2"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="osafree.html#E7:53_2_2"><i><font color="Green" title="E46">A19</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E9:53_2_2"/><i><font color="Green" title="E48">A21</font></i>: 
( <font color="Maroon" title="c18">x2</font> is    <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> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c17">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> <font color="Maroon" title="c18">x2</font> )
 
<b>by </b><i><a class="ref" href="osafree.html#T13" target="_self" title="OSAFREE:th.13">Th13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">x3</font> = <font color="Maroon" title="c18">x2</font> as    <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> <b>by </b><i><a class="ref" href="osafree.html#T13" target="_self" title="OSAFREE:th.13">Th13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">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="E12:53_2_2"/><i><font color="Green" title="E49">A22</font></i>: 
(  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c17">o2</font>,<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="c22">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; <font color="Maroon" title="c21">x3</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="c22">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> <font color="Maroon" title="c17">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="c21">x3</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="c22">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> <font color="Maroon" title="c21">x3</font> &amp; ( for <font color="Olive" title="b1">s2</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> <font color="Maroon" title="c17">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="c21">x3</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">s2</font> iff  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c22">o3</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">s2</font> ) ) )
 <b>by </b><i><a class="txt" href="osafree.html#E9:53_2_2"><i><font color="Green" title="E48">A21</font></i></a>, <a class="ref" href="osafree.html#T12" target="_self" title="OSAFREE:th.12">Th12</a></i>;<br/>
<a NAME="E13:53_2_2"/><i><font color="Green" title="E50">A23</font></i>: 
<font color="Maroon" title="c17">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">o3</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E12:53_2_2"><i><font color="Green" title="E49">A22</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E14:53_2_2"/><i><font color="Green" title="E51">A24</font></i>: 
 <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="c17">o2</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> <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="c10">o</font></span>)</span>
 
<b>by </b><i><a class="txt" href="osafree.html#E7:53_2_2"><i><font color="Green" title="E46">A19</font></i></a>, <a class="txt" href="osafree.html#E8:53_2_2"><i><font color="Green" title="E47">A20</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c23">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> <b> such that </b><br/><a NAME="E16:53_2_2"/><i><font color="Green" title="E52">A25</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">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> <font color="Maroon" title="c13">x</font> &amp; ( 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> <font color="Maroon" title="c13">x</font> 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="c18">x2</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="c23">w3</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> <font color="Maroon" title="c13">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) )
 <b>by </b><i><a class="txt" href="osafree.html#E4:53_2_2"><i><font color="Green" title="E44">A17</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">xy</font> = <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="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c21">x3</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p3">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>,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="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ;<br/>
<a NAME="E18:53_2_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">x3</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="E19:53_2_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">x3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E20:53_2_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">xy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">x3</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T33" title="FINSEQ_2:th.33">FINSEQ_2:33</a></i>;<br/>
<a NAME="E21:53_2_2"/><b>then </b><i><font color="Green" title="E53">A26</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">x3</font> <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="c24">xy</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E22:53_2_2"/><i><font color="Green" title="E54">A27</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">x2</font> <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="c13">x</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E9:53_2"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="osafree.html#E14:53_2_2"><i><font color="Green" title="E51">A24</font></i></a>, <a class="ref" href="msualg_3.html#T6" title="MSUALG_3:th.6">MSUALG_3:6</a></i>;<br/>
<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="c15">y</font> = 
 <a href="osafree.html#K18" title="OSAFREE:func.18">@</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c17">o2</font>,<font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c24">xy</font>
<b>by </b><i><a class="txt" href="osafree.html#E5:53_2_2"><i><font color="Green" title="E45">A18</font></i></a>, <a class="txt" href="osafree.html#E9:53_2_2"><i><font color="Green" title="E48">A21</font></i></a>, <a class="txt" href="osafree.html#E12:53_2_2"><i><font color="Green" title="E49">A22</font></i></a>, <a class="txt" href="osafree.html#E13:53_2_2"><i><font color="Green" title="E50">A23</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">o4</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">x4</font></span>)</span>,<font color="Olive" title="b3">s4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">o4</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">x4</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">o4</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">s4</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/>(  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c17">o2</font>,<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="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">o4</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">o4</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">s4</font> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b1">o4</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">s4</font> ) &amp;  ex <font color="Olive" title="b4">w4</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">w4</font> <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="c24">xy</font> &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> <font color="Maroon" title="c24">xy</font> 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">x4</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">w4</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> <font color="Maroon" title="c24">xy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">y</font> ) ) ) </span>}</span> 

