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

<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>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>thus </b><a NAME="E1:15"/>
( <font color="Maroon">X</font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">p</font> implies  not <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> is <a href="henmodel.html#V1" target="_self">Consistent</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1"/>
<font color="Maroon">X</font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">p</font>
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E3:15_1"/><i><font color="Green">E47</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="calcul_1.html#R4">|-</a> <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">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E4:15_1"/><i><font color="Green">E48</font></i>: 
 <a href="calcul_1.html#R4">|-</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"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">f1</font> = <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"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/>
<a NAME="E5:15_1"/><i><font color="Green">E49</font></i>: 
(  <a href="calcul_1.html#K1">Ant</a> <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"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <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"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp;  <a href="calcul_1.html#K2">Suc</a> <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"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="calcul_1.html#T5">CALCUL_1:5</a></i>;<br/>
<a NAME="E6:15_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="calcul_1.html#K1">Ant</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></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">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="calcul_1.html#K2">Suc</a> <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"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/>
<a NAME="E7:15_1"/>
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#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:15_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/>
<a NAME="E9:15_1"/><b>then </b>
<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 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1">Ant</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T41">FINSEQ_1:41</a></i>;<br/>
<a NAME="E10:15_1"/><b>then </b>
 <a href="calcul_1.html#K2">Suc</a> <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"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="calcul_1.html#R1">is_tail_of</a>  <a href="calcul_1.html#K1">Ant</a> <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"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="calcul_1.html#D3">CALCUL_1:def 3</a></i>;<br/>
<a NAME="E11:15_1"/><b>then </b>
 <a href="calcul_1.html#R4">|-</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"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="calcul_1.html#T33">CALCUL_1:33</a></i>;<br/>
<a NAME="E12:15_1"/><b>then </b>
not <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"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> is <a href="henmodel.html#V2" target="_self">Consistent</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:15_1"/><b>then </b><i><font color="Green">E51</font></i>: 
(  not <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> is <a href="henmodel.html#V1" target="_self">Consistent</a> or  not  <a href="relset_1.html#K5">rng</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"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">f2</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"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/>
<a NAME="E14:15_1"/><i><font color="Green">E52</font></i>: 
(  <a href="calcul_1.html#K1">Ant</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"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="calcul_1.html#K2">Suc</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"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="calcul_1.html#T5">CALCUL_1:5</a></i>;<br/>
<a NAME="E15:15_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <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"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="calcul_1.html#T10">CALCUL_1:10</a></i>;<br/>
<a NAME="E16:15_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&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"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
;<br/>
<a NAME="E17:15_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</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"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="calcul_1.html#T3">CALCUL_1:3</a></i>;<br/>
<b>hence </b><a NAME="E18:15_1"/>
not <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> is <a href="henmodel.html#V1" target="_self">Consistent</a>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/>


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

<b>thus </b><a NAME="E2:15"/>
( not <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> is <a href="henmodel.html#V1" target="_self">Consistent</a> implies <font color="Maroon">X</font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">p</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_2"/>
not <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> is <a href="henmodel.html#V1" target="_self">Consistent</a>
 ;<br/>

<a NAME="E2:15_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E4:15_2"/><i><font color="Green">E54</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="calcul_1.html#R4">|-</a> <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">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_2_1"/>
not  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:15_2_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> <font color="Maroon">f</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">B</font> =  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>set </b><font color="Maroon">n</font> =  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font>;<br/><b>set </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>;<br/><b>set </b><font color="Maroon">h</font> = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span>;<br/><a NAME="E6:15_2_1"/><i><font color="Green">E56</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/><a NAME="E7:15_2_1"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="calcul_2.html#K2">seq</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></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> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="calcul_2.html#K2">seq</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> ) );<br/><a NAME="E8:15_2_1"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">b1</font>, <font color="Olive">b2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">k</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">b1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">b2</font>] holds <br/><font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <font color="Olive">b2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E9:15_2_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span></span>)</span> holds <br/> ex <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">a</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:15_2_1_1"/>
<font color="Maroon">k</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/>

<div><i><font color="Green">E60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_2_1_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">k</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> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:15_2_1_1_1"/><i><font color="Green">E67</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="calcul_2.html#K2">seq</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:15_2_1_1_1"/><i><font color="Green">E68</font></i>: 
not <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="calcul_2.html#K2">seq</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>take </b><font color="Maroon">a</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>;<br/><b>thus </b><a NAME="E4:15_2_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">a</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E3:15_2_1_1"/>
 ex <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Olive">a</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">F</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E11:15_2_1"/><i><font color="Green">E69</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span></span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S1">FINSEQ_1:sch 1</a>(, );<br/></i><i><font color="Green">E70</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></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">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_3.html#T66">FINSEQ_3:66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>

