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

<b>let </b><font color="Maroon">F</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/><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> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span>;<br/><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> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span>;<br/>





<b>set </b><font color="Maroon">G</font> = <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font>;<br/>
<b>set </b><font color="Maroon">H</font> = <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t'</font>;<br/>
<b>assume </b><a NAME="E1:42"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">t</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">t'</font>
 ;<br/>

<a NAME="E2:42"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">t</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">t'</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E3:42"/><b>then </b>
<span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t'</font> <a href="qc_lang2.html#R2">is_subformula_of</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:42"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t'</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1"/><i><font color="Green">E32</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t'</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span>)</span>
 ;<br/><b>consider </b><font color="Maroon">r</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:42_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">t'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="trees_1.html#T8">TREES_1:8</a></i>;<br/><b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="finseq_1.html#T50">FINSEQ_1:50</a></i>;<br/><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">t1</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> ex <font color="Olive">r1</font> being  <a href="finseq_1.html#NM1">FinSequence</a> ex <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Olive">m</font></span>)</span> &amp; <font color="Olive">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">r1</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Olive">t1</font> );<br/><a NAME="E5:42_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 ;<br/><a NAME="E6:42_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font></span>)</span> holds <br/> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:42_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font></span>)</span>
 ;<br/>

<a NAME="E2:42_1_1"/>
<font color="Maroon">r</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> is   <a href="finseq_1.html#NM1">FinSequence</a>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T19">FINSEQ_1:19</a></i>;<br/>
<b>then consider </b><font color="Maroon">r1</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E4:42_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span>
 ;<br/>
<a NAME="E5:42_1_1"/>
<font color="Maroon">r1</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E9:42_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="trees_1.html#D1">TREES_1:def 1</a></i>;<br/>
<a NAME="E6:42_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r1</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="finseq_6.html#T15">FINSEQ_6:15</a></i>;<br/>
<a NAME="E7:42_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="trees_1.html#T45">TREES_1:45</a></i>;<br/>
<b>then consider </b><font color="Maroon">t1</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span><b> such that </b><br/><a NAME="E9:42_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r1</font>
 ;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:42_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font>
 ;<br/>
<b>thus </b><a NAME="E12:42_1_1"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Olive">x</font>]
 <b>by </b><i><a class="txt" href="qc_lang4.html#E9:42_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="qc_lang4.html#E10:42_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="qc_lang4.html#T7" target="_self">Th7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:42_1"/>
 ex <font color="Olive">L</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K4">dom</a> <font color="Olive">L</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font></span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S1">FINSEQ_1:sch 1</a>(, );<br/></i><b>then consider </b><font color="Maroon">L</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E9:42_1"/><i><font color="Green">E44</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font></span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font></span>)</span> holds <br/> ex <font color="Olive">t1</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> ex <font color="Olive">r1</font> being  <a href="finseq_1.html#NM1">FinSequence</a> ex <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> &amp; <font color="Olive">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Olive">m</font></span>)</span> &amp; <font color="Olive">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">r1</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Olive">t1</font> ) ) )
 ;<br/><a NAME="E10:42_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> holds <br/> ex <font color="Olive">t1</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> ex <font color="Olive">r1</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Olive">k</font></span>)</span> &amp; <font color="Olive">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">r1</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Olive">t1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:42_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> )
 ;<br/>

<a NAME="E2:42_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>then consider </b><font color="Maroon">t1</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span>, <font color="Maroon">r1</font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">m</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="E4:42_1_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E9:42_1"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:42_1_2"/>
 ex <font color="Olive">t1</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> ex <font color="Olive">r1</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Olive">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">r1</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Olive">t1</font> )
 <b>by </b><i><a class="ref" href="qc_lang4.html#T7" target="_self">Th7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:42_1"/><i><font color="Green">E75</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:42_1_3"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E5:42_1"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t1</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <b>by </b><i><a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/>
<a NAME="E3:42_1_3"/>
<font color="Maroon">t1</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E4:42_1_3"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">t1</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E5:42_1_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E6:42_1_3"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="trees_1.html#T24">TREES_1:24</a></i>;<br/>
<a NAME="E7:42_1_3"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E8:42_1_3"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font>
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">t1</font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span>, <font color="Maroon">r1</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E13:42_1"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span> &amp; <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E10:42_1"><i><font color="Green">E45</font></i></a></i>;<br/><b>set </b><font color="Maroon">H1</font> = <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font>;<br/><a NAME="E14:42_1"/><i><font color="Green">E80</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:42_1_4"/>
<font color="Maroon">t</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">t1</font>
 
<b>by </b><i>, <a class="ref" href="trees_1.html#T8">TREES_1:8</a></i>;<br/>
<a NAME="E2:42_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font> <a href="qc_lang2.html#R2">is_subformula_of</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E3:42_1"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E3:42_1_4"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:42_1_4"/>
<font color="Maroon">r1</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="trees_1.html#D1">TREES_1:def 1</a></i>;<br/>
<a NAME="E5:42_1_4"/><b>then </b>
<font color="Maroon">t1</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">t'</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_6.html#T15">FINSEQ_6:15</a></i>;<br/>
<a NAME="E6:42_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t'</font> <a href="qc_lang2.html#R2">is_subformula_of</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E3:42_1"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E7:42_1_4"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t'</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:42_1_4"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:42_1"/>
<span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_5"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">m</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="E2:42_1_5"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="qc_lang4.html#T7" target="_self">Th7</a>, , </i>;<br/>
<b>thus </b><a NAME="E3:42_1_5"/>
<span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t1</font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E16:42_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="qc_lang2.html#T71">QC_LANG2:71</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:42"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t'</font></span>)</span></span>)</span> <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="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">F</font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div>
