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

<b>let </b><font color="Maroon">S</font> be  <a href="osalg_4.html#V2">locally_directed</a> <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">X</font> be  <a href="relat_1.html#V2">V5</a> <a href="struct_0.html#NM8">ManySortedSet</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">R</font> be   <a href="msualg_4.html#NM4">ManySortedRelation</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span>;<br/>





<b>set </b><font color="Maroon">PTA</font> =  <a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">SPTA</font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">D</font> =  <a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">OU</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p2">(<span class="default"><a href="msafree.html#K3">coprod</a> <font color="Maroon">X</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">C</font> =  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p3">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>set </b><font color="Maroon">F</font> =  <a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>] <b>means </b>for <font color="Olive">s1</font>, <font color="Olive">s2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font> holds <br/>( ( <font color="Olive">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s2</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">s1</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">s1</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">s2</font> ) &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">s1</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">s2</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<span class="p2">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">s1</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">s2</font> ) holds <br/>( <font color="Olive">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s2</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">s1</font></span><a href="tarski.html#K4">]</a></span> ) ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>] <b>means </b>for <font color="Olive">o1</font>, <font color="Olive">o2</font> being  <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font><br/> for <font color="Olive">x1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Olive">o1</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span><br/> for <font color="Olive">x2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Olive">o2</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span><br/> for <font color="Olive">s3</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o1</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o2</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">s3</font> iff ( <font color="Olive">o1</font> <a href="osalg_1.html#R1">~=</a> <font color="Olive">o2</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Olive">o1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Olive">o2</font></span>)</span> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Olive">o1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s3</font> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Olive">o2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s3</font> &amp;  ex <font color="Olive">w3</font> being   <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a>  st <br/>(  <a href="finseq_1.html#K4">dom</a> <font color="Olive">w3</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">x1</font> &amp; ( for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">w3</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">x2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span> ) ) ) );<br/>
<a NAME="E1:58"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">R1</font>, <font color="Olive">R2</font> being  <a href="msualg_4.html#NM4">ManySortedRelation</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">R1</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">R1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">R2</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">R2</font>] holds <br/><font color="Olive">R1</font> <a href="pboole.html#R6">=</a> <font color="Olive">R2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">R1</font> be   <a href="msualg_4.html#NM4">ManySortedRelation</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">R2</font> be   <a href="msualg_4.html#NM4">ManySortedRelation</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:58_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">R1</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">R1</font>] )
 <b>and </b><br/><a NAME="E2:58_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">R2</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">R2</font>] )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="msualg_4.html#K1">.</a> <font color="Olive">s</font> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> <a href="msualg_4.html#K1">.</a> <font color="Olive">s</font> );<br/>
<a NAME="E3:58_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">s</font> being  <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>  st <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6">Terminals</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> holds <br/><font color="Maroon">S<sub>3</sub></font>[ <a href="trees_4.html#K2">root-tree</a> <font color="Olive">s</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:58_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">sy</font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6">Terminals</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">s</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:58_1_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">sy</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, </i>;<br/>
<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">s1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:58_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">sy</font></span>)</span>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s1</font>
 ;<br/><a NAME="E2:58_1_1_1"/><b>then </b>
( <font color="Maroon">s</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, <a class="ref" href="osafree.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E3:58_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">sy</font></span>)</span>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s1</font>
 <b>by </b><i>, <a class="ref" href="osafree.html#D4" target="_self">Def4</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E5:58_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">sy</font></span>)</span>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s1</font>
 ;<br/>

<a NAME="E6:58_1_1"/><b>then </b>
( <font color="Maroon">s</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i>, <a class="ref" href="osafree.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E7:58_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">sy</font></span>)</span>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s1</font>
 <b>by </b><i>, <a class="ref" href="osafree.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:58_1"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">nt</font> being  <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span><br/> for <font color="Olive">ts</font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>  st <font color="Olive">nt</font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Olive">ts</font> &amp; ( for <font color="Olive">t</font> being  <a href="trees_2.html#NM6">DecoratedTree</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>  st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">ts</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">t</font>] ) holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Olive">ts</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">nt</font> be   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">ts</font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:58_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">nt</font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">ts</font>
 <b>and </b><br/><a NAME="E2:58_1_2"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">t</font> being  <a href="trees_2.html#NM6">DecoratedTree</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>  st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">t</font>]
 ;<br/>

