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

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">q</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:49"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">F</font> where <font color="Olive">F</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  :  ex <font color="Olive">f</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" target="_self">Proof_Step_Kinds</a> </span><a href="zfmisc_1.html#K2">:]</a></span> st <br/>( <font color="Olive">f</font> <a href="cqc_the1.html#R2" target="_self">is_a_proof_wrt</a> <font color="Maroon">X</font> &amp;  <a href="cqc_the1.html#K3" target="_self">Effect</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E2:49"/><i><font color="Green">E30</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E3:49"/>
 ex <font color="Olive">t</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font> &amp;  ex <font color="Olive">f</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" target="_self">Proof_Step_Kinds</a> </span><a href="zfmisc_1.html#K2">:]</a></span> st <br/>( <font color="Olive">f</font> <a href="cqc_the1.html#R2" target="_self">is_a_proof_wrt</a> <font color="Maroon">X</font> &amp;  <a href="cqc_the1.html#K3" target="_self">Effect</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> ) )
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">f</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" target="_self">Proof_Step_Kinds</a> </span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E5:49"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">f</font> <a href="cqc_the1.html#R2" target="_self">is_a_proof_wrt</a> <font color="Maroon">X</font> &amp;  <a href="cqc_the1.html#K3" target="_self">Effect</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font> )
 ;<br/>
<a NAME="E6:49"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">qq</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">q</font></span>)</span></span>)</span>,8</span><a href="tarski.html#K4">]</a></span> as    <a href="subset_1.html#M1">Element</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" target="_self">Proof_Step_Kinds</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="ref" href="cqc_the1.html#T10" target="_self">Th10</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>set </b><font color="Maroon">h</font> = <font color="Maroon">f</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">qq</font></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/>
<i><font color="Green">E44</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">qq</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">qq</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/>
<a NAME="E9:49"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">qq</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> holds <br/><font color="Maroon">f</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">qq</font></span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Olive">n</font> <a href="cqc_the1.html#R1" target="_self">is_a_correct_step_wrt</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:49_2"/><i><font color="Green">E46</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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">qq</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_2_1_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">qq</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> )
 </span><b>by </b><i>, , <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_1"><b>suppose </b></a><a NAME="E1:49_2_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:49_2_1_1_1"/><b>then </b>
<font color="Maroon">f</font>,<font color="Maroon">n</font> <a href="cqc_the1.html#R1" target="_self">is_a_correct_step_wrt</a> <font color="Maroon">X</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:49_2_1_1_1"/>
<font color="Maroon">f</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">qq</font></span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">n</font> <a href="cqc_the1.html#R1" target="_self">is_a_correct_step_wrt</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="txt" href="cqc_the1.html#E1:49_2_1_1_1"><i><font color="Green">E49</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_2"><b>suppose </b></a><a NAME="E1:49_2_1_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">qq</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:49_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">qq</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">qq</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/><a NAME="E3:49_2_1_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">qq</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <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> 8 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">qq</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <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">p</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">x</font>,<font color="Maroon">q</font></span>)</span> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><a NAME="E4:49_2_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, </i>;<br/><a NAME="E5:49_2_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/><a NAME="E6:49_2_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><i><font color="Green">E52</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">qq</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">f</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">f</font></span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font>
<b>by </b><i>, <a class="txt" href="cqc_the1.html#E1:49_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="cqc_the1.html#E5:49"><i><font color="Green">E31</font></i></a></i>
;<br/><a NAME="E8:49_2_1_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 <b>by </b><i>, , , , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E9:49_2_1_1_2"/>
<font color="Maroon">f</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">qq</font></span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">n</font> <a href="cqc_the1.html#R1" target="_self">is_a_correct_step_wrt</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="cqc_the1.html#E8:49"><i><font color="Green">E44</font></i></a>, <a class="txt" href="cqc_the1.html#E1:49_2_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="cqc_the1.html#E3:49_2_1_1_2"><i><font color="Green">E51</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:49_2"/>
<font color="Maroon">f</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">qq</font></span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">n</font> <a href="cqc_the1.html#R1" target="_self">is_a_correct_step_wrt</a> <font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:49"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">f</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">qq</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="cqc_the1.html#R2" target="_self">is_a_proof_wrt</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
 <a href="cqc_the1.html#K3" target="_self">Effect</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">qq</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">qq</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="txt" href="cqc_the1.html#E5:49"><i><font color="Green">E31</font></i></a></i>
<br/>.= 
<font color="Maroon">qq</font> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
<br/>.= 
<font color="Maroon">p</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">x</font>,<font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
<b>hence </b><a NAME="E12:49"/>
<font color="Maroon">p</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">x</font>,<font color="Maroon">q</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">G</font> where <font color="Olive">G</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  :  ex <font color="Olive">f</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" target="_self">Proof_Step_Kinds</a> </span><a href="zfmisc_1.html#K2">:]</a></span> st <br/>( <font color="Olive">f</font> <a href="cqc_the1.html#R2" target="_self">is_a_proof_wrt</a> <font color="Maroon">X</font> &amp;  <a href="cqc_the1.html#K3" target="_self">Effect</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> ) </span>}</span> 
 <b>by </b><i/>;<br/>


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