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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>



<b>thus </b><a NAME="E1:76"/>
 <a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:76_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E3:76_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:76_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="calcul_1.html#D5" target="_self">Def5</a></i>;<br/>
<div><i><font color="Green">E70</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:76_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E3:76_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:76_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="calcul_1.html#E1:76_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="calcul_1.html#E2:76_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="calcul_1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E4:76_1_1"/>
 <a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E5:76_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:76_1_1"><i><font color="Green">E73</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E1:76_1_2"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E2:76_1_2"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:76_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E3:76_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:76_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="calcul_1.html#E1:76_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="calcul_1.html#E2:76_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="calcul_1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E4:76_1_2"/>
 <a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E5:76_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:76_1_2"><i><font color="Green">E73</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:76_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:76_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="calcul_1.html#E4:76_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:76"/>
<span class="p1">(<span class="default"><a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:76_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<div><i><font color="Green">E69</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E3:76_2_1"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="calcul_1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E4:76_2_1"/><i><font color="Green">E71</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T39">FINSEQ_1:39</a></i>;<br/><a NAME="E5:76_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:76_2_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E6:76_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:76_2_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="calcul_1.html#E4:76_2_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="calcul_1.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E3:76_2_2"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="calcul_1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E4:76_2_2"/><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:76_2_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_1.html#T41">FINSEQ_1:41</a></i>;<br/><a NAME="E5:76_2_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:76_2_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E6:76_2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:76_2_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="calcul_1.html#E4:76_2_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="calcul_1.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:76_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:76_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="calcul_1.html#E2:76_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>


</div>