<a NAME="E3:58_1_2"/>
<font color="Maroon">nt</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> :  ex <font color="Olive">n</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <font color="Olive">s</font> <a href="lang1.html#R1">==&gt;</a> <font color="Olive">n</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="osafree.html#D4" target="_self">Def4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">nt1</font> = <font color="Maroon">nt</font> as   <a href="dtconstr.html#NM3">NonTerminal</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="lang1.html#D3">LANG1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">tss</font> = <font color="Maroon">ts</font> as    <a href="dtconstr.html#M2">SubtreeSeq</a> of <font color="Maroon">nt1</font> <b>by </b><i><a class="ref" href="osafree.html#D4" target="_self">Def4</a>, <a class="ref" href="dtconstr.html#D9">DTCONSTR:def 9</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">s</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>



<i><font color="Green">E65</font></i>: the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> = 
 <a href="osafree.html#K4" target="_self">ParsedTerms</a> <font color="Maroon">X</font>,<font color="Maroon">s</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">a1</font> where <font color="Olive">a1</font> is    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> : (  ex <font color="Olive">s1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font> &amp; <font color="Olive">a1</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">s1</font></span><a href="tarski.html#K4">]</a></span> ) or  ex <font color="Olive">o</font> being  <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">a1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Olive">o</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> ) ) </span>}</span> 

;<br/>
<a NAME="E7:58_1_2"/><i><font color="Green">E67</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">nt</font>,<span class="p2">(<span class="default"><a href="dtconstr.html#K1">roots</a> <font color="Maroon">ts</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="lang1.html#U1">Rules</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="osafree.html#D4" target="_self">Def4</a>, <a class="ref" href="lang1.html#D1">LANG1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">sy</font> = <font color="Maroon">nt</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p2">(<span class="default"><a href="msafree.html#K3">coprod</a> <font color="Maroon">X</font></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">rt</font> =  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">ts</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p3">(<span class="default"><a href="msafree.html#K3">coprod</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E10:58_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">sy</font>,<font color="Maroon">rt</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="osafree.html#K1" target="_self">OSREL</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="osafree.html#D4" target="_self">Def4</a>, <a class="ref" href="lang1.html#D1">LANG1:def 1</a></i>;<br/>
<a NAME="E11:58_1_2"/><b>then </b>
<font color="Maroon">sy</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">o</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>, <font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K1">}</a></span><b> such that </b><br/><a NAME="E13:58_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">sy</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E14:58_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E15:58_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">tss</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>
<a NAME="E16:58_1_2"/><i><font color="Green">E71</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts</font> <a href="tarski.html#R1">c=</a>  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:58_1_2_2"/><i><font color="Green">E72</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts</font></span>)</span>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/><a NAME="E2:58_1_2_2"/><b>then </b>
( <font color="Maroon">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon">a1</font> being    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E4:58_1_2_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font>
 <b>and </b><br/><a NAME="E5:58_1_2_2"/><i><font color="Green">E75</font></i>: 
(  ex <font color="Olive">s1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">s1</font></span><a href="tarski.html#K4">]</a></span> ) or  ex <font color="Olive">o</font> being  <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Olive">o</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> ) )
 <b>by </b><i/>;<br/><a NAME="E6:58_1_2_2"/>
for <font color="Olive">s1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><br/> for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> or  not <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font> or  not <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">s1</font></span><a href="tarski.html#K4">]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2_2_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">s1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:58_1_2_2_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/>

<a NAME="E2:58_1_2_2_1"/>
<font color="Maroon">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, , , <a class="ref" href="osafree.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E3:58_1_2_2_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">nt</font> &amp; <font color="Maroon">ts</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="trees_4.html#T17">TREES_4:17</a></i>;<br/>
<a NAME="E4:58_1_2_2_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E5:58_1_2_2_1"/><b>then </b>
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">s1</font>
 
