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

<b>let </b><font color="Maroon">CX</font> be  <a href="henmodel.html#V1">Consistent</a> <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>

<b>assume </b><a NAME="E1:36"/><i><font color="Green">E54</font></i>: 
 <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <font color="Maroon">CX</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:36"/><i><font color="Green">E55</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="goedelcp.html#K1" target="_self">ExCl</a>  <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="cqc_lang.html#K7">CQC-WFF</a>  <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></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> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">K</font>, <font color="Olive">L</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a>  &amp; <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p4">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p5">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p0">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">L</font> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">K</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p0">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Olive">L</font> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> );<br/>
<a NAME="E5:36"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">C</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>  ex <font color="Olive">D</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">C</font>,<font color="Olive">D</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">C</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>



<b>set </b><font color="Maroon">K</font> = <font color="Maroon">C</font> <a href="mcart_1.html#K2">`2</a> ;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:36_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="cqc_lang.html#K7">CQC-WFF</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="qc_lang1.html#K2">bound_QC-variables</a>  &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">K</font> = <font color="Maroon">C</font> <a href="mcart_1.html#K2">`2</a>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  <b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>set </b><font color="Maroon">L</font> = <font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/>
<b>set </b><font color="Maroon">D</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p4">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p5">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p0">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p0">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>take </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p4">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p5">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p0">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p0">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
;<br/>

<b>thus </b><a NAME="E4:36_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">C</font>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p4">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p5">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p0">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p0">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>]
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">A</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,0</span>)</span> <a href="substut2.html#K4">.</a> <span class="p4">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,0</span>)</span>,<span class="p4">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p5">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p0">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">CX</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"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p3">(<span class="default"><font color="Maroon">CX</font> <a href="subset_1.html#K4">\/</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,0</span>)</span> <a href="substut2.html#K4">.</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,0</span>)</span>,<span class="p1">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p2">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p3">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">CX</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"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<b>consider </b><font color="Maroon">h</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E8:36"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>,<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S2">RECDEF_1:sch 2</a>();<br/></i>
<b>set </b><font color="Maroon">CY</font> = <font color="Maroon">CX</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><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="E1:36_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CX</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_2_1"/>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CX</font>
 ;<br/><a NAME="E2:36_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">n</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="E4:36_2_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 ;<br/><b>consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:36_2_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="cqc_lang.html#K7">CQC-WFF</a>  &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="qc_lang1.html#K2">bound_QC-variables</a>  &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>thus </b><a NAME="E7:36_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="cqc_lang.html#K7">CQC-WFF</a> 
 <b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:36_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="cqc_lang.html#K7">CQC-WFF</a> 
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">CY</font> = <font color="Maroon">CX</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a>  <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><a NAME="E1:36_3"/>
 <a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K1" target="_self">ExCl</a> 
 <b>by </b><i/>;<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="E3:36_3"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">a</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">a</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">r'</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/><a NAME="E6:36_3"/>
(  <a href="goedelcp.html#K2" target="_self">Ex-bound_in</a> <font color="Maroon">r'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="goedelcp.html#K3" target="_self">Ex-the_scope_of</a> <font color="Maroon">r'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E7:36_3"/><b>then </b><i><font color="Green">E64</font></i>: 
(  <a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E8:36_3"/><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 ;<br/><a NAME="E9:36_3"/><i><font color="Green">E67</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">CY</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><div><i><font color="Green">E70</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_3_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>take </b><font color="Maroon">y</font> =  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">CX</font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r'</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span>;<br/><a NAME="E2:36_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">r'</font></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span> <a href="substut2.html#K4">.</a> <span class="p2">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><b>hence </b><a NAME="E3:36_3_1"/>
<font color="Maroon">CY</font> <a href="henmodel.html#R1">|-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="substut2.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_3_2"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:36_3_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</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="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:36_3_2"/><i><font color="Green">E94</font></i>: 
 ex <font color="Olive">K</font>, <font color="Olive">L</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  &amp; <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r'</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">r'</font></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span> <a href="substut2.html#K4">.</a> <span class="p4">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span>,<span class="p4">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p5">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p0">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">L</font> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">K</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">r'</font></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span> <a href="substut2.html#K4">.</a> <span class="p0">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span>,<span class="p0">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Olive">L</font> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> )
 <b>by </b><i>, </i>;<br/><b>set </b><font color="Maroon">K</font> = <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K2">`2</a> ;<br/><b>set </b><font color="Maroon">L</font> = <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r'</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span>;<br/><b>take </b><font color="Maroon">y</font> =  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r'</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> </span>}</span> </span>)</span>;<br/><a NAME="E6:36_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">r'</font></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span> <a href="substut2.html#K4">.</a> <span class="p2">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="goedelcp.html#D1" target="_self">Def1</a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><b>hence </b><a NAME="E7:36_3_2"/>
<font color="Maroon">CY</font> <a href="henmodel.html#R1">|-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="substut2.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:36_3"/>
 ex <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a> st <font color="Maroon">CY</font> <a href="henmodel.html#R1">|-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="substut2.html#K4">.</a> <font color="Maroon">x</font>,<font color="Olive">y</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E12:36"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">CY</font> is <a href="goedelcp.html#V2" target="_self">with_examples</a>
 
