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

<b>let </b><font color="Maroon">t1</font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> , <font color="Maroon">t2</font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:32_1_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; ( for <font color="Olive">x</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t1</font> holds   <a href="trees_9.html#K2">succ</a> <font color="Maroon">t1</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="trees_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> ) )
 <b>and </b><br/><a NAME="E2:32_1_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">t2</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; ( for <font color="Olive">x</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t2</font> holds   <a href="trees_9.html#K2">succ</a> <font color="Maroon">t2</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <span class="p1">(<span class="default"><font color="Maroon">t2</font> <a href="trees_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> ) )
 ;<br/>

<a NAME="E3:32_1_2"/>
for <font color="Olive">t1</font>, <font color="Olive">t2</font> being <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>   st <font color="Olive">t1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; ( for <font color="Olive">x</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Olive">t1</font> holds   <a href="trees_9.html#K2">succ</a> <font color="Olive">t1</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <span class="p1">(<span class="default"><font color="Olive">t1</font> <a href="trees_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> ) &amp; <font color="Olive">t2</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; ( for <font color="Olive">x</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Olive">t2</font> holds   <a href="trees_9.html#K2">succ</a> <font color="Olive">t2</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <span class="p1">(<span class="default"><font color="Olive">t2</font> <a href="trees_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> ) holds <br/><font color="Olive">t1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">t2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t1</font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> , <font color="Maroon">t2</font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:32_1_2_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; ( for <font color="Olive">x</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t1</font> holds   <a href="trees_9.html#K2">succ</a> <font color="Maroon">t1</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="trees_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> ) )
 <b>and </b><br/><a NAME="E2:32_1_2_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">t2</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; ( for <font color="Olive">x</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t2</font> holds   <a href="trees_9.html#K2">succ</a> <font color="Maroon">t2</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <span class="p1">(<span class="default"><font color="Maroon">t2</font> <a href="trees_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t2</font> &amp; <font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E3:32_1_2_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<b>by </b><i>, , <a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/>
<a NAME="E4:32_1_2_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">t</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t1</font><br/> for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t</font>] &amp; <font color="Olive">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">n</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">t1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</font> be    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t1</font>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:32_1_2_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">t</font>] &amp; <font color="Maroon">t</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">n</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">t1</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">t'</font> = <font color="Maroon">t</font> as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t2</font> ;<br/>
<i><font color="Green">E74</font></i>:  <a href="trees_9.html#K2">succ</a> <font color="Maroon">t1</font>,<font color="Maroon">t</font> = 
 <a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <span class="p1">(<span class="default"><font color="Maroon">t2</font> <a href="trees_2.html#K3">.</a> <font color="Maroon">t'</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="trees_9.html#K2">succ</a> <font color="Maroon">t2</font>,<font color="Maroon">t'</font>
<b>by </b><i/>
;<br/>
<a NAME="E4:32_1_2_1_1"/>
<font color="Maroon">t</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">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:32_1_2_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="trees_9.html#T7">TREES_9:7</a></i>;<br/>
<a NAME="E6:32_1_2_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="trees_9.html#K1">succ</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="trees_9.html#D5">TREES_9:def 5</a></i>;<br/>
<a NAME="E7:32_1_2_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="trees_9.html#K2">succ</a> <font color="Maroon">t1</font>,<font color="Maroon">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="trees_9.html#T29">TREES_9:29</a></i>;<br/>
<a NAME="E8:32_1_2_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">t'</font> <a href="trees_9.html#K1">succ</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="trees_9.html#T29">TREES_9:29</a></i>;<br/>
<a NAME="E9:32_1_2_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="trees_2.html#K1">succ</a> <font color="Maroon">t'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="trees_9.html#D5">TREES_9:def 5</a></i>;<br/>
<a NAME="E10:32_1_2_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">t'</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">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">t'</font>
 
<b>by </b><i><a class="ref" href="trees_9.html#T7">TREES_9:7</a></i>;<br/>
<b>hence </b><a NAME="E11:32_1_2_1_1"/>
<font color="Maroon">t</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">n</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">t2</font>
 ;<br/>

<b>thus </b><font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="trees_9.html#K2">succ</a> <font color="Maroon">t2</font>,<font color="Maroon">t'</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">t2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:32_1_2_1"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">t</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t1</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t</font>]
 
<b>from </b><i><a class="ref" href="trees_2.html#S1">TREES_2:sch 1</a>(, );<br/></i>
<a NAME="E6:32_1_2_1"/><b>then </b>
for <font color="Olive">t</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t1</font> holds <br/><font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t2</font>
 
;<br/>
<a NAME="E7:32_1_2_1"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">t1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:32_1_2_1"/>
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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t1</font> holds <br/><font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:32_1_2_1"/>
<font color="Maroon">t1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">t2</font>
 <b>by </b><i>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:32_1_2"/><b>then </b>
( <font color="Maroon">t1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">t2</font> &amp; <font color="Maroon">t2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">t1</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:32_1_2"/>
<font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