;<br/>
<b>then consider </b><font color="Maroon" title="c25">o5</font> being   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c26">x5</font> being    <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="Maroon" title="c25">o5</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="Maroon" title="c27">s5</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E25:53_2_2"/><i><font color="Green" title="E55">A28</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">z</font>,<font color="Maroon" title="c14">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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"><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> <font color="Maroon" title="c25">o5</font>,<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="c26">x5</font></span>)</span>,<font color="Maroon" title="c27">s5</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E26:53_2_2"/><i><font color="Green" title="E56">A29</font></i>: 
(  ex <font color="Olive" title="b1">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/>(  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c17">o2</font>,<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="Olive" title="b1">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="b1">o1</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c25">o5</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="b1">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="Maroon" title="c25">o5</font></span>)</span> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b1">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c27">s5</font> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c25">o5</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c27">s5</font> ) &amp;  ex <font color="Olive" title="b1">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="b1">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> <font color="Maroon" title="c24">xy</font> &amp; ( for <font color="Olive" title="b2">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">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> <font color="Maroon" title="c24">xy</font> 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="c26">x5</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">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> <font color="Maroon" title="c24">xy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> ) ) )
 <b>by </b><i><a class="txt" href="osafree.html#E1:53_2_2"><i><font color="Green" title="E42">A15</font></i></a></i>;<br/>
<a NAME="E27:53_2_2"/><i><font color="Green" title="E57">A30</font></i>: 
( <font color="Maroon" title="c16">z</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="c25">o5</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_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x5</font> &amp; <font color="Maroon" title="c14">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">s5</font> )
 
<b>by </b><i><a class="txt" href="osafree.html#E25:53_2_2"><i><font color="Green" title="E55">A28</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c28">o6</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="E29:53_2_2"/><i><font color="Green" title="E58">A31</font></i>: 
(  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c17">o2</font>,<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="c28">o6</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="c28">o6</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c25">o5</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="Maroon" title="c28">o6</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="Maroon" title="c25">o5</font></span>)</span> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c28">o6</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c27">s5</font> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c25">o5</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c27">s5</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E26:53_2_2"><i><font color="Green" title="E56">A29</font></i></a></i>;<br/>
<a NAME="E30:53_2_2"/><i><font color="Green" title="E59">A32</font></i>: 
<font color="Maroon" title="c28">o6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">o2</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E29:53_2_2"><i><font color="Green" title="E58">A31</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E31:53_2_2"/><i><font color="Green" title="E60">A33</font></i>: 
( <font color="Maroon" title="c26">x5</font> is    <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> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c25">o5</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> <font color="Maroon" title="c26">x5</font> )
 