;<br/><a NAME="E13:15_2_1"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">B</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:15_2_1_3"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><a NAME="E2:15_2_1_3"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:15_2_1_3"/>
<font color="Maroon">b</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#D3">FINSEQ_1:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E16:15_2_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</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="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E17:15_2_1"/><b>then </b><i><font color="Green">E74</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T45">FINSEQ_3:45</a></i>;<br/><a NAME="E18:15_2_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T167">RELAT_1:167</a></i>;<br/><b>then </b><i><font color="Green">E75</font></i>:  <a href="card_1.html#K4">card</a> <font color="Maroon">C</font> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">B</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#T63">CARD_2:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</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="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_3.html#T66">FINSEQ_3:66</a></i>
;<br/><a NAME="E20:15_2_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T167">RELAT_1:167</a></i>;<br/><a NAME="E21:15_2_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</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#D3">FINSEQ_1:def 3</a></i>;<br/><div><i><font color="Green">E77</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:15_2_1_5"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="calcul_2.html#K2">seq</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> as   <a href="int_1.html#NM1">Integer</a> ;<br/><a NAME="E3:15_2_1_5"/><i><font color="Green">E79</font></i>: 
( 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</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">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="calcul_2.html#T1">CALCUL_2:1</a></i>;<br/><a NAME="E4:15_2_1_5"/><b>then </b><i><font color="Green">E80</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E5:15_2_1_5"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E7:15_2_1_5"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E8:15_2_1_5"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T45">FINSEQ_3:45</a></i>;<br/></div><b>end;</b></div><a NAME="E23:15_2_1"/><i><font color="Green">E81</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_6_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="E1:15_2_1_6_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/><b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:15_2_1_6_1"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><a NAME="E4:15_2_1_6_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="calcul_2.html#K2">seq</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 <b>by </b><i>, , , </i>;<br/><div><i><font color="Green">E102</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_6_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_2_1_6_1_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">i</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> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:15_2_1_6_1_1"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:15_2_1_6_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="finseq_3.html#T45">FINSEQ_3:45</a></i>;<br/><a NAME="E4:15_2_1_6_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E5:15_2_1_6_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><b>hence </b><a NAME="E6:15_2_1_6_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_6_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_2_1_6_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="calcul_2.html#K2">seq</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span>
 ;<br/><a NAME="E2:15_2_1_6_1_2"/><b>then </b><i><font color="Green">E105</font></i>: 
( <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, , </i>;<br/><a NAME="E3:15_2_1_6_1_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E4:15_2_1_6_1_2"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E5:15_2_1_6_1_2"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T167">RELAT_1:167</a></i>;<br/><b>hence </b><a NAME="E6:15_2_1_6_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:15_2_1_6_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:15_2_1_6"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">a</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="E3:15_2_1_6"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<div><i><font color="Green">E108</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_6_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_2_1_6_2"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:15_2_1_6_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:15_2_1_6_2"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><b>set </b><font color="Maroon">m</font> = <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>;<br/><a NAME="E5:15_2_1_6_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a></i>;<br/><a NAME="E6:15_2_1_6_2"/><b>then </b><i><font color="Green">E111</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:15_2_1_6_2"/><b>then </b><i><font color="Green">E112</font></i>: 
( 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</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">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E8:15_2_1_6_2"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="calcul_2.html#K2">seq</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="calcul_2.html#T1">CALCUL_2:1</a></i>;<br/><a NAME="E9:15_2_1_6_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E10:15_2_1_6_2"/><b>then </b><i><font color="Green">E114</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></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">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/><a NAME="E11:15_2_1_6_2"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E12:15_2_1_6_2"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E13:15_2_1_6_2"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E14:15_2_1_6_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><span class="p0">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E15:15_2_1_6_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E16:15_2_1_6_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_6_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_2_1_6_3"/>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:15_2_1_6_3"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_6_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:15_2_1_6_3_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><a NAME="E2:15_2_1_6_3_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:15_2_1_6_3_1"/>
<font color="Maroon">b</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#D3">FINSEQ_1:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E4:15_2_1_6_3"/><b>then </b>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</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="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E5:15_2_1_6_3"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:15_2_1_6_3"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">C</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><a NAME="E8:15_2_1_6_3"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">C</font></span>)</span> <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_3.html#K1">-</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:15_2_1_6_3"/><b>then </b><i><font color="Green">E119</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</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_3.html#K1">-</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:15_2_1_6_3"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">i</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> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E11:15_2_1_6_3"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="calcul_1.html#T6">CALCUL_1:6</a></i>;<br/><a NAME="E12:15_2_1_6_3"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:15_2_1_6_3"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">i</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E14:15_2_1_6_3"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E15:15_2_1_6_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:15_2_1_6"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">B</font>,<font color="Maroon">B</font> <b>by </b><i>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/><a NAME="E25:15_2_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">B</font>,<font color="Maroon">B</font> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E27:15_2_1"/>
<font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a1</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon">a2</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:15_2_1_7"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a2</font> )
 ;<br/>

