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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:109"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V6">closed</a>
 <b>and </b><br/><a NAME="E2:109"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:109"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="cqc_the1.html#K1">Cn</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="cqc_the2.html#E2:109"><i><font color="Green">E74</font></i></a>, <a class="ref" href="cqc_the1.html#D9">CQC_THE1:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="cqc_the1.html#K2">Proof_Step_Kinds</a> </span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E5:109"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="cqc_the1.html#R2">is_a_proof_wrt</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E6:109"/><i><font color="Green">E76</font></i>: 
 <a href="cqc_the1.html#K3">Effect</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T70">CQC_THE1:70</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> implies for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a>  holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Olive">b<sub>1</sub></font> );<br/>
<a NAME="E7:109"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:109_1"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a>  holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:109_1"/><i><font color="Green">E79</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>

<b>assume </b><a NAME="E3:109_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 ;<br/>

<a NAME="E4:109_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="cqc_the1.html#R1">is_a_correct_step_wrt</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="cqc_the2.html#E5:109"><i><font color="Green">E75</font></i></a>, <a class="txt" href="cqc_the2.html#E2:109_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="cqc_the1.html#D5">CQC_THE1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:109_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 0 or <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 1 or <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 2 or <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 3 or <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 4 or <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 5 or <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 6 or <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 7 or <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 8 or <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 9 )
 </span><b>by </b><i><a class="txt" href="cqc_the2.html#E2:109_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="cqc_the1.html#T45">CQC_THE1:45</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1_1"><b>suppose </b></a><a NAME="E1:109_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:109_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E3:109_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="cqc_the2.html#E4:109_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="cqc_the1.html#D4">CQC_THE1:def 4</a></i>;<br/><a NAME="E3:109_1_1_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:109_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="cqc_the2.html#E3:109_1_1_1_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:109_1_1_1_1_1_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:109_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="cqc_the1.html#K1">Cn</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T38">CQC_THE1:38</a></i>;<br/><a NAME="E3:109_1_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="cqc_the1.html#K1">Cn</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> is <a href="cqc_the1.html#V2">valid</a> )
 <b>by </b><i><a class="txt" href="cqc_the2.html#E1:109_1_1_1_1_1_1_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="lukasi_1.html#T45">LUKASI_1:45</a></i>;<br/><a NAME="E4:109_1_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>6</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="cqc_the1.html#T98">CQC_THE1:98</a>, <a class="ref" href="cqc_the1.html#D9">CQC_THE1:def 9</a></i>;<br/><b>hence </b><a NAME="E5:109_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T92">CQC_THE1:92</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:109_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:109_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>6</sub></font> is <a href="cqc_the1.html#V2">valid</a>
 <b>by </b><i><a class="ref" href="lukasi_1.html#T44">LUKASI_1:44</a></i>;<br/><b>hence </b><a NAME="E3:109_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T98">CQC_THE1:98</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:109_1_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1_2"><b>suppose </b></a><a NAME="E1:109_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:109_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K9">VERUM</a> 
 <b>by </b><i><a class="txt" href="cqc_the2.html#E3:109_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="cqc_the2.html#E4:109_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="cqc_the1.html#D4">CQC_THE1:def 4</a></i>;<br/><a NAME="E3:109_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>6</sub></font> is <a href="cqc_the1.html#V2">valid</a>
 <b>by </b><i><a class="ref" href="lukasi_1.html#T50">LUKASI_1:50</a></i>;<br/><b>hence </b><a NAME="E4:109_1_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T98">CQC_THE1:98</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1_3"><b>suppose </b></a><a NAME="E1:109_1_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:109_1_1_1_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E3:109_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="cqc_the2.html#E4:109_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="cqc_the1.html#D4">CQC_THE1:def 4</a></i>;<br/><a NAME="E3:109_1_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="cqc_the1.html#V2">valid</a>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T100">CQC_THE1:100</a></i>;<br/><a NAME="E4:109_1_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T98">CQC_THE1:98</a></i>;<br/><b>hence </b><a NAME="E5:109_1_1_1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="lukasi_1.html#T78">LUKASI_1:78</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1_4"><b>suppose </b></a><a NAME="E1:109_1_1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><a