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

<span class="kw">deffunc </span><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> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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/>
<span class="kw">consider </span><font color="Maroon" title="c2">W</font> being   <a href="trees_9.html#V3" title="TREES_9:attr.3">finite-branching</a>  <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> <span class="kw"> such that </span><br/><a NAME="E2:6_1_1"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
( <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> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="trees_9.html#S2" title="TREES_9:sch.2">TREES_9:sch 2</a>();<br/></span>
<a NAME="E3:6_1_1"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
 for <font color="Olive" title="b1">t</font>, <font color="Olive" title="b2">t9</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">t9</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">t9</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="6_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">t</font>, <font color="Maroon" title="c4">t9</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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="Maroon" title="c4">t9</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="Olive" title="b1">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">t9</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">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>)</span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">t9</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">t9</font> implies <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>) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:6_1_1_1"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
<font color="Maroon" title="c4">t9</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>
 <span class="kw">and </span><br/><a NAME="E2:6_1_1_1"/><span class="lab"><font color="Green" title="E8">A4</font></span>: 
<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">t9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/>

<span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E4:6_1_1_1"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
<font color="Maroon" title="c4">t9</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>
 <span class="kw">and </span><br/><a NAME="E5:6_1_1_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E10">A6</font></span>: <font color="Maroon" title="c2">W</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c4">t9</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6_1_1_1"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="trees_9.html#T40" title="TREES_9:th.40">TREES_9:40</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>
;<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_9.html#T38" title="TREES_9:th.38">TREES_9:38</a></span>
;<br/>
<a NAME="E8:6_1_1_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6_1_1_1"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="trees_9.html#T39" title="TREES_9:th.39">TREES_9:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:6_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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6_1_1_1"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="txt" href="#E6:6_1_1_1"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="qc_lang4.html#T3" target="_self" title="QC_LANG4:th.3">Th30</a>, <a class="ref" href="qc_lang2.html#T51" title="QC_LANG2:th.51">QC_LANG2:51</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:6_1_1"/>
<font color="Maroon" title="c2">W</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="trees_9.html#S3" title="TREES_9:sch.3">TREES_9:sch 3</a>(<a class="txt" href="#E3:6_1_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>);<br/></span>
<span class="kw">then </span><span class="kw">reconsider </span><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#NM5" title="TREES_2:NM.5">DecoratedTree</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a>  ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">W</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> ) )</span><br/>

<span class="kw">thus </span><a NAME="E6:6_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> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