;<br/>
<b>hence </b><a NAME="E6:58_1_2_2_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">o1</font> being   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E8:58_1_2_2"/><i><font color="Green">E77</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">ts1</font> being    <a href="dtconstr.html#M2">SubtreeSeq</a> of  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o1</font>,<font color="Maroon">X</font><b> such that </b><br/><a NAME="E10:58_1_2_2"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o1</font>,<font color="Maroon">X</font></span>)</span> <a href="dtconstr.html#K9">-tree</a> <font color="Maroon">ts1</font> &amp;  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o1</font>,<font color="Maroon">X</font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">ts1</font> &amp; <font color="Maroon">ts1</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ts1</font> )
 <b>by </b><i><a class="ref" href="osafree.html#T2" target="_self">Th2</a>, </i>;<br/><b>consider </b><font color="Maroon">ts2</font> being    <a href="dtconstr.html#M2">SubtreeSeq</a> of  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font><b> such that </b><br/><a NAME="E12:58_1_2_2"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">nt1</font> <a href="dtconstr.html#K9">-tree</a> <font color="Maroon">tss</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font></span>)</span> <a href="dtconstr.html#K9">-tree</a> <font color="Maroon">ts2</font> &amp;  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">ts2</font> &amp; <font color="Maroon">ts2</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">nt1</font> <a href="dtconstr.html#K9">-tree</a> <font color="Maroon">tss</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ts2</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E13:58_1_2_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">ts2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">tss</font>
 <b>by </b><i>, <a class="ref" href="trees_4.html#T15">TREES_4:15</a></i>;<br/><b>reconsider </b><font color="Maroon">tsa</font> = <font color="Maroon">ts1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="osafree.html#T3" target="_self">Th3</a></i>;<br/><b>reconsider </b><font color="Maroon">tsb</font> = <font color="Maroon">ts2</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/><a NAME="E16:58_1_2_2"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">o</font> <a href="osalg_1.html#R1">~=</a> <font color="Maroon">o1</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font></span>)</span> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> &amp;  ex <font color="Olive">w3</font> being   <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a>  st <br/>(  <a href="finseq_1.html#K4">dom</a> <font color="Olive">w3</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">tsb</font> &amp; ( for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">w3</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">tsb</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">tsa</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="msualg_4.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span> ) ) )
 <b>by </b><i>, , , <a class="ref" href="osafree.html#T3" target="_self">Th3</a>, </i>;<br/><b>consider </b><font color="Maroon">w3</font> being    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E18:58_1_2_2"/><i><font color="Green">E105</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w3</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">tsb</font> &amp; ( for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w3</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">tsb</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">tsa</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="msualg_4.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span> ) )
 <b>by </b><i>, , , <a class="ref" href="osafree.html#T3" target="_self">Th3</a>, </i>;<br/><a NAME="E19:58_1_2_2"/>
for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w3</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">tsb</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">tsa</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> <a href="msualg_4.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:58_1_2_2_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w3</font>
 ;<br/>

<a NAME="E2:58_1_2_2_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">tsb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts</font>
 
<b>by </b><i>, , <a class="txt" href="osafree.html#E4:58_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">tsb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> as    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E4:58_1_2_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t</font>,<span class="p2">(<span class="default"><font color="Maroon">tsa</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="msualg_4.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="osafree.html#E4:58_1"><i><font color="Green">E61</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:58_1_2_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">tsb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">tsa</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> <a href="msualg_4.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="osafree.html#E1:58_1_2"><i><font color="Green">E62</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E20:58_1_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts</font></span>)</span>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, , <a class="ref" href="osafree.html#T3" target="_self">Th3</a>, , , </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E18:58_1_2"/><i><font color="Green">E108</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts</font></span>)</span>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E19:58_1_2"/><b>then </b>
( <font color="Maroon">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">a1</font> being    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E21:58_1_2"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font>
 <b>and </b><br/><a NAME="E22:58_1_2"/><i><font color="Green">E110</font></i>: 
(  ex <font color="Olive">s1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">s1</font></span><a href="tarski.html#K4">]</a></span> ) or  ex <font color="Olive">o</font> being  <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Olive">o</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E23:58_1_2"/>
for <font color="Olive">s1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><br/> for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> or  not <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font> or  not <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">s1</font></span><a href="tarski.html#K4">]</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2_3"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">s1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:58_1_2_3"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/>

<a NAME="E2:58_1_2_3"/>
<font color="Maroon">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, , , <a class="ref" href="osafree.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E3:58_1_2_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">nt</font> &amp; <font color="Maroon">ts</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="trees_4.html#T17">TREES_4:17</a></i>;<br/>
<a NAME="E4:58_1_2_3"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E5:58_1_2_3"/><b>then </b>
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">s1</font>
 
