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

<b>consider </b><font color="Maroon" title="c2">W</font> being  <a href="trees_9.html#V4" title="TREES_9:attr.4">finite-branching</a> <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> <b> such that </b><br/><a NAME="E2:32_1_1"/><i><font color="Green" title="E28">A1</font></i>: 
( <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">W</font><br/> for <font color="Olive" title="b2">w</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a>   st <font color="Olive" title="b2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> <a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> <font color="Maroon" title="c2">W</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <font color="Olive" title="b2">w</font> ) )
 <b>from </b><i><a class="ref" href="qc_lang4.html#S1" target="_self" title="QC_LANG4:sch.1">QC_LANG4:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> $1</span>)</span>;<br/>
<a NAME="E3:32_1_1"/><i><font color="Green" title="E29">A2</font></i>: 
for <font color="Olive" title="b1">t</font>, <font color="Olive" title="b2">t'</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">W</font><br/> for <font color="Olive" title="b3">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a>   st <font color="Olive" title="b2">t'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b3">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">t'</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b3">d</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">W</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">t</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">t</font>, <font color="Maroon" title="c4">t'</font> be    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">W</font>;<br/><b>let </b><font color="Maroon" title="c5">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:32_1_1_1"/><i><font color="Green" title="E30">A3</font></i>: 
( <font color="Maroon" title="c4">t'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c3">t</font> &amp; <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c4">t'</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:32_1_1_1"/><i><font color="Green" title="E31">A4</font></i>: 
( <font color="Maroon" title="c4">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp; <font color="Maroon" title="c3">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">W</font> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E1:32_1_1_1"><i><font color="Green" title="E30">A3</font></i></a></i>;<br/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c3">t</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> <font color="Maroon" title="c2">W</font>,<font color="Maroon" title="c3">t</font></span>)</span>
<b>by </b><i><a class="txt" href="qc_lang4.html#E2:32_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a> </span>)</span>
<b>by </b><i><a class="ref" href="qc_lang4.html#T11" target="_self" title="QC_LANG4:th.11">Th11</a></i>
;<br/>
<a NAME="E5:32_1_1_1"/><b>then </b><i><font color="Green" title="E32">A5</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c3">t</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E3:32_1_1_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="qc_lang4.html#T12" target="_self" title="QC_LANG4:th.12">Th12</a></i>;<br/>
<font color="Maroon" title="c2">W</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c4">t'</font> = 
<span class="p1">(<span class="default"><a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> <font color="Maroon" title="c2">W</font>,<font color="Maroon" title="c3">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="qc_lang4.html#E3:32_1_1_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="qc_lang4.html#T13" target="_self" title="QC_LANG4:th.13">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c3">t</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="qc_lang4.html#E2:32_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E7:32_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">d</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">W</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c3">t</font>)
 <b>by </b><i><a class="txt" href="qc_lang4.html#E1:32_1_1_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="qc_lang4.html#E5:32_1_1_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="qc_lang4.html#T30" target="_self" title="QC_LANG4:th.30">Th30</a>, <a class="ref" href="qc_lang2.html#T71" title="QC_LANG2:th.71">QC_LANG2:71</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:32_1_1"/>
<font color="Maroon" title="c2">W</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>from </b><i><a class="ref" href="qc_lang4.html#S2" target="_self" title="QC_LANG4:sch.2">QC_LANG4:sch 2</a>(<a class="txt" href="qc_lang4.html#E3:32_1_1"><i><font color="Green" title="E29">A2</font></i></a>);<br/></i>
<b>then reconsider </b><font color="Maroon" title="c3">W</font> = <font color="Maroon" title="c2">W</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c3">W</font>
;<br/>

<b>thus </b><a NAME="E6:32_1_1"/>
( <font color="Maroon" title="c3">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">W</font> holds   <a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> <font color="Maroon" title="c3">W</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E2:32_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>


</div>