<b>by </b><i/>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">CX</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E14:36"/><i><font color="Green">E96</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp; ( for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E15:36"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">CY</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:36_4"/>
<font color="Maroon">CY</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_1"><b>proof </b></a><div class="add">

<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="E1:36_4_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CY</font>
 ;<br/>

<a NAME="E2:36_4_1"/><b>then </b><i><font color="Green">E98</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CX</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><i><font color="Green">E99</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_4_1_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CX</font>
 ;<br/><a NAME="E2:36_4_1_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">CX</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="hidden.html#R2">in</a> 0 </span>}</span> 
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_1_1_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="E1:36_4_1_1_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="hidden.html#R2">in</a> 0 </span>}</span> 
 ;<br/><b>consider </b><font color="Maroon">n</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="E3:36_4_1_1_1"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> 0 )
 <b>by </b><i><a class="ref" href="goedelcp.html#D2" target="_self">Def2</a></i>;<br/><b>thus </b><a NAME="E4:36_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="goedelcp.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><a NAME="E4:36_4_1_1"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="hidden.html#R2">in</a> 0 </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E5:36_4_1_1"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">CX</font>
 <b>by </b><i><a class="ref" href="goedelcp.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E6:36_4_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 0 <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/><b>hence </b><a NAME="E7:36_4_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="goedelcp.html#D2" target="_self">Def2</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_4_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">n</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="E3:36_4_1_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 ;<br/><a NAME="E4:36_4_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:36_4_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/><a NAME="E6:36_4_1_2"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E7:36_4_1_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">CX</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E8:36_4_1_2"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E9:36_4_1_2"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="goedelcp.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E10:36_4_1_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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/><b>hence </b><a NAME="E11:36_4_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="goedelcp.html#D2" target="_self">Def2</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:36_4_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


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

<b>thus </b><a NAME="E2:36_4"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">CY</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_2"><b>proof </b></a><div class="add">

<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="E1:36_4_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_4_2"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</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="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:36_4_2"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">c</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, </i>;<br/>
<a NAME="E7:36_4_2"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CX</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> </span>}</span> 
 
<b>by </b><i>, , </i>;<br/>
<div><i><font color="Green">E111</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_4_2_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CX</font>
 ;<br/><a NAME="E2:36_4_2_1"/>
<font color="Maroon">CX</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">CY</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E3:36_4_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CY</font>
 <b>by </b><i><a class="ref" href="goedelcp.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_4_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">m</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="E3:36_4_2_2"/><i><font color="Green">E144</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> )
 ;<br/><a NAME="E4:36_4_2_2"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="goedelcp.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E5:36_4_2_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">CY</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E6:36_4_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CY</font>
 <b>by </b><i><a class="ref" href="goedelcp.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:36_4_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CY</font>
 <b>by </b><i>, <a class="ref" href="goedelcp.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E16:36"/><i><font color="Green">E146</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_5"><b>proof </b></a><div class="add">

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



<b>assume </b><a NAME="E1:36_5"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> )
 ;<br/>