<a NAME="E2:15_2_1_7"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="calcul_2.html#K2">seq</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="calcul_2.html#K2">seq</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span> )
 
<b>by </b><i>, , , </i>;<br/>
<div><i><font color="Green">E124</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_2_1_7_1"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">a1</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> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:15_2_1_7_1"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:15_2_1_7_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="finseq_3.html#T45">FINSEQ_3:45</a></i>;<br/><a NAME="E4:15_2_1_7_1"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E5:15_2_1_7_1"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><div><i><font color="Green">E129</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_7_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_2_1_7_1_1"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">a2</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> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:15_2_1_7_1_1"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:15_2_1_7_1_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="finseq_3.html#T45">FINSEQ_3:45</a></i>;<br/><a NAME="E4:15_2_1_7_1_1"/><i><font color="Green">E133</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a2</font>
 <b>by </b><i>, , , </i>;<br/><a NAME="E5:15_2_1_7_1_1"/>
 <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">C</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T99">FINSEQ_3:99</a></i>;<br/><b>hence </b><a NAME="E6:15_2_1_7_1_1"/>
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_7_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_2_1_7_1_2"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a>  <a href="calcul_2.html#K2">seq</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">a2</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:15_2_1_7_1_2"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:15_2_1_7_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E5:15_2_1_7_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E6:15_2_1_7_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><b>hence </b><a NAME="E7:15_2_1_7_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:15_2_1_7_1"/>
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_7_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_2_1_7_2"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="calcul_2.html#K2">seq</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">a1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:15_2_1_7_2"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:15_2_1_7_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E5:15_2_1_7_2"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E6:15_2_1_7_2"/><b>then </b><i><font color="Green">E137</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><div><i><font color="Green">E138</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_7_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_2_1_7_2_1"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a>  <a href="calcul_2.html#K2">seq</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">i1</font> = <font color="Maroon">a2</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:15_2_1_7_2_1"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E4:15_2_1_7_2_1"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:15_2_1_7_2_1"/><i><font color="Green">E142</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , , </i>;<br/><a NAME="E6:15_2_1_7_2_1"/>
 <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T99">FINSEQ_3:99</a></i>;<br/><a NAME="E7:15_2_1_7_2_1"/><b>then </b>
<font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E8:15_2_1_7_2_1"/><b>then </b>
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 ;<br/><b>hence </b><a NAME="E9:15_2_1_7_2_1"/>
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_7_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_2_1_7_2_2"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">a2</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> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:15_2_1_7_2_2"/><b>then </b><i><font color="Green">E144</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:15_2_1_7_2_2"/>
<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="finseq_3.html#T45">FINSEQ_3:45</a></i>;<br/><a NAME="E4:15_2_1_7_2_2"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E5:15_2_1_7_2_2"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><b>hence </b><a NAME="E6:15_2_1_7_2_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:15_2_1_7_2"/>
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:15_2_1_7"/>
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T83">FUNCT_2:83</a></i>;<br/><a NAME="E29:15_2_1"/><i><font color="Green">E145</font></i>: 
 <a href="calcul_2.html#K3">Per</a> <font color="Maroon">f</font>,<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_8"><b>proof </b></a><div class="add">