;<br/>
<b>hence </b><a NAME="E6:58_1_2_3"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">o1</font> being   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E25:58_1_2"/><i><font color="Green">E113</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">ts1</font> being    <a href="dtconstr.html#M2">SubtreeSeq</a> of  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o1</font>,<font color="Maroon">X</font><b> such that </b><br/><a NAME="E27:58_1_2"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o1</font>,<font color="Maroon">X</font></span>)</span> <a href="dtconstr.html#K9">-tree</a> <font color="Maroon">ts1</font> &amp;  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o1</font>,<font color="Maroon">X</font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">ts1</font> &amp; <font color="Maroon">ts1</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ts1</font> )
 <b>by </b><i><a class="ref" href="osafree.html#T2" target="_self">Th2</a>, </i>;<br/>
<b>consider </b><font color="Maroon">ts2</font> being    <a href="dtconstr.html#M2">SubtreeSeq</a> of  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font><b> such that </b><br/><a NAME="E29:58_1_2"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">nt1</font> <a href="dtconstr.html#K9">-tree</a> <font color="Maroon">tss</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font></span>)</span> <a href="dtconstr.html#K9">-tree</a> <font color="Maroon">ts2</font> &amp;  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">ts2</font> &amp; <font color="Maroon">ts2</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">nt1</font> <a href="dtconstr.html#K9">-tree</a> <font color="Maroon">tss</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ts2</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E30:58_1_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">ts2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">tss</font>
 
<b>by </b><i>, <a class="ref" href="trees_4.html#T15">TREES_4:15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">tsa</font> = <font color="Maroon">ts1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="osafree.html#T3" target="_self">Th3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">tsb</font> = <font color="Maroon">ts2</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E33:58_1_2"/><i><font color="Green">E176</font></i>: 
( <font color="Maroon">o</font> <a href="osalg_1.html#R1">~=</a> <font color="Maroon">o1</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font></span>)</span> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> &amp;  ex <font color="Olive">w3</font> being   <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a>  st <br/>(  <a href="finseq_1.html#K4">dom</a> <font color="Olive">w3</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">tsb</font> &amp; ( for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">w3</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">tsb</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">tsa</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> <a href="msualg_4.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span> ) ) )
 
<b>by </b><i>, , , <a class="ref" href="osafree.html#T3" target="_self">Th3</a>, </i>;<br/>
<b>consider </b><font color="Maroon">w3</font> being    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E35:58_1_2"/><i><font color="Green">E177</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w3</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">tsb</font> &amp; ( for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w3</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">tsb</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">tsa</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> <a href="msualg_4.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span> ) )
 <b>by </b><i>, , , <a class="ref" href="osafree.html#T3" target="_self">Th3</a>, </i>;<br/>
<a NAME="E36:58_1_2"/>
for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w3</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">tsb</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">tsa</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="msualg_4.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:58_1_2_4"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w3</font>
 ;<br/>

<a NAME="E2:58_1_2_4"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">tsb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts</font>
 
<b>by </b><i>, , <a class="txt" href="osafree.html#E4:58_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">tsb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> as    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E4:58_1_2_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t</font>,<span class="p2">(<span class="default"><font color="Maroon">tsa</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> <a href="msualg_4.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="osafree.html#E4:58_1"><i><font color="Green">E61</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:58_1_2_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">tsb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">tsa</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="msualg_4.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="osafree.html#E1:58_1_2"><i><font color="Green">E62</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E37:58_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts</font></span>)</span>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, , <a class="ref" href="osafree.html#T3" target="_self">Th3</a>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:58_1"/><i><font color="Green">E195</font></i>: 
for <font color="Olive">t</font> being  <a href="trees_2.html#NM6">DecoratedTree</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>  st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">t</font>]
 
<b>from </b><i><a class="ref" href="dtconstr.html#S7">DTCONSTR:sch 7</a>(, );<br/></i>
<a NAME="E6:58_1"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/><font color="Maroon">R1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:58_1_3"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E3:58_1_3"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_3_1"><b>proof </b></a><div class="add">

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

<i><font color="Green">E197</font></i>: the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> = 
 <a href="osafree.html#K4" target="_self">ParsedTerms</a> <font color="Maroon">X</font>,<font color="Maroon">s</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">a1</font> where <font color="Olive">a1</font> is    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> : (  ex <font color="Olive">s1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font> &amp; <font color="Olive">a1</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">s1</font></span><a href="tarski.html#K4">]</a></span> ) or  ex <font color="Olive">o</font> being  <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">a1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Olive">o</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> ) ) </span>}</span> 

