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

<b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">g</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:53"/><i><font color="Green">E52</font></i>: 
(  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font> <a href="calcul_1.html#R2" target="_self">is_Subsequence_of</a>  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">g</font> &amp;  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">g</font> &amp;  <a href="calcul_1.html#R4" target="_self">|-</a> <font color="Maroon">f</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">PR</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="calcul_1.html#K4" target="_self">set_of_CQC-WFF-seq</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="E3:53"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">PR</font> is <a href="calcul_1.html#V1" target="_self">a_proof</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">PR</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">PR</font></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:53"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="calcul_1.html#K4" target="_self">set_of_CQC-WFF-seq</a>  &amp; 2 <a href="hidden.html#R2">in</a>  <a href="cqc_the1.html#K2">Proof_Step_Kinds</a>  )
 
<b>by </b><i>, <a class="ref" href="cqc_the1.html#T43">CQC_THE1:43</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:53_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,2</span><a href="domain_1.html#K1">]</a></span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/><a NAME="E2:53_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,2</span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="calcul_1.html#E4:53"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><a NAME="E3:53_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,2</span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:53_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="calcul_1.html#K4" target="_self">set_of_CQC-WFF-seq</a> ,<a href="cqc_the1.html#K2">Proof_Step_Kinds</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:53"><i><font color="Green">E53</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:53"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,2</span><a href="domain_1.html#K1">]</a></span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="calcul_1.html#K4" target="_self">set_of_CQC-WFF-seq</a> ,<a href="cqc_the1.html#K2">Proof_Step_Kinds</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">PR1</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,2</span><a href="domain_1.html#K1">]</a></span></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="calcul_1.html#K4" target="_self">set_of_CQC-WFF-seq</a> ,<a href="cqc_the1.html#K2">Proof_Step_Kinds</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">PR2</font> = <font color="Maroon">PR</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">PR1</font>;<br/>
<b>reconsider </b><font color="Maroon">PR2</font> = <font color="Maroon">PR</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">PR1</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="calcul_1.html#K4" target="_self">set_of_CQC-WFF-seq</a> ,<a href="cqc_the1.html#K2">Proof_Step_Kinds</a> </span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<a NAME="E9:53"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">PR</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="calcul_1.html#E1:53"><i><font color="Green">E52</font></i></a>, </i>;<br/>
<a NAME="E10:53"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">PR2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T48">FINSEQ_1:48</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:53_2"/><i><font color="Green">E62</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">PR2</font> )
 ;<br/><a NAME="E2:53_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">PR2</font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E9:53"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><div><i><font color="Green">E66</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_2_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">PR</font>
 ;<br/><a NAME="E2:53_2_1"/><b>then </b><i><font color="Green">E67</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">PR</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:53_2_1"/><b>then </b>
<font color="Maroon">PR</font>,<font color="Maroon">n</font> <a href="calcul_1.html#R3" target="_self">is_a_correct_step</a> 
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:53"><i><font color="Green">E52</font></i></a>, </i>;<br/><b>hence </b><a NAME="E4:53_2_1"/>
<font color="Maroon">PR2</font>,<font color="Maroon">n</font> <a href="calcul_1.html#R3" target="_self">is_a_correct_step</a> 
 <b>by </b><i><a class="ref" href="calcul_1.html#D4" target="_self">Def4</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:53_2_2"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">PR1</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">PR</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> )
 ;<br/><a NAME="E2:53_2_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i><a class="ref" href="calcul_1.html#D4" target="_self">Def4</a>, <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><a NAME="E3:53_2_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:53_2_2"/><b>then </b>
<font color="Maroon">PR1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,2</span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E5:53_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">PR1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">PR1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><a NAME="E6:53_2_2"/><b>then </b><i><font color="Green">E83</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">PR2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">PR2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="calcul_1.html#D4" target="_self">Def4</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E7:53_2_2"/><i><font color="Green">E84</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">PR</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">PR</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E4:53"><i><font color="Green">E54</font></i></a>, <a class="ref" href="calcul_1.html#D4" target="_self">Def4</a>, <a class="ref" href="calcul_1.html#T1" target="_self">Th1</a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:53_2_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">PR</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">PR</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:53_2_2"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">PR2</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">PR</font></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:53"><i><font color="Green">E52</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E10:53_2_2"/>
<font color="Maroon">PR2</font>,<font color="Maroon">n</font> <a href="calcul_1.html#R3" target="_self">is_a_correct_step</a> 
 <b>by </b><i><a class="ref" href="calcul_1.html#D1" target="_self">Def1</a>, <a class="txt" href="calcul_1.html#E10:53"><i><font color="Green">E61</font></i></a>, <a class="txt" href="calcul_1.html#E1:53_2"><i><font color="Green">E62</font></i></a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:53_2"/>
<font color="Maroon">PR2</font>,<font color="Maroon">n</font> <a href="calcul_1.html#R3" target="_self">is_a_correct_step</a> 
 <b>by </b><i><a class="ref" href="calcul_1.html#D2" target="_self">Def2</a>, <a class="ref" href="calcul_1.html#D3" target="_self">Def3</a>, <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:53"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">PR2</font> is <a href="calcul_1.html#V1" target="_self">a_proof</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:53"/>
<font color="Maroon">PR2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">PR2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">PR2</font> <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">PR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">PR1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E14:53"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">PR2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">PR2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">PR2</font> <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">PR</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E15:53"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E16:53"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">PR1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/>
<a NAME="E17:53"/><b>then </b>
<font color="Maroon">PR2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">PR2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">PR1</font> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i><a class="ref" href="calcul_1.html#D2" target="_self">Def2</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<a NAME="E18:53"/><b>then </b>
<font color="Maroon">PR2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">PR2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,2</span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E19:53"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">PR2</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">PR2</font></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E20:53"/>
 <a href="calcul_1.html#R4" target="_self">|-</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E9:53"><i><font color="Green">E56</font></i></a>, </i>;<br/>


</div>