<a NAME="E2:36_5"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CX</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:36_5"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CX</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_5_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="E1:36_5_1"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/><div><i><font color="Green">E151</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_5_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_5_1_1"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CX</font>
 ;<br/><a NAME="E2:36_5_1_1"/>
<font color="Maroon">CX</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="goedelcp.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E3:36_5_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="goedelcp.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_5_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_5_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> </span>}</span> 
 ;<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="E3:36_5_1_2"/><i><font color="Green">E153</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> )
 ;<br/><a NAME="E4:36_5_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="goedelcp.html#T2" target="_self">Th2</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/><a NAME="E5:36_5_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:36_5_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/><a NAME="E7:36_5_1_2"/><b>then </b><i><font color="Green">E154</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="goedelcp.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E8:36_5_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="goedelcp.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E9:36_5_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:36_5_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="goedelcp.html#D2" target="_self">Def2</a>, <a class="ref" href="goedelcp.html#T1" target="_self">Th1</a>, <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="E5:36_5"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:36"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="cqc_lang.html#K7">CQC-WFF</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_6"><b>proof </b></a><div class="add">

<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="E1:36_6"/><i><font color="Green">E155</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">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_6"/><i><font color="Green">E156</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, </i>;<br/>
<a NAME="E5:36_6"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CX</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:36_6_1"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> </span>}</span> 
 ;<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="E3:36_6_1"/><i><font color="Green">E159</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="goedelcp.html#D2" target="_self">Def2</a></i>;<br/><b>thus </b><a NAME="E4:36_6_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="cqc_lang.html#K7">CQC-WFF</a> 
 <b>by </b><i><a class="ref" href="goedelcp.html#T1" target="_self">Th1</a>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:36_6"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="cqc_lang.html#K7">CQC-WFF</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:36_6"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="cqc_lang.html#K7">CQC-WFF</a> 
 
<b>by </b><i><a class="ref" href="goedelcp.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E9:36_6"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="cqc_lang.html#K7">CQC-WFF</a> 
 <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="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <a href="cqc_lang.html#K7">CQC-WFF</a>  <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E19:36"/><i><font color="Green">E160</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_7"><b>proof </b></a><div class="add">

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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_7_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="E1:36_7_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 ;<br/><b>then 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="E3:36_7_1"/><i><font color="Green">E161</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><a NAME="E4:36_7_1"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/><a NAME="E5:36_7_1"/><b>then </b>
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:36_7_1"/><b>then </b><i><font color="Green">E162</font></i>: 
( <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><div><i><font color="Green">E163</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_7_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_7_1_1"/>
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:36_7_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><a NAME="E3:36_7_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E4:36_7_1_1"/><b>then </b><i><font color="Green">E164</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:36_7_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E6:36_7_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="goedelcp.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_7_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_7_1_2"/>
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:36_7_1_2"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:36_7_1_2"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/><a NAME="E4:36_7_1_2"/><b>then </b><i><font color="Green">E165</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E5:36_7_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E6:36_7_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="goedelcp.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:36_7_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="goedelcp.html#D1" target="_self">Def1</a>, <a class="ref" href="goedelcp.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:36_7"/><b>then </b><i><font color="Green">E166</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_7_2"><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="E1:36_7_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:36_7_2"/><b>then </b><i><font color="Green">E167</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> </span>}</span>  or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div><i><font color="Green">E168</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_7_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_7_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> </span>}</span> 
 ;<br/><b>then 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="E3:36_7_2_1"/><i><font color="Green">E169</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> )
 ;<br/><a NAME="E4:36_7_2_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="goedelcp.html#T1" target="_self">Th1</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E5:36_7_2_1"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:36_7_2_1"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/><b>hence </b><a NAME="E7:36_7_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 <b>by </b><i><a class="ref" href="goedelcp.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_7_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_7_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:36_7_2_2"/><b>then </b><i><font color="Green">E170</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:36_7_2_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:36_7_2_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/><b>hence </b><a NAME="E5:36_7_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 <b>by </b><i><a class="ref" href="goedelcp.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:36_7_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 <b>by </b><i><a class="ref" href="goedelcp.html#D1" target="_self">Def1</a>, <a class="ref" href="goedelcp.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:36_7"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:36_7"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E6:36_7"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">CX</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:36_7"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">CX</font> <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<b>hence </b><a NAME="E8:36_7"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  is <a href="finset_1.html#V1">finite</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  is   <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  );<br/>
<a NAME="E20:36"/><i><font color="Green">E171</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>  holds <br/> ( <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  is <a href="finset_1.html#V1">finite</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  is   <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_8"><b>proof </b></a><div class="add">