;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:58_1_3_1_2"/><i><font color="Green">E198</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/><a NAME="E2:58_1_3_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon">a1</font> being    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E4:58_1_3_1_2"/><i><font color="Green">E199</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font>
 <b>and </b><br/><a NAME="E5:58_1_3_1_2"/>
(  ex <font color="Olive">s1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">s1</font></span><a href="tarski.html#K4">]</a></span> ) or  ex <font color="Olive">o</font> being  <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Olive">o</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> ) )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E6:58_1_3_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="osafree.html#D4" target="_self">Def4</a>, , </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:58_1_3_1"/><i><font color="Green">E200</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E4:58_1_3_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">a1</font> being    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E6:58_1_3_1"/><i><font color="Green">E217</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font>
 <b>and </b><br/><a NAME="E7:58_1_3_1"/>
(  ex <font color="Olive">s1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">s1</font></span><a href="tarski.html#K4">]</a></span> ) or  ex <font color="Olive">o</font> being  <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Olive">o</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> ) )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E8:58_1_3_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="osafree.html#D4" target="_self">Def4</a>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:58_1_3"/>
<font color="Maroon">R1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:58_1"/>
<font color="Maroon">R1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">R2</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">P</font> =  <a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font>;<br/>
<a NAME="E2:58"/><i><font color="Green">E218</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">s2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:58_2"/><i><font color="Green">E219</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>
 ;<br/>

<a NAME="E2:58_2"/><i><font color="Green">E220</font></i>: 
<span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">x1</font> is    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>, <font color="Olive">y</font> is    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> : <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x1</font>,<font color="Maroon">s2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="osafree.html#E1:58"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">sy</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> as   <a href="dtconstr.html#NM2">Terminal</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i>, </i>;<br/>
<a NAME="E4:58_2"/><i><font color="Green">E221</font></i>: 
 <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E222</font></i>: <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K8">root-tree</a> <font color="Maroon">sy</font></span>)</span> = 
 <a href="osafree.html#K17" target="_self">@</a> <font color="Maroon">sy</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K8">root-tree</a> <font color="Maroon">sy</font></span>)</span>,<font color="Olive">s3</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">s3</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> :  ex <font color="Olive">s4</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s4</font> &amp; <font color="Maroon">sy</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">s4</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">s4</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s3</font> ) </span>}</span> 

;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:58_2_2"/><i><font color="Green">E223</font></i>: 
<font color="Maroon">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s2</font>
 ;<br/><a NAME="E2:58_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K8">root-tree</a> <font color="Maroon">sy</font></span>)</span>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K8">root-tree</a> <font color="Maroon">sy</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:58_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K8">root-tree</a> <font color="Maroon">sy</font></span>)</span>,<font color="Maroon">s2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K8">root-tree</a> <font color="Maroon">sy</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:58_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s2</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<a NAME="E7:58_2"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/>
<a NAME="E8:58_2"/><b>then </b><i><font color="Green">E224</font></i>: 
<span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s2</font> <a href="relat_2.html#R3">is_symmetric_in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D11">RELAT_2:def 11</a></i>;<br/>
<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E9:58_2"/><i><font color="Green">E225</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s2</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s2</font> )
 ;<br/>

<a NAME="E10:58_2"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> &amp;  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E11:58_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><a href="dtconstr.html#K8">root-tree</a> <font color="Maroon">sy</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s2</font>
 
<b>by </b><i>, , <a class="ref" href="relat_2.html#D3">RELAT_2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">y1</font> being    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">r1</font> being    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E13:58_2"/><i><font color="Green">E227</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><a href="dtconstr.html#K8">root-tree</a> <font color="Maroon">sy</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">r1</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E14:58_2"/><i><font color="Green">E228</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">s2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">r1</font>
 <b>by </b><i/>;<br/>
<a NAME="E15:58_2"/><i><font color="Green">E229</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp;  <a href="dtconstr.html#K8">root-tree</a> <font color="Maroon">sy</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>then consider </b><font color="Maroon">s3</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E17:58_2"/><i><font color="Green">E230</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">s2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K8">root-tree</a> <font color="Maroon">sy</font></span>)</span>,<font color="Maroon">s3</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E18:58_2"/><i><font color="Green">E231</font></i>: 
 ex <font color="Olive">s4</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s4</font> &amp; <font color="Maroon">sy</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">s4</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">s4</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s3</font> )
 <b>by </b><i><a class="ref" href="osafree.html#D4" target="_self">Def4</a>, </i>;<br/>
<b>consider </b><font color="Maroon">s4</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E20:58_2"/><i><font color="Green">E232</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s4</font> &amp; <font color="Maroon">sy</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s4</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">s4</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s3</font> )
 <b>by </b><i/>;<br/>
<a NAME="E21:58_2"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K8">root-tree</a> <font color="Maroon">sy</font> &amp; <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s3</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E22:58_2"/>
( <font color="Maroon">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s1</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:58"/>
<font color="Maroon">S<sub>2</sub></font>[ <a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o1</font> be   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>, <font color="Maroon">o2</font> be   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">x1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">x2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o2</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">s3</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>