NAME="E2:109_1_1_1_4"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E3:109_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="cqc_the2.html#E4:109_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="cqc_the1.html#D4">CQC_THE1:def 4</a></i>;<br/><a NAME="E3:109_1_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="cqc_the1.html#V2">valid</a>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T101">CQC_THE1:101</a></i>;<br/><a NAME="E4:109_1_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T98">CQC_THE1:98</a></i>;<br/><b>hence </b><a NAME="E5:109_1_1_1_4"/>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="lukasi_1.html#T78">LUKASI_1:78</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1_5"><b>suppose </b></a><a NAME="E1:109_1_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><a NAME="E2:109_1_1_1_5"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E3:109_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="cqc_the2.html#E4:109_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="cqc_the1.html#D4">CQC_THE1:def 4</a></i>;<br/><a NAME="E3:109_1_1_1_5"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="cqc_the1.html#V2">valid</a>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T102">CQC_THE1:102</a></i>;<br/><a NAME="E4:109_1_1_1_5"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T98">CQC_THE1:98</a></i>;<br/><b>hence </b><a NAME="E5:109_1_1_1_5"/>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="lukasi_1.html#T78">LUKASI_1:78</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1_6"><b>suppose </b></a><a NAME="E1:109_1_1_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 5
 ;<br/><div class="add"><a NAME="E2:109_1_1_1_6"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E3:109_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="cqc_the2.html#E4:109_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="cqc_the1.html#D4">CQC_THE1:def 4</a></i>;<br/><a NAME="E3:109_1_1_1_6"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="cqc_the1.html#V2">valid</a>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T103">CQC_THE1:103</a></i>;<br/><a NAME="E4:109_1_1_1_6"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T98">CQC_THE1:98</a></i>;<br/><b>hence </b><a NAME="E5:109_1_1_1_6"/>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="lukasi_1.html#T78">LUKASI_1:78</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1_7"><b>suppose </b></a><a NAME="E1:109_1_1_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 6
 ;<br/><div class="add"><a NAME="E2:109_1_1_1_7"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ex <font color="Olive">b<sub>2</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a> st <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E3:109_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="cqc_the2.html#E4:109_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="cqc_the1.html#D4">CQC_THE1:def 4</a></i>;<br/><a NAME="E3:109_1_1_1_7"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="cqc_the1.html#V2">valid</a>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T105">CQC_THE1:105</a></i>;<br/><a NAME="E4:109_1_1_1_7"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T98">CQC_THE1:98</a></i>;<br/><b>hence </b><a NAME="E5:109_1_1_1_7"/>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="lukasi_1.html#T78">LUKASI_1:78</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1_8"><b>suppose </b></a><a NAME="E1:109_1_1_1_8"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 7
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</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="E3:109_1_1_1_8"/><i><font color="Green">E82</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>and </b><br/><a NAME="E4:109_1_1_1_8"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="cqc_the2.html#E3:109_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="cqc_the2.html#E4:109_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="cqc_the1.html#D4">CQC_THE1:def 4</a></i>;<br/><a NAME="E5:109_1_1_1_8"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="cqc_the2.html#E3:109_1_1_1_8"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:109_1_1_1_8"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="cqc_the2.html#E2:109_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="cqc_the2.html#E3:109_1_1_1_8"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:109_1_1_1_8"/><b>then </b><i><font color="Green">E84</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="cqc_the2.html#E1:109_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="cqc_the2.html#E4:109_1_1_1_8"><i><font color="Green">E83</font></i></a></i>;<br/><a NAME="E8:109_1_1_1_8"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> is <a href="cqc_the1.html#V2">valid</a>
 <b>by </b><i><a class="ref" href="lukasi_1.html#T54">LUKASI_1:54</a></i>;<br/><a NAME="E9:109_1_1_1_8"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T98">CQC_THE1:98</a></i>;<br/><a NAME="E10:109_1_1_1_8"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E7:109_1_1_1_8"><i><font color="Green">E84</font></i></a>, <a class="ref" href="cqc_the1.html#T92">CQC_THE1:92</a></i>;<br/><b>hence </b><a NAME="E11:109_1_1_1_8"/>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E4:109_1_1_1_8"><i><font color="Green">E83</font></i></a>, <a class="txt" href="cqc_the2.html#E7:109_1_1_1_8"><i><font color="Green">E84</font></i></a>, <a class="ref" href="cqc_the1.html#T92">CQC_THE1:92</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1_9"><b>suppose </b></a><a NAME="E1:109_1_1_1_9"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 8
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , <font color="Maroon">c<sub>10</sub></font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a><b> such that </b><br/><a NAME="E3:109_1_1_1_9"/><i><font color="Green">E82</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>and </b><br/><a NAME="E4:109_1_1_1_9"/><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E5:109_1_1_1_9"/><i><font color="Green">E84</font></i>: 
( not <font color="Maroon">c<sub>10</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>8</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="cqc_the2.html#E3:109_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="cqc_the2.html#E4:109_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="cqc_the1.html#D4">CQC_THE1:def 4</a></i>;<br/><a NAME="E6:109_1_1_1_9"/><i><font color="Green">E85</font></i>: 