<b>by </b><i><a class="ref" href="osafree.html#T13" target="_self" title="OSAFREE:th.13">Th13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c29">x6</font> = <font color="Maroon" title="c26">x5</font> as    <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> <b>by </b><i><a class="ref" href="osafree.html#T13" target="_self" title="OSAFREE:th.13">Th13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c30">o7</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="E34:53_2_2"/><i><font color="Green" title="E61">A34</font></i>: 
(  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c25">o5</font>,<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="c30">o7</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="c29">x6</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="c30">o7</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> <font color="Maroon" title="c25">o5</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="c29">x6</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="c30">o7</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="c29">x6</font> &amp; ( for <font color="Olive" title="b1">s2</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> <font color="Maroon" title="c25">o5</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="c29">x6</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">s2</font> iff  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c30">o7</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">s2</font> ) ) )
 <b>by </b><i><a class="txt" href="osafree.html#E31:53_2_2"><i><font color="Green" title="E60">A33</font></i></a>, <a class="ref" href="osafree.html#T12" target="_self" title="OSAFREE:th.12">Th12</a></i>;<br/>
<a NAME="E35:53_2_2"/><i><font color="Green" title="E62">A35</font></i>: 
<font color="Maroon" title="c25">o5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">o7</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E34:53_2_2"><i><font color="Green" title="E61">A34</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E36:53_2_2"/><i><font color="Green" title="E63">A36</font></i>: 
 <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="c25">o5</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> <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="c17">o2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="osafree.html#E29:53_2_2"><i><font color="Green" title="E58">A31</font></i></a>, <a class="txt" href="osafree.html#E30:53_2_2"><i><font color="Green" title="E59">A32</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c31">w5</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> <b> such that </b><br/><a NAME="E38:53_2_2"/><i><font color="Green" title="E64">A37</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c31">w5</font> <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="c24">xy</font> &amp; ( 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> <font color="Maroon" title="c24">xy</font> 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="c26">x5</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="c31">w5</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> <font color="Maroon" title="c24">xy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) )
 <b>by </b><i><a class="txt" href="osafree.html#E26:53_2_2"><i><font color="Green" title="E56">A29</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c32">xz</font> = <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="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c29">x6</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p3">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>,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="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ;<br/>
<a NAME="E40:53_2_2"/><i><font color="Green" title="E65">A38</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">x6</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> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">x3</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="E41:53_2_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">x6</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E42:53_2_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">xz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">x6</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T33" title="FINSEQ_2:th.33">FINSEQ_2:33</a></i>;<br/>
<a NAME="E43:53_2_2"/><b>then </b><i><font color="Green" title="E66">A39</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c29">x6</font> <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="c32">xz</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<i><font color="Green" title="E67">A40</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c26">x5</font> = 
 <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="c17">o2</font></span>)</span>
<b>by </b><i><a class="txt" href="osafree.html#E36:53_2_2"><i><font color="Green" title="E63">A36</font></i></a>, <a class="ref" href="msualg_3.html#T6" title="MSUALG_3:th.6">MSUALG_3:6</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c24">xy</font>
<b>by </b><i><a class="txt" href="osafree.html#E21:53_2_2"><i><font color="Green" title="E53">A26</font></i></a>, <a class="ref" href="msualg_3.html#T6" title="MSUALG_3:th.6">MSUALG_3:6</a></i>
;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</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"><font color="Maroon" title="c11">ts1</font> <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> <font color="Maroon" title="c32">xz</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<a NAME="E45:53_2_2"/><i><font color="Green" title="E68">A41</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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> <font color="Maroon" title="c32">xz</font> holds <br/> ex <font color="Olive" title="b2">sy</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">sy</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>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">sy</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c33">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:53_2_2_3"/><i><font color="Green" title="E69">A42</font></i>: 
<font color="Maroon" title="c33">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> <font color="Maroon" title="c32">xz</font>
 ;<br/>

<a NAME="E2:53_2_2_3"/><i><font color="Green" title="E70">A43</font></i>: 
( <font color="Maroon" title="c33">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> <font color="Maroon" title="c11">ts1</font> &amp; <font color="Maroon" title="c33">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> <font color="Maroon" title="c26">x5</font> &amp; <font color="Maroon" title="c33">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> <font color="Maroon" title="c18">x2</font> &amp; <font color="Maroon" title="c33">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> <font color="Maroon" title="c13">x</font> )
 
<b>by </b><i><a class="txt" href="osafree.html#E9:53_2"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="osafree.html#E14:53_2_2"><i><font color="Green" title="E51">A24</font></i></a>, <a class="txt" href="osafree.html#E21:53_2_2"><i><font color="Green" title="E53">A26</font></i></a>, <a class="txt" href="osafree.html#E43:53_2_2"><i><font color="Green" title="E66">A39</font></i></a>, <a class="txt" href="osafree.html#E44:53_2_2"><i><font color="Green" title="E67">A40</font></i></a>, <a class="txt" href="osafree.html#E1:53_2_2_3"><i><font color="Green" title="E69">A42</font></i></a>, <a class="ref" href="msualg_3.html#T6" title="MSUALG_3:th.6">MSUALG_3:6</a></i>;<br/>
<a NAME="E3:53_2_2_3"/><i><font color="Green" title="E71">A44</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="c33">y</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> &amp; <font color="Maroon" title="c26">x5</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">y</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="c29">x6</font> &amp; <font color="Maroon" title="c18">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">y</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="c21">x3</font> )
 
