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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">d</font> being  <a href="trees_2.html#NM4">DecoratedTree</a> ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_9.html#K9" target="_self">Subtrees</a> <font color="Maroon">X</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">d</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">p</font> );<br/>
<a NAME="E1:63_1_1"/><i><font color="Green">E25</font></i>: 
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">C</font> <a href="trees_9.html#K12" target="_self">-Subtrees</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_9.html#K9" target="_self">Subtrees</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_2.html#K3">*</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:63_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="trees_9.html#K12" target="_self">-Subtrees</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">x</font> as    <a href="trees_3.html#M3">Element</a> of  <a href="trees_9.html#K9" target="_self">Subtrees</a> <font color="Maroon">X</font> ;<br/>
<b>consider </b><font color="Maroon">t</font> being    <a href="trees_3.html#M3">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">sp</font> being   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">t</font><b> such that </b><br/><a NAME="E4:63_1_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="trees_2.html#K5">|</a> <font color="Maroon">sp</font>
 <b>by </b><i/>;<br/>
<a NAME="E5:63_1_1_1"/>
<font color="Maroon">t</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i/>;<br/>
<a NAME="E6:63_1_1_1"/><b>then </b>
<font color="Maroon">s</font> is  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>
 
<b>by </b><i><a class="txt" href="trees_9.html#E8:63_1_1_1"><i><font color="Green">E28</font></i></a>, </i>;<br/>
<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">p</font> being  <a href="trees_3.html#V6">DTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E8:63_1_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/>
<a NAME="E9:63_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="trees_9.html#K9" target="_self">Subtrees</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:63_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:63_1_1_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E4:63_1_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="trees_2.html#K5">|</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i>, <a class="txt" href="trees_9.html#E3:63_1_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">e</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="trees_4.html#NM1">Node</a> of <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="trees_2.html#K5">|</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <b>by </b><i><a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/>
<a NAME="E6:63_1_1_1_1"/>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/>
<a NAME="E7:63_1_1_1_1"/><b>then </b>
( <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="trees_2.html#K5">|</a> <font color="Maroon">sp</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font></span>)</span> <a href="trees_1.html#K4">|</a> <font color="Maroon">sp</font> )
 
<b>by </b><i>, <a class="txt" href="trees_9.html#E3:63_1_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="trees_9.html#E4:63_1_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="trees_2.html#D11">TREES_2:def 11</a>, <a class="ref" href="trees_4.html#T11">TREES_4:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">sp</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span> as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">t</font> <b>by </b><i><a class="txt" href="trees_9.html#E8:63_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="trees_1.html#D9">TREES_1:def 9</a></i>;<br/>
<a NAME="E9:63_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="trees_2.html#K5">|</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="txt" href="trees_9.html#E8:63_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="trees_9.html#E4:63_1_1_1_1"><i><font color="Green">E31</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E10:63_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_9.html#K9" target="_self">Subtrees</a> <font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_9.html#K9" target="_self">Subtrees</a> <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">p</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E12:63_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_9.html#K9" target="_self">Subtrees</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_2.html#K3">*</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>

<b>reconsider </b><font color="Maroon">d</font> = <font color="Maroon">s</font> as   <a href="trees_2.html#NM4">DecoratedTree</a> ;<br/>
<b>take </b>
<font color="Maroon">d</font>
;<br/>

<b>take </b>
<font color="Maroon">p</font>
;<br/>

<b>thus </b><a NAME="E14:63_1_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:63_1_1"/><i><font color="Green">E33</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="trees_9.html#K12" target="_self">-Subtrees</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="trees_9.html#K9" target="_self">Subtrees</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_2.html#K3">*</a>  &amp; ( 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">C</font> <a href="trees_9.html#K12" target="_self">-Subtrees</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">C</font> <a href="trees_9.html#K12" target="_self">-Subtrees</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="trees_9.html#K9" target="_self">Subtrees</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>let </b><font color="Maroon">d</font> be   <a href="trees_2.html#NM4">DecoratedTree</a>;<br/>

<b>assume </b><a NAME="E5:63_1_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="trees_9.html#K12" target="_self">-Subtrees</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E6:63_1_1"/><b>then </b>
 ex <font color="Olive">d'</font> being  <a href="trees_2.html#NM4">DecoratedTree</a> ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_9.html#K9" target="_self">Subtrees</a> <font color="Maroon">X</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> &amp; <font color="Olive">d'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; <font color="Olive">d'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">d'</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">p</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:63_1_1"/>
for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="trees_9.html#K9" target="_self">Subtrees</a> <font color="Maroon">X</font>  st <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> holds <br/><font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="trees_4.html#K4">-tree</a> <font color="Olive">p</font>
 ;<br/>


</div>