not <font color="Maroon">c<sub>10</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>2</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E1:109"><i><font color="Green">E73</font></i></a>, <a class="ref" href="qc_lang1.html#D30">QC_LANG1:def 30</a></i>;<br/><a NAME="E7:109_1_1_1_9"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E2:109_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="cqc_the2.html#E3:109_1_1_1_9"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:109_1_1_1_9"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E1:109_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="cqc_the2.html#E3:109_1_1_1_9"><i><font color="Green">E82</font></i></a>, <a class="txt" href="cqc_the2.html#E4:109_1_1_1_9"><i><font color="Green">E83</font></i></a></i>;<br/><a NAME="E9:109_1_1_1_9"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_the2.html#T94" target="_self">Th94</a></i>;<br/><a NAME="E10:109_1_1_1_9"/>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> is <a href="cqc_the1.html#V2">valid</a>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E6:109_1_1_1_9"><i><font color="Green">E85</font></i></a>, <a class="ref" href="cqc_the2.html#T78" target="_self">Th78</a></i>;<br/><a NAME="E11:109_1_1_1_9"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T98">CQC_THE1:98</a></i>;<br/><a NAME="E12:109_1_1_1_9"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E9:109_1_1_1_9"><i><font color="Green">E86</font></i></a>, <a class="ref" href="cqc_the1.html#T92">CQC_THE1:92</a></i>;<br/><a NAME="E13:109_1_1_1_9"/>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E5:109_1_1_1_9"><i><font color="Green">E84</font></i></a>, <a class="ref" href="cqc_the2.html#T95" target="_self">Th95</a></i>;<br/><b>hence </b><a NAME="E14:109_1_1_1_9"/>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E5:109_1_1_1_9"><i><font color="Green">E84</font></i></a>, <a class="txt" href="cqc_the2.html#E12:109_1_1_1_9"><i><font color="Green">E87</font></i></a>, <a class="ref" href="lukasi_1.html#T76">LUKASI_1:76</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1_10"><b>suppose </b></a><a NAME="E1:109_1_1_1_10"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 9
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a>, <font color="Maroon">c<sub>10</sub></font> being   <a href="qc_lang1.html#NM8">QC-formula</a><b> such that </b><br/><a NAME="E3:109_1_1_1_10"/><i><font color="Green">E82</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>and </b><br/><a NAME="E4:109_1_1_1_10"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="cqc_lang.html#K7">CQC-WFF</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="cqc_lang.html#K7">CQC-WFF</a>  )
 <b>and </b><br/><a NAME="E5:109_1_1_1_10"/><i><font color="Green">E84</font></i>: 
( not <font color="Maroon">c<sub>8</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>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="cqc_the2.html#E3:109_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="cqc_the2.html#E4:109_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="cqc_the1.html#D4">CQC_THE1:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>9</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  <b>by </b><i><a class="txt" href="cqc_the2.html#E4:109_1_1_1_10"><i><font color="Green">E83</font></i></a></i>;<br/><a NAME="E7:109_1_1_1_10"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E2:109_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="cqc_the2.html#E3:109_1_1_1_10"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:109_1_1_1_10"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E1:109_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="cqc_the2.html#E3:109_1_1_1_10"><i><font color="Green">E82</font></i></a>, <a class="txt" href="cqc_the2.html#E5:109_1_1_1_10"><i><font color="Green">E84</font></i></a></i>;<br/><a NAME="E9:109_1_1_1_10"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_the2.html#T94" target="_self">Th94</a></i>;<br/><a NAME="E10:109_1_1_1_10"/>
not <font color="Maroon">c<sub>8</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>2</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E1:109"><i><font color="Green">E73</font></i></a>, <a class="ref" href="qc_lang1.html#D30">QC_LANG1:def 30</a></i>;<br/><a NAME="E11:109_1_1_1_10"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_the2.html#T95" target="_self">Th95</a></i>;<br/><a NAME="E12:109_1_1_1_10"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E9:109_1_1_1_10"><i><font color="Green">E85</font></i></a>, <a class="ref" href="cqc_the1.html#T92">CQC_THE1:92</a></i>;<br/><a NAME="E13:109_1_1_1_10"/>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>12</sub></font> is <a href="cqc_the1.html#V2">valid</a>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E5:109_1_1_1_10"><i><font color="Green">E84</font></i></a>, <a class="ref" href="cqc_the2.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E14:109_1_1_1_10"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T98">CQC_THE1:98</a></i>;<br/><b>hence </b><a NAME="E15:109_1_1_1_10"/>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E5:109_1_1_1_10"><i><font color="Green">E84</font></i></a>, <a class="txt" href="cqc_the2.html#E12:109_1_1_1_10"><i><font color="Green">E86</font></i></a>, <a class="ref" href="lukasi_1.html#T76">LUKASI_1:76</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:109_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:109"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4">NAT_1:sch 4</a>(<a class="txt" href="cqc_the2.html#E7:109"><i><font color="Green">E77</font></i></a>);<br/></i>
<a NAME="E9:109"/><i><font color="Green">E79</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="cqc_the2.html#E5:109"><i><font color="Green">E75</font></i></a>, <a class="ref" href="cqc_the1.html#T58">CQC_THE1:58</a></i>;<br/>
<a NAME="E10:109"/>
<font color="Maroon">c<sub>4</sub></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="cqc_the2.html#E5:109"><i><font color="Green">E75</font></i></a>, <a class="ref" href="cqc_the1.html#D5">CQC_THE1:def 5</a></i>;<br/>
<a NAME="E11:109"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 
<b>by </b><i><a class="txt" href="cqc_the2.html#E6:109"><i><font color="Green">E76</font></i></a>, <a class="ref" href="cqc_the1.html#D6">CQC_THE1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E12:109"/>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E8:109"><i><font color="Green">E78</font></i></a>, <a class="txt" href="cqc_the2.html#E9:109"><i><font color="Green">E79</font></i></a></i>;<br/>


</div>