<b>by </b><i><a class="txt" href="osafree.html#E9:53_2"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="osafree.html#E21:53_2_2"><i><font color="Green" title="E53">A26</font></i></a>, <a class="txt" href="osafree.html#E22:53_2_2"><i><font color="Green" title="E54">A27</font></i></a>, <a class="txt" href="osafree.html#E43:53_2_2"><i><font color="Green" title="E66">A39</font></i></a>, <a class="txt" href="osafree.html#E44:53_2_2"><i><font color="Green" title="E67">A40</font></i></a>, <a class="txt" href="osafree.html#E1:53_2_2_3"><i><font color="Green" title="E69">A42</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="c34">t1</font> = <font color="Maroon" title="c11">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">y</font>, <font color="Maroon" title="c35">t2</font> = <font color="Maroon" title="c21">x3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">y</font>, <font color="Maroon" title="c36">t3</font> = <font color="Maroon" title="c26">x5</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">y</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#E7:53_2"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="osafree.html#E40:53_2_2"><i><font color="Green" title="E65">A38</font></i></a></i>;<br/>
<a NAME="E5:53_2_2_3"/><i><font color="Green" title="E72">A45</font></i>: 
<font color="Maroon" title="c33">y</font> is   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>
 
<b>by </b><i><a class="txt" href="osafree.html#E1:53_2_2_3"><i><font color="Green" title="E69">A42</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E6:53_2_2_3"/><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"><font color="Maroon" title="c26">x5</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c31">w5</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c33">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> <font color="Maroon" title="c24">xy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">y</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E38:53_2_2"><i><font color="Green" title="E64">A37</font></i></a>, <a class="txt" href="osafree.html#E43:53_2_2"><i><font color="Green" title="E66">A39</font></i></a>, <a class="txt" href="osafree.html#E44:53_2_2"><i><font color="Green" title="E67">A40</font></i></a>, <a class="txt" href="osafree.html#E1:53_2_2_3"><i><font color="Green" title="E69">A42</font></i></a></i>;<br/>
<a NAME="E7:53_2_2_3"/><b>then </b><i><font color="Green" title="E73">A46</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c36">t3</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c31">w5</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c33">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"><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="c35">t2</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E21:53_2_2"><i><font color="Green" title="E53">A26</font></i></a>, <a class="txt" href="osafree.html#E43:53_2_2"><i><font color="Green" title="E66">A39</font></i></a>, <a class="txt" href="osafree.html#E44:53_2_2"><i><font color="Green" title="E67">A40</font></i></a>, <a class="txt" href="osafree.html#E1:53_2_2_3"><i><font color="Green" title="E69">A42</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<a NAME="E8:53_2_2_3"/><b>then </b><i><font color="Green" title="E74">A47</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c35">t2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c31">w5</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c33">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"><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="c35">t2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c35">t2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c31">w5</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c33">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"><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="c36">t3</font> )
 
<b>by </b><i><a class="ref" href="osafree.html#T20" target="_self" title="OSAFREE:th.20">Th20</a>, <a class="ref" href="osafree.html#T21" target="_self" title="OSAFREE:th.21">Th21</a></i>;<br/>
<a NAME="E9:53_2_2_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c36">t3</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c31">w5</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c33">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"><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="c36">t3</font>
 
<b>by </b><i><a class="ref" href="osafree.html#T21" target="_self" title="OSAFREE:th.21">Th21</a></i>;<br/>
<a NAME="E10:53_2_2_3"/><b>then </b><i><font color="Green" title="E75">A48</font></i>: 
( <font color="Maroon" title="c35">t2</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"><font color="Maroon" title="c31">w5</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c33">y</font></span>)</span> &amp; <font color="Maroon" title="c36">t3</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"><font color="Maroon" title="c31">w5</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c33">y</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="osafree.html#E8:53_2_2_3"><i><font color="Green" title="E74">A47</font></i></a>, <a class="ref" href="osafree.html#T20" target="_self" title="OSAFREE:th.20">Th20</a></i>;<br/>
<a NAME="E11:53_2_2_3"/><b>then </b><i><font color="Green" title="E76">A49</font></i>: 
 <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c35">t2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c31">w5</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c33">y</font>
 