<a NAME="E1:58_3"/><i><font color="Green">E233</font></i>: 
<span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s3</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x3</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">x3</font> is    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>, <font color="Olive">y</font> is    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> : <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x3</font>,<font color="Maroon">s3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="osafree.html#E1:58"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E2:58_3"/><i><font color="Green">E234</font></i>: 
( <font color="Maroon">x1</font> is    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o1</font>,<font color="Maroon">X</font> <a href="lang1.html#R1">==&gt;</a>  <a href="trees_3.html#K16">roots</a> <font color="Maroon">x1</font> )
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">ts1</font> = <font color="Maroon">x1</font> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<i><font color="Green">E253</font></i>: <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K7" target="_self">PTOper</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_4.html#K5">.</a> <font color="Maroon">o1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="osafree.html#K6" target="_self">PTDenOp</a> <font color="Maroon">o1</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ts1</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o1</font>,<font color="Maroon">X</font></span>)</span> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts1</font>
<b>by </b><i>, </i>
;<br/>
<a NAME="E5:58_3"/><i><font color="Green">E254</font></i>: 
( <font color="Maroon">x2</font> is    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o2</font>,<font color="Maroon">X</font> <a href="lang1.html#R1">==&gt;</a>  <a href="trees_3.html#K16">roots</a> <font color="Maroon">x2</font> )
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">ts2</font> = <font color="Maroon">x2</font> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<i><font color="Green">E255</font></i>: <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K7" target="_self">PTOper</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_4.html#K5">.</a> <font color="Maroon">o2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="osafree.html#K6" target="_self">PTDenOp</a> <font color="Maroon">o2</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ts2</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o2</font>,<font color="Maroon">X</font></span>)</span> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts2</font>
<b>by </b><i><a class="ref" href="osafree.html#D4" target="_self">Def4</a>, </i>
;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="lang1.html#K15">*</a> <font color="Maroon">ts1</font>, <font color="Maroon">y</font> = <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="lang1.html#K15">*</a> <font color="Maroon">ts2</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p3">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<a NAME="E9:58_3"/><i><font color="Green">E256</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts1</font> <a href="tarski.html#R1">c=</a>  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts2</font> <a href="tarski.html#R1">c=</a>  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:58_3"/><b>then </b><i><font color="Green">E257</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">ts1</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">ts2</font> )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T33">FINSEQ_2:33</a></i>;<br/>
<a NAME="E11:58_3"/><b>then </b><i><font color="Green">E258</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ts1</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ts2</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<i><font color="Green">E259</font></i>: <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o1</font>,<font color="Maroon">X</font></span>)</span> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts1</font></span>)</span> = 
 <a href="osafree.html#K18" target="_self">@</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o1</font>,<font color="Maroon">X</font></span>)</span>,<font color="Maroon">x</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o3</font>,<span class="p5">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x3</font></span>)</span>,<font color="Olive">s4</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">o3</font> is   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>, <font color="Olive">x3</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Olive">o3</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span>, <font color="Olive">s4</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : (  ex <font color="Olive">o4</font> being  <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>(  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o1</font>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o4</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">o4</font> <a href="osalg_1.html#R1">~=</a> <font color="Olive">o3</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Olive">o4</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Olive">o3</font></span>)</span> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Olive">o4</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s4</font> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Olive">o3</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s4</font> ) &amp;  ex <font color="Olive">w3</font> being   <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a>  st <br/>(  <a href="finseq_1.html#K4">dom</a> <font color="Olive">w3</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) ) ) </span>}</span> 

;<br/>
<a NAME="E13:58_3"/><i><font color="Green">E260</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x1</font> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x2</font> )
 
