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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font></span>)</span>;<br/>







<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>assume </b><a NAME="E1:51"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V4">conjunctive</a> )
 ;<br/>

<a NAME="E2:51"/><i><font color="Green">E46</font></i>: 
 <a href="trees_9.html#K2">succ</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="qc_lang4.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E3:51"/><i><font color="Green">E47</font></i>: 
 <a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E1:51"><i><font color="Green">E45</font></i></a>, <a class="ref" href="qc_lang4.html#D1" target="_self">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></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">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:51"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="trees_9.html#K2">succ</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</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">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="trees_9.html#K1">succ</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="trees_9.html#D6">TREES_9:def 6</a></i>;<br/>
<a NAME="E6:51"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="trees_9.html#K1">succ</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E2:51"><i><font color="Green">E46</font></i></a>, <a class="txt" href="qc_lang4.html#E3:51"><i><font color="Green">E47</font></i></a>, <a class="txt" href="qc_lang4.html#E5:51"><i><font color="Green">E48</font></i></a>, <a class="ref" href="qc_lang4.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E7:51"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E1:51"><i><font color="Green">E45</font></i></a>, <a class="ref" href="qc_lang4.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E8:51"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E9:51"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:51"/><b>then </b><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E7:51"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="qc_lang4.html#E10:51"><i><font color="Green">E50</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1"><b>suppose </b></a><a NAME="E1:51_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="trees_9.html#K2">succ</a> <span class="p2">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="qc_lang4.html#E1:51"><i><font color="Green">E45</font></i></a>, <a class="ref" href="qc_lang4.html#T13" target="_self">Th13</a></i>
<br/>.= 
 <a href="qc_lang1.html#K18">the_left_argument_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="qc_lang4.html#E2:51"><i><font color="Green">E46</font></i></a>, <a class="txt" href="qc_lang4.html#E3:51"><i><font color="Green">E47</font></i></a>, <a class="txt" href="qc_lang4.html#E1:51_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><b>hence </b><a NAME="E3:51_1_1"/>
( ( <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K18">the_left_argument_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 ) or ( <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K19">the_right_argument_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E1:51_1_1"><i><font color="Green">E51</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_2"><b>suppose </b></a><a NAME="E1:51_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="trees_9.html#K2">succ</a> <span class="p2">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="qc_lang4.html#E1:51"><i><font color="Green">E45</font></i></a>, <a class="ref" href="qc_lang4.html#T13" target="_self">Th13</a></i>
<br/>.= 
 <a href="qc_lang1.html#K19">the_right_argument_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="qc_lang4.html#E2:51"><i><font color="Green">E46</font></i></a>, <a class="txt" href="qc_lang4.html#E3:51"><i><font color="Green">E47</font></i></a>, <a class="txt" href="qc_lang4.html#E1:51_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><b>hence </b><a NAME="E3:51_1_2"/>
( ( <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K18">the_left_argument_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 ) or ( <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K19">the_right_argument_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang4.html#K2" target="_self">tree_of_subformulae</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E1:51_1_2"><i><font color="Green">E51</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