<b>by </b><i><a class="ref" href="osafree.html#D12" target="_self" title="OSAFREE:def.12">Def12</a></i>;<br/>
<a NAME="E12:53_2_2_3"/><i><font color="Green" title="E77">A50</font></i>: 
<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="c18">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c33">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> <font color="Maroon" title="c13">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">y</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E5:53_2_2_3"><i><font color="Green" title="E72">A45</font></i></a>, <a class="txt" href="osafree.html#E16:53_2_2"><i><font color="Green" title="E52">A25</font></i></a>, <a class="txt" href="osafree.html#E21:53_2_2"><i><font color="Green" title="E53">A26</font></i></a>, <a class="txt" href="osafree.html#E22:53_2_2"><i><font color="Green" title="E54">A27</font></i></a>, <a class="txt" href="osafree.html#E43:53_2_2"><i><font color="Green" title="E66">A39</font></i></a>, <a class="txt" href="osafree.html#E44:53_2_2"><i><font color="Green" title="E67">A40</font></i></a>, <a class="txt" href="osafree.html#E1:53_2_2_3"><i><font color="Green" title="E69">A42</font></i></a></i>;<br/>
<a NAME="E13:53_2_2_3"/><b>then </b><i><font color="Green" title="E78">A51</font></i>: 
<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="c18">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c33">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"><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="c33">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="osafree.html#E9:53_2"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="osafree.html#E21:53_2_2"><i><font color="Green" title="E53">A26</font></i></a>, <a class="txt" href="osafree.html#E22:53_2_2"><i><font color="Green" title="E54">A27</font></i></a>, <a class="txt" href="osafree.html#E43:53_2_2"><i><font color="Green" title="E66">A39</font></i></a>, <a class="txt" href="osafree.html#E44:53_2_2"><i><font color="Green" title="E67">A40</font></i></a>, <a class="txt" href="osafree.html#E1:53_2_2_3"><i><font color="Green" title="E69">A42</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<a NAME="E14:53_2_2_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c35">t2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c33">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"><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="c34">t1</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E2:53_2_2_3"><i><font color="Green" title="E70">A43</font></i></a>, <a class="txt" href="osafree.html#E12:53_2_2_3"><i><font color="Green" title="E77">A50</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<a NAME="E15:53_2_2_3"/><b>then </b><i><font color="Green" title="E79">A52</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c34">t1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c33">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"><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="c35">t2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c34">t1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c33">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"><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="c34">t1</font> )
 
<b>by </b><i><a class="ref" href="osafree.html#T20" target="_self" title="OSAFREE:th.20">Th20</a>, <a class="ref" href="osafree.html#T21" target="_self" title="OSAFREE:th.21">Th21</a></i>;<br/>
<a NAME="E16:53_2_2_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c35">t2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c33">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"><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="c35">t2</font>
 
<b>by </b><i><a class="ref" href="osafree.html#T21" target="_self" title="OSAFREE:th.21">Th21</a></i>;<br/>
<a NAME="E17:53_2_2_3"/><b>then </b><i><font color="Green" title="E80">A53</font></i>: 
( <font color="Maroon" title="c35">t2</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"><font color="Maroon" title="c23">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c33">y</font></span>)</span> &amp; <font color="Maroon" title="c34">t1</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"><font color="Maroon" title="c23">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c33">y</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="osafree.html#E15:53_2_2_3"><i><font color="Green" title="E79">A52</font></i></a>, <a class="ref" href="osafree.html#T20" target="_self" title="OSAFREE:th.20">Th20</a></i>;<br/>
<a NAME="E18:53_2_2_3"/><b>then </b>
 <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c35">t2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c23">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c33">y</font>
 