<b>by </b><i><a class="ref" href="msualg_3.html#T6">MSUALG_3:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">tx</font> = <span class="p1">(<span class="default"><a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o1</font>,<font color="Maroon">X</font></span>)</span> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts1</font>, <font color="Maroon">ty</font> = <span class="p1">(<span class="default"><a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o2</font>,<font color="Maroon">X</font></span>)</span> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts2</font> as    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i>, <a class="ref" href="osafree.html#D4" target="_self">Def4</a>, </i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:58_3_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s3</font>
 ;<br/><b>then consider </b><font color="Maroon">t1</font> being    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">t2</font> being    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E3:58_3_4"/><i><font color="Green">E263</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t1</font>,<font color="Maroon">t2</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E4:58_3_4"/><i><font color="Green">E264</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t1</font>,<font color="Maroon">s3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t2</font>
 <b>by </b><i/>;<br/><a NAME="E5:58_3_4"/><i><font color="Green">E265</font></i>: 
( <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E6:58_3_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t2</font>,<font color="Maroon">s3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t1</font>
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">o3</font> being   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>, <font color="Maroon">x3</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o3</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">s4</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E8:58_3_4"/><i><font color="Green">E266</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t2</font>,<font color="Maroon">s3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o3</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x3</font></span>)</span>,<font color="Maroon">s4</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E9:58_3_4"/><i><font color="Green">E267</font></i>: 
 ex <font color="Olive">o4</font> being  <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> st <br/>(  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o1</font>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o4</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">o4</font> <a href="osalg_1.html#R1">~=</a> <font color="Maroon">o3</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Olive">o4</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o3</font></span>)</span> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Olive">o4</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s4</font> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o3</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s4</font> )
 <b>and </b><br/><a NAME="E10:58_3_4"/><i><font color="Green">E268</font></i>: 
 ex <font color="Olive">w3</font> being   <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a>  st <br/>(  <a href="finseq_1.html#K4">dom</a> <font color="Olive">w3</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) )
 <b>by </b><i>, , </i>;<br/><b>consider </b><font color="Maroon">o4</font> being   <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E12:58_3_4"/><i><font color="Green">E269</font></i>: 
(  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o1</font>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o4</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">o4</font> <a href="osalg_1.html#R1">~=</a> <font color="Maroon">o3</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o4</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o3</font></span>)</span> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o4</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s4</font> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o3</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s4</font> )
 <b>by </b><i><a class="ref" href="osafree.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E13:58_3_4"/><i><font color="Green">E270</font></i>: 
( <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o4</font> &amp; <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o3</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x3</font> &amp; <font color="Maroon">s3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s4</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E14:58_3_4"/><i><font color="Green">E271</font></i>: 
( <font color="Maroon">x3</font> is    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o3</font>,<font color="Maroon">X</font> <a href="lang1.html#R1">==&gt;</a>  <a href="trees_3.html#K16">roots</a> <font color="Maroon">x3</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">ts3</font> = <font color="Maroon">x3</font> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/><span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o3</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x3</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K7" target="_self">PTOper</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_4.html#K5">.</a> <font color="Maroon">o3</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x3</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="osafree.html#K6" target="_self">PTDenOp</a> <font color="Maroon">o3</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ts3</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o3</font>,<font color="Maroon">X</font></span>)</span> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts3</font>
<b>by </b><i>, </i>
;<br/><a NAME="E17:58_3_4"/><b>then </b><i><font color="Green">E273</font></i>: 
(  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o3</font>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o2</font>,<font color="Maroon">X</font> &amp; <font color="Maroon">ts3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ts2</font> )
 <b>by </b><i>, , , <a class="ref" href="trees_4.html#T15">TREES_4:15</a></i>;<br/><a NAME="E18:58_3_4"/><b>then </b><i><font color="Green">E274</font></i>: 
( <font color="Maroon">o3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> &amp; <font color="Maroon">x3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>thus </b><a NAME="E19:58_3_4"/>
( <font color="Maroon">o1</font> <a href="osalg_1.html#R1">~=</a> <font color="Maroon">o2</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font></span>)</span> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s3</font> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s3</font> )
 <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E20:58_3_4"/><i><font color="Green">E275</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>consider </b><font color="Maroon">w3</font> being    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E22:58_3_4"/><i><font color="Green">E276</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w3</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E23:58_3_4"/><i><font color="Green">E277</font></i>: 
for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font>
 <b>by </b><i><a class="ref" href="osafree.html#T3" target="_self">Th3</a></i>;<br/><b>take </b><font color="Maroon">w3</font> = <font color="Maroon">w3</font>;<br/><b>thus </b><a NAME="E24:58_3_4"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w3</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x1</font>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E25:58_3_4"/><i><font color="Green">E278</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w3</font>
 ;<br/><a NAME="E26:58_3_4"/><i><font color="Green">E279</font></i>: 
<span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_4.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x5</font>,<font color="Olive">y5</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">x5</font> is    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>, <font color="Olive">y5</font> is    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> : <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x5</font>,<span class="p2">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y5</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="osafree.html#E1:58"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E27:58_3_4"/>
( <font color="Maroon">ts1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts1</font> &amp; <font color="Maroon">ts2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts2</font> )
 <b>by </b><i>, , , , , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">t22</font> = <font color="Maroon">ts2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>, <font color="Maroon">t11</font> = <font color="Maroon">ts1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> as    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/><a NAME="E29:58_3_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E30:58_3_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ts2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ts1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E31:58_3_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t11</font>,<span class="p2">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t22</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E32:58_3_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_4.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E16:58_3"/><i><font color="Green">E282</font></i>: 
( <font color="Maroon">o1</font> <a href="osalg_1.html#R1">~=</a> <font color="Maroon">o2</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font></span>)</span> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s3</font> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">o2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s3</font> &amp;  ex <font color="Olive">w3</font> being   <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a>  st <br/>(  <a href="finseq_1.html#K4">dom</a> <font color="Olive">w3</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x1</font> &amp; ( for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">w3</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_4.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span> ) ) )
 ;<br/>

