<?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>assume </b><a NAME="E1:41"/>
<font color="Maroon">t</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">t'</font>
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:41"/><i><font color="Green">E22</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>consider </b><font color="Maroon">n</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="E6:41"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font>
 ;<br/>
<a NAME="E7:41"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<a NAME="E8:41"/><b>then </b><i><font color="Green">E30</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</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">G</font> being  <a href="qc_lang1.html#NM8">QC-formula</a> ex <font color="Olive">m</font>, <font color="Olive">k1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">r1</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &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">k1</font></span>)</span> &amp; <font color="Olive">G</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="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">r1</font></span>)</span> );<br/>
<a NAME="E9:41"/><i><font color="Green">E32</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"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</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="E10:41"/><i><font color="Green">E35</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"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</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="41_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="E1:41_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"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E2:41_1"/><b>then </b>
( 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> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 
<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">k1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:41_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">k1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E6:41_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">k1</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="E8:41_1"/><i><font color="Green">E39</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">k1</font></span>)</span>
 ;<br/>
<a NAME="E9:41_1"/>
<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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">q</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E2:41_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">q</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">k1</font></span>)</span> &amp; <font color="Maroon">q</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">r</font> )
 <b>by </b><i/>;<br/>
<a NAME="E3:41_1_1"/>
<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="txt" href="qc_lang4.html#E12:41_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="qc_lang4.html#E13:41"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_6.html#T15">FINSEQ_6:15</a></i>;<br/>
<b>hence </b><a NAME="E4:41_1_1"/>
<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/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">t1</font> = <font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r1</font> 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> ;<br/>
<b>consider </b><font color="Maroon">G</font> being   <a href="qc_lang1.html#NM8">QC-formula</a><b> such that </b><br/><a NAME="E12:41_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">G</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="E13:41_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#E12:41_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="qc_lang4.html#E13:41"><i><font color="Green">E43</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:41"/>
 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"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</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"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</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="E13:41"/><i><font color="Green">E43</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"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</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"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/> ex <font color="Olive">G</font> being  <a href="qc_lang1.html#NM8">QC-formula</a> ex <font color="Olive">m</font>, <font color="Olive">k1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">r1</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> &amp; <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> &amp; <font color="Olive">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &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">k1</font></span>)</span> &amp; <font color="Olive">G</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="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">r1</font></span>)</span> ) ) )
 ;<br/>
<a NAME="E14:41"/><i><font color="Green">E44</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E15:41"/><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> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 holds <br/> ex <font color="Olive">G</font> being  <a href="qc_lang1.html#NM8">QC-formula</a> ex <font color="Olive">k1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">r1</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &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">k1</font></span>)</span> &amp; <font color="Olive">G</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="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">r1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_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="E1:41_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> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<a NAME="E2:41_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"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</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">G</font> being   <a href="qc_lang1.html#NM8">QC-formula</a>, <font color="Maroon">m</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">r1</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E4:41_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &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">k1</font></span>)</span> &amp; <font color="Maroon">G</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="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r1</font></span>)</span> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E5:41_2"/>
 ex <font color="Olive">G</font> being  <a href="qc_lang1.html#NM8">QC-formula</a> ex <font color="Olive">k1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">r1</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &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">k1</font></span>)</span> &amp; <font color="Olive">G</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="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">r1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E14:41"><i><font color="Green">E44</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">G</font> being   <a href="qc_lang1.html#NM8">QC-formula</a>, <font color="Maroon">k1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">r1</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E17:41"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> 1 &amp; 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &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">k1</font></span>)</span> &amp; <font color="Maroon">G</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="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r1</font></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E18:41"/>
<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E14:41"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E19:41"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E20:41"/><b>then </b><i><font color="Green">E76</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">r1</font>
 
