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

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></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">E28</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> holds  <a href="trees_9.html#K2">succ</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font> <a href="trees_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 <b>and </b><br/><a NAME="E2:32_1_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> holds  <a href="trees_9.html#K2">succ</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font> <a href="trees_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 ;<br/>

<a NAME="E3:32_1_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font> holds  <a href="trees_9.html#K2">succ</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></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">b<sub>1</sub></font> <a href="trees_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> ) &amp; <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>2</sub></font> holds  <a href="trees_9.html#K2">succ</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></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">b<sub>2</sub></font> <a href="trees_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> ) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></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">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></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">E30</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> holds  <a href="trees_9.html#K2">succ</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font> <a href="trees_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 <b>and </b><br/><a NAME="E2:32_1_2_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> holds  <a href="trees_9.html#K2">succ</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="trees_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ( <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">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></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">E32</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E1:32_1_2_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="qc_lang4.html#E2:32_1_2_1"><i><font color="Green">E31</font></i></a>, <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">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Olive">b<sub>1</sub></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">b<sub>2</sub></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">c<sub>4</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></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">b<sub>2</sub></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">c<sub>6</sub></font> be    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:32_1_2_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>] &amp; <font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></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">c<sub>4</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> ;<br/>
<i><font color="Green">E35</font></i>:  <a href="trees_9.html#K2">succ</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> = 
 <a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="qc_lang4.html#E1:32_1_2_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="qc_lang4.html#E1:32_1_2_1_1"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
 <a href="trees_9.html#K2">succ</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="qc_lang4.html#E2:32_1_2_1"><i><font color="Green">E31</font></i></a></i>
;<br/>
<a NAME="E4:32_1_2_1_1"/>
<font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></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">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E1:32_1_2_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E5:32_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></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">c<sub>6</sub></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">c<sub>7</sub></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">c<sub>6</sub></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">c<sub>7</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></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">c<sub>8</sub></font> <a href="trees_9.html#K1">succ</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E3:32_1_2_1_1"><i><font color="Green">E35</font></i></a>, <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">c<sub>7</sub></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">c<sub>8</sub></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">E36</font></i>: 
<font color="Maroon">c<sub>8</sub></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">c<sub>7</sub></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">c<sub>8</sub></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">c<sub>6</sub></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">c<sub>7</sub></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">c<sub>5</sub></font>
 ;<br/>

<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></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">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="qc_lang4.html#E1:32_1_2_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="qc_lang4.html#E3:32_1_2_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="qc_lang4.html#T13" target="_self">Th13</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="qc_lang4.html#E10:32_1_2_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="qc_lang4.html#T13" target="_self">Th13</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:32_1_2_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="trees_2.html#S1">TREES_2:sch 1</a>(<a class="txt" href="qc_lang4.html#E3:32_1_2_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="qc_lang4.html#E4:32_1_2_1"><i><font color="Green">E33</font></i></a>);<br/></i>
<a NAME="E6:32_1_2_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
;<br/>
<a NAME="E7:32_1_2_1"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></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">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E5:32_1_2_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:32_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E7:32_1_2_1"><i><font color="Green">E35</font></i></a>, <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">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E1:32_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="qc_lang4.html#E2:32_1_2"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:32_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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