<b>then consider </b><font color="Maroon">w3</font> being    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E18:58_3"/><i><font color="Green">E283</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w3</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x1</font>
 <b>and </b><br/><a NAME="E19:58_3"/><i><font color="Green">E284</font></i>: 
for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w3</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_4.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span>
 ;<br/>
<a NAME="E20:58_3"/>
for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_3_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:58_3_5"/><i><font color="Green">E285</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:58_3_5"/><i><font color="Green">E286</font></i>: 
<span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_4.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x5</font>,<font color="Olive">y5</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">x5</font> is    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>, <font color="Olive">y5</font> is    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span> : <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x5</font>,<span class="p2">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y5</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="osafree.html#E1:58"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E3:58_3_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_4.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i>, , , </i>;<br/>
<b>then consider </b><font color="Maroon">x5</font> being    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">y5</font> being    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" target="_self">DTConOSA</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E5:58_3_5"/><i><font color="Green">E287</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x5</font>,<font color="Maroon">y5</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E6:58_3_5"/><i><font color="Green">E288</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x5</font>,<span class="p2">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y5</font>
 <b>by </b><i><a class="ref" href="osafree.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E7:58_3_5"/><i><font color="Green">E289</font></i>: 
( <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x5</font> &amp; <font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y5</font> )
 
<b>by </b><i><a class="ref" href="osafree.html#T3" target="_self">Th3</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E8:58_3_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y5</font>,<span class="p2">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x5</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:58_3_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:58_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span>,<font color="Maroon">s3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K9" target="_self">OSSym</a> <font color="Maroon">o1</font>,<font color="Maroon">X</font></span>)</span> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts1</font></span>)</span>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E22:58_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">tx</font>,<font color="Maroon">s3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" target="_self">PTClasses</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">ty</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E23:58_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s3</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:58"/>
( <font color="Maroon">R</font> <a href="pboole.html#R6">=</a>  <a href="osafree.html#K20" target="_self">PTCongruence</a> <font color="Maroon">X</font> iff ( ( for <font color="Olive">s1</font>, <font color="Olive">s2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font> holds <br/>( ( <font color="Olive">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s2</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">s1</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">s1</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Olive">s2</font> ) &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">s1</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Olive">s2</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<span class="p2">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">s1</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Olive">s2</font> ) holds <br/>( <font color="Olive">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s2</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">s1</font></span><a href="tarski.html#K4">]</a></span> ) ) ) ) &amp; ( for <font color="Olive">o1</font>, <font color="Olive">o2</font> being  <a href="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font><br/> for <font color="Olive">x1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Olive">o1</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span><br/> for <font color="Olive">x2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Olive">o2</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span><br/> for <font color="Olive">s3</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o1</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o2</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" target="_self">ParsedTermsOSA</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Olive">s3</font> iff ( <font color="Olive">o1</font> <a href="osalg_1.html#R1">~=</a> <font color="Olive">o2</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Olive">o1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Olive">o2</font></span>)</span> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Olive">o1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s3</font> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Olive">o2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s3</font> &amp;  ex <font color="Olive">w3</font> being   <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="finseq_2.html#K3">*</a>  st <br/>(  <a href="finseq_1.html#K4">dom</a> <font color="Olive">w3</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">x1</font> &amp; ( for <font color="Olive">y</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">w3</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">x2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">w3</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">y</font></span>)</span> ) ) ) ) ) ) )
 <b>by </b><i>, </i>;<br/>


</div>