<b>by </b><i><a class="ref" href="osafree.html#D12" target="_self" title="OSAFREE:def.12">Def12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c37">s7</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E20:53_2_2_3"/><i><font color="Green" title="E81">A54</font></i>: 
( <font color="Maroon" title="c31">w5</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c33">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c37">s7</font> &amp; <font color="Maroon" title="c23">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c33">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c37">s7</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E11:53_2_2_3"><i><font color="Green" title="E76">A49</font></i></a>, <a class="ref" href="osalg_4.html#T12" title="OSALG_4:th.12">OSALG_4:12</a></i>;<br/>
<a NAME="E21:53_2_2_3"/><i><font color="Green" title="E82">A55</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c35">t2</font>,<font color="Maroon" title="c37">s7</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"><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="c34">t1</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E13:53_2_2_3"><i><font color="Green" title="E78">A51</font></i></a>, <a class="txt" href="osafree.html#E17:53_2_2_3"><i><font color="Green" title="E80">A53</font></i></a>, <a class="txt" href="osafree.html#E20:53_2_2_3"><i><font color="Green" title="E81">A54</font></i></a>, <a class="ref" href="osafree.html#T22" target="_self" title="OSAFREE:th.22">Th22</a></i>;<br/>
<a NAME="E22:53_2_2_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c36">t3</font>,<font color="Maroon" title="c37">s7</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"><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="c35">t2</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E7:53_2_2_3"><i><font color="Green" title="E73">A46</font></i></a>, <a class="txt" href="osafree.html#E10:53_2_2_3"><i><font color="Green" title="E75">A48</font></i></a>, <a class="txt" href="osafree.html#E20:53_2_2_3"><i><font color="Green" title="E81">A54</font></i></a>, <a class="ref" href="osafree.html#T22" target="_self" title="OSAFREE:th.22">Th22</a></i>;<br/>
<a NAME="E23:53_2_2_3"/><b>then </b><i><font color="Green" title="E83">A56</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c34">t1</font>,<font color="Maroon" title="c37">s7</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"><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="c36">t3</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E2:53_2"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="osafree.html#E3:53_2_2_3"><i><font color="Green" title="E71">A44</font></i></a>, <a class="txt" href="osafree.html#E21:53_2_2_3"><i><font color="Green" title="E82">A55</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c37">s7</font>
;<br/>

<b>thus </b><a NAME="E24:53_2_2_3"/>
<font color="Maroon" title="c37">s7</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>
 ;<br/>

<b>thus </b><a NAME="E25:53_2_2_3"/>
<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="c11">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">y</font></span>)</span>,<font color="Maroon" title="c37">s7</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">xz</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">y</font>
 <b>by </b><i><a class="txt" href="osafree.html#E1:53_2_2_3"><i><font color="Green" title="E69">A42</font></i></a>, <a class="txt" href="osafree.html#E23:53_2_2_3"><i><font color="Green" title="E83">A56</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c33">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c32">xz</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E47:53_2_2"/><i><font color="Green" title="E69">A57</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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> <font color="Maroon" title="c32">xz</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c33">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="osafree.html#E45:53_2_2"><i><font color="Green" title="E68">A41</font></i></a>);<br/></i>
<a NAME="E48:53_2_2"/><i><font color="Green" title="E70">A58</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c33">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> <font color="Maroon" title="c32">xz</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E49:53_2_2"/><b>then </b>
 ex <font color="Olive" title="b1">n</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>  st  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c33">f</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="Olive" title="b1">n</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/>
<b>then reconsider </b><font color="Maroon" title="c34">f1</font> = <font color="Maroon" title="c33">f</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E51:53_2_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">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>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E52:53_2_2"/><b>then </b>
<font color="Maroon" title="c34">f1</font> is    <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="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c35">f</font> = <font color="Maroon" title="c33">f</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="E54:53_2_2"/><i><font color="Green" title="E71">A59</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c35">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> <font color="Maroon" title="c32">xz</font> &amp; ( 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> <font color="Maroon" title="c32">xz</font> 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="c11">ts1</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="c35">f</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> <font color="Maroon" title="c32">xz</font> <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="53_2_2_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:53_2_2_4"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c35">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> <font color="Maroon" title="c32">xz</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>

<b>let </b><font color="Maroon" title="c36">y</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E2:53_2_2_4"/><i><font color="Green" title="E72">A60</font></i>: 
<font color="Maroon" title="c36">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> <font color="Maroon" title="c32">xz</font>
 ;<br/>