<b>by </b><i>, <a class="txt" href="qc_lang4.html#E14:41"><i><font color="Green">E44</font></i></a>, <a class="ref" href="relat_1.html#T98">RELAT_1:98</a></i>;<br/>
<a NAME="E21:41"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">L</font> <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> <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">t</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">G</font> being   <a href="qc_lang1.html#NM8">QC-formula</a>, <font color="Maroon">k1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">r1</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E2:41_3"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <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> &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &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">k1</font></span>)</span> &amp; <font color="Maroon">G</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="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r1</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="qc_lang4.html#E13:41"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E3:41_3"/>
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="relat_1.html#T110">RELAT_1:110</a></i>;<br/>
<b>hence </b><a NAME="E4:41_3"/>
<font color="Maroon">L</font> <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> <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">t</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:41"/>
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#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 holds <br/> ex <font color="Olive">H1</font>, <font color="Olive">G1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  st <br/>( <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">H1</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">G1</font> &amp; <font color="Olive">H1</font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a> <font color="Olive">G1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4"><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="E1:41_4"/><i><font color="Green">E80</font></i>: 
( 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#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<b>then consider </b><font color="Maroon">H1</font> being   <a href="qc_lang1.html#NM8">QC-formula</a>, <font color="Maroon">k1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">r1</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:41_4"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &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">k1</font></span>)</span> &amp; <font color="Maroon">H1</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="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E13:41"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E4:41_4"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>then consider </b><font color="Maroon">G1</font> being   <a href="qc_lang1.html#NM8">QC-formula</a>, <font color="Maroon">k2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">r2</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E6:41_4"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">r2</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">k2</font></span>)</span> &amp; <font color="Maroon">G1</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="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E13:41"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E7:41_4"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:41_4"/>
<font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r2</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">q</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E2:41_4_1"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">q</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">k2</font></span>)</span> &amp; <font color="Maroon">q</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">r</font> )
 <b>by </b><i/>;<br/>
<a NAME="E3:41_4_1"/>
<font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r2</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/>
<b>hence </b><a NAME="E4:41_4_1"/>
<font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r2</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/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">t2</font> = <font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r2</font> 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> ;<br/>
<a NAME="E10:41_4"/>
<font color="Maroon">k2</font> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4_2"><b>proof </b></a><div class="add">

<a NAME="E1:41_4_2"/>
<font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E2:41_4_2"/>
<font color="Maroon">k2</font> <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="qc_lang4.html#T8" target="_self">Th8</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>


</div><b>end;</b></div>
<b>then 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="E12:41_4"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> <a href="finseq_1.html#K7">^</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>, , </i>;<br/>
<a NAME="E13:41_4"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">t2</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">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">r1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/>
<a NAME="E14:41_4"/>
<font color="Maroon">t2</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">m</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> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">q</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E2:41_4_3"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">q</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">k1</font></span>)</span> &amp; <font color="Maroon">q</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">r</font> )
 <b>by </b><i/>;<br/>
<a NAME="E3:41_4_3"/>
<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="qc_lang4.html#T10" target="_self">Th10</a>, <a class="ref" href="finseq_6.html#T15">FINSEQ_6:15</a></i>;<br/>
<b>hence </b><a NAME="E4:41_4_3"/>
<font color="Maroon">t2</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">m</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> <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="qc_lang4.html#T9" target="_self">Th9</a>, <a class="ref" href="trees_1.html#T45">TREES_1:45</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:41_4"/><b>then </b>
<font color="Maroon">H1</font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a> <font color="Maroon">G1</font>
 
<b>by </b><i>, , <a class="ref" href="qc_lang4.html#T9" target="_self">Th9</a>, </i>;<br/>
<b>hence </b><a NAME="E16:41_4"/>
 ex <font color="Olive">H1</font>, <font color="Olive">G1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  st <br/>( <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">H1</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">G1</font> &amp; <font color="Olive">H1</font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a> <font color="Olive">G1</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E23:41"/>
<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>, , <a class="txt" href="qc_lang4.html#E14:41"><i><font color="Green">E44</font></i></a>, <a class="txt" href="qc_lang4.html#E15:41"><i><font color="Green">E45</font></i></a>, <a class="ref" href="qc_lang4.html#T7" target="_self">Th7</a>, <a class="ref" href="qc_lang2.html#D21">QC_LANG2:def 21</a></i>;<br/>


</div>