<i><font color="Green">E146</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="calcul_2.html#K3">Per</a> <font color="Maroon">f</font>,<font color="Maroon">F</font></span>)</span> = 
<font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="calcul_2.html#T19">CALCUL_2:19</a></i>
<br/>.= 
 <a href="relset_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/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_8_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:15_2_1_8_2"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:15_2_1_8_2"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">k</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><div><i><font color="Green">E149</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_8_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_2_1_8_2_1"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 ;<br/><a NAME="E2:15_2_1_8_2_1"/><b>then </b><i><font color="Green">E151</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:15_2_1_8_2_1"/>
<font color="Maroon">k</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> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:15_2_1_8_2_1"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:15_2_1_8_2_1"/>
<font color="Maroon">k</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">F</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="calcul_2.html#D2">CALCUL_2:def 2</a></i>;<br/><a NAME="E6:15_2_1_8_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</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#K14">Sgm</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="calcul_2.html#K3">Per</a> <font color="Maroon">f</font>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">f</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#K14">Sgm</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="calcul_2.html#D2">CALCUL_2:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">C</font></span>)</span> <a href="relset_1.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="finseq_3.html#D1">FINSEQ_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_8_2_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:15_2_1_8_2_2"/><i><font color="Green">E153</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> )
 ;<br/><a NAME="E2:15_2_1_8_2_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/><a NAME="E3:15_2_1_8_2_2"/><b>then </b><i><font color="Green">E154</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:15_2_1_8_2_2"/><b>then </b>
( 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</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">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:15_2_1_8_2_2"/><b>then </b><i><font color="Green">E155</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="calcul_2.html#K2">seq</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="calcul_2.html#T1">CALCUL_2:1</a></i>;<br/><a NAME="E6:15_2_1_8_2_2"/><b>then </b><i><font color="Green">E156</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E7:15_2_1_8_2_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E8:15_2_1_8_2_2"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E9:15_2_1_8_2_2"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E10:15_2_1_8_2_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E11:15_2_1_8_2_2"/><b>then </b><i><font color="Green">E157</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E12:15_2_1_8_2_2"/>
<font color="Maroon">k</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">F</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="calcul_2.html#D2">CALCUL_2:def 2</a></i>;<br/><a NAME="E13:15_2_1_8_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E14:15_2_1_8_2_2"/><b>then </b><i><font color="Green">E158</font></i>: 
<span class="p1">(<span class="default"><a href="calcul_2.html#K3">Per</a> <font color="Maroon">f</font>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="calcul_2.html#D2">CALCUL_2:def 2</a></i>;<br/><a NAME="E15:15_2_1_8_2_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="calcul_2.html#K3">Per</a> <font color="Maroon">f</font>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E5:15_2_1_8_2"/>
<span class="p1">(<span class="default"><a href="calcul_2.html#K3">Per</a> <font color="Maroon">f</font>,<font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:15_2_1_8"/>
 <a href="calcul_2.html#K3">Per</a> <font color="Maroon">f</font>,<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">h</font> = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/><a NAME="E31:15_2_1"/><i><font color="Green">E159</font></i>: 
 <a href="calcul_1.html#R4">|-</a> <font color="Maroon">h</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">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i>, , <a class="ref" href="calcul_2.html#T30">CALCUL_2:30</a></i>;<br/><a NAME="E32:15_2_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or <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">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></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/><a NAME="E33:15_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E35:15_2_1"/><i><font color="Green">E160</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><b>consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E37:15_2_1"/><i><font color="Green">E161</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><a NAME="E38:15_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E39:15_2_1"/><b>then </b><i><font color="Green">E162</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="calcul_2.html#D2">CALCUL_2:def 2</a></i>;<br/><div><i><font color="Green">E163</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_9"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_2_1_9"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 ;<br/><a NAME="E2:15_2_1_9"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E3:15_2_1_9"/><b>then </b><i><font color="Green">E165</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:15_2_1_9"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T72">FINSEQ_3:72</a></i>;<br/><b>hence </b><a NAME="E5:15_2_1_9"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E41:15_2_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/><a NAME="E42:15_2_1"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font>
 <b>by </b><i>, <a class="ref" href="calcul_2.html#T19">CALCUL_2:19</a></i>;<br/><b>then consider </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><br/><a NAME="E44:15_2_1"/><i><font color="Green">E166</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="finseq_3.html#K1">-</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  <b>by </b><i><a class="ref" href="finseq_3.html#T93">FINSEQ_3:93</a></i>;<br/><a NAME="E46:15_2_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E47:15_2_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="calcul_2.html#K6">IdFinS</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E48:15_2_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/><a NAME="E49:15_2_1"/><b>then </b><i><font color="Green">E167</font></i>: 
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="calcul_2.html#T32">CALCUL_2:32</a></i>;<br/><a NAME="E50:15_2_1"/>
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">g</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">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="calcul_2.html#T21">CALCUL_2:21</a></i>;<br/><a NAME="E51:15_2_1"/><b>then </b><i><font color="Green">E168</font></i>: 
 <a href="calcul_1.html#R4">|-</a> <font color="Maroon">g</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">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="calcul_2.html#T26">CALCUL_2:26</a></i>;<br/><a NAME="E52:15_2_1"/><i><font color="Green">E169</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T72">FINSEQ_3:72</a></i>;<br/><a NAME="E53:15_2_1"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/><a NAME="E54:15_2_1"/><b>then </b><i><font color="Green">E170</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T40">XBOOLE_1:40</a></i>;<br/><a NAME="E55:15_2_1"/>
<font color="Maroon">X</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><a NAME="E56:15_2_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E57:15_2_1"/>
<font color="Maroon">X</font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">p</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:15_2"/>
<font color="Maroon">X</font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">p</font>
 <b>by </b><i>, </i>;<br/>


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


</div>