<a NAME="E3:53_2_2_4"/>
<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="c11">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c35">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">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> <font color="Maroon" title="c32">xz</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">y</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E47:53_2_2"><i><font color="Green" title="E69">A57</font></i></a>, <a class="txt" href="osafree.html#E2:53_2_2_4"><i><font color="Green" title="E72">A60</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:53_2_2_4"/>
<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="c11">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c35">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c36">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> <font color="Maroon" title="c32">xz</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">y</font>
 <b>by </b><i><a class="txt" href="osafree.html#E48:53_2_2"><i><font color="Green" title="E70">A58</font></i></a>, <a class="txt" href="osafree.html#E2:53_2_2_4"><i><font color="Green" title="E72">A60</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green" title="E72">A61</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="c16">z</font> = 
 <a href="osafree.html#K18" title="OSAFREE:func.18">@</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c25">o5</font>,<font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c32">xz</font>
<b>by </b><i><a class="txt" href="osafree.html#E27:53_2_2"><i><font color="Green" title="E57">A30</font></i></a>, <a class="txt" href="osafree.html#E31:53_2_2"><i><font color="Green" title="E60">A33</font></i></a>, <a class="txt" href="osafree.html#E34:53_2_2"><i><font color="Green" title="E61">A34</font></i></a>, <a class="txt" href="osafree.html#E35:53_2_2"><i><font color="Green" title="E62">A35</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">o4</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">x4</font></span>)</span>,<font color="Olive" title="b3">s4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">o4</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">x4</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">o4</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">s4</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/>(  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c25">o5</font>,<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="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">o4</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">o4</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">s4</font> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b1">o4</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">s4</font> ) &amp;  ex <font color="Olive" title="b4">w4</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">w4</font> <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="c32">xz</font> &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> <font color="Maroon" title="c32">xz</font> 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">x4</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">w4</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> <font color="Maroon" title="c32">xz</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">y</font> ) ) ) </span>}</span> 

;<br/>
<a NAME="E56:53_2_2"/>
(  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c25">o5</font>,<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="c25">o5</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="c25">o5</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c10">o</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="Maroon" title="c25">o5</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="Maroon" title="c10">o</font></span>)</span> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c25">o5</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c14">s1</font> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c10">o</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c14">s1</font> )
 
<b>by </b><i><a class="txt" href="osafree.html#E3:53_2_2"><i><font color="Green" title="E43">A16</font></i></a>, <a class="txt" href="osafree.html#E7:53_2_2"><i><font color="Green" title="E46">A19</font></i></a>, <a class="txt" href="osafree.html#E8:53_2_2"><i><font color="Green" title="E47">A20</font></i></a>, <a class="txt" href="osafree.html#E25:53_2_2"><i><font color="Green" title="E55">A28</font></i></a>, <a class="txt" href="osafree.html#E29:53_2_2"><i><font color="Green" title="E58">A31</font></i></a>, <a class="txt" href="osafree.html#E30:53_2_2"><i><font color="Green" title="E59">A32</font></i></a>, <a class="ref" href="osalg_1.html#T2" title="OSALG_1:th.2">OSALG_1:2</a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E57:53_2_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="c8">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c9">ts</font></span>)</span>,<font color="Maroon" title="c14">s1</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"><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="c16">z</font>
 <b>by </b><i><a class="txt" href="osafree.html#E4:53_2"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="osafree.html#E54:53_2_2"><i><font color="Green" title="E71">A59</font></i></a>, <a class="txt" href="osafree.html#E55:53_2_2"><i><font color="Green" title="E72">A61</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E3:53"/>
for <font color="Olive" title="b1">t</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">t</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>1</sub></font>[<font color="Olive" title="b1">t</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#E1:53"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="osafree.html#E2:53"><i><font color="Green" title="E35">A8</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:53"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</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><br/> for <font color="Olive" title="b4">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b4">s</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"><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="Olive" title="b1">x</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">z</font>,<font color="Olive" title="b4">s</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"><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="Olive" title="b2">y</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b4">s</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"><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="Olive" title="b3">z</font>
 ;<br/>


</div>