<a NAME="E1:36_8"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_8_1"><b>proof </b></a><div class="add">

<a NAME="E1:36_8_1"/><i><font color="Green">E173</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">CX</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"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,0</span>)</span> <a href="substut2.html#K4">.</a> <span class="p5">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,0</span>)</span>,<span class="p5">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p0">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">CX</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"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,0</span>)</span> <a href="substut2.html#K4">.</a> <span class="p2">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,0</span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p3">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p4">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">CX</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"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/>
<a NAME="E3:36_8_1"/>
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#T20">CQC_SIM1:20</a></i>;<br/>
<a NAME="E4:36_8_1"/><b>then </b>
 <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i/>;<br/>
<a NAME="E5:36_8_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <font color="Maroon">CX</font></span>)</span> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="finset_1.html#T14">FINSET_1:14</a></i>;<br/>
<b>hence </b><a NAME="E6:36_8_1"/>
<font color="Maroon">S<sub>2</sub></font>[0]
 <b>by </b><i><a class="ref" href="goedelcp.html#D2" target="_self">Def2</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:36_8"/><i><font color="Green">E174</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="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_8_2"><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="E1:36_8_2"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E2:36_8_2"/><i><font color="Green">E176</font></i>: 
 ex <font color="Olive">K</font>, <font color="Olive">L</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  &amp; <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p4">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p5">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p0">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">L</font> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">K</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p0">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Olive">L</font> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">K</font> = <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K2">`2</a> ;<br/>
<b>reconsider </b><font color="Maroon">K</font> = <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  <b>by </b><i><a class="ref" href="goedelcp.html#T1" target="_self">Th1</a></i>;<br/>
<b>set </b><font color="Maroon">L</font> = <font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/>
<a NAME="E4:36_8_2"/><i><font color="Green">E177</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p4">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p5">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p0">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="goedelcp.html#T2" target="_self">Th2</a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>set </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p2">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p3">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p4">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E5:36_8_2"/>
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p3">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p4">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p5">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#T20">CQC_SIM1:20</a></i>;<br/>
<b>hence </b><a NAME="E6:36_8_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="ref" href="goedelcp.html#T1" target="_self">Th1</a>, , <a class="ref" href="finset_1.html#T14">FINSET_1:14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:36_8"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="goedelcp.html#D1" target="_self">Def1</a>, <a class="ref" href="goedelcp.html#D2" target="_self">Def2</a>);<br/></i>
<b>hence </b><a NAME="E4:36_8"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  is <a href="finset_1.html#V1">finite</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  is   <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  )
 ;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>(  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  &amp; not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> </span>}</span> </span>)</span> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> );<br/>
<a NAME="E21:36"/><i><font color="Green">E178</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>  holds <br/> (  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  &amp; not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> </span>}</span> </span>)</span> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_9"><b>proof </b></a><div class="add">

<a NAME="E1:36_9"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_9_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,0</span>)</span> <a href="substut2.html#K4">.</a> <span class="p2">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,0</span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p3">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p4">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">CX</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"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">A1</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,0</span>)</span> <a href="substut2.html#K4">.</a> <span class="p4">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,0</span>)</span>,<span class="p4">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p5">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p0">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">CX</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"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/>
<a NAME="E2:36_9_1"/><i><font color="Green">E180</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">CX</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"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,0</span>)</span> <a href="substut2.html#K4">.</a> <span class="p5">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,0</span>)</span>,<span class="p5">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p0">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">CX</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"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">B</font> = <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="mcart_1.html#K2">`2</a>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  ;<br/>
<b>reconsider </b><font color="Maroon">C</font> = <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <a href="qc_lang1.html#K2">bound_QC-variables</a>  ;<br/>
<a NAME="E5:36_9_1"/>
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#T20">CQC_SIM1:20</a></i>;<br/>
<a NAME="E6:36_9_1"/><b>then </b>
(  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span> is <a href="finset_1.html#V1">finite</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="mcart_1.html#K2">`2</a>  is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:36_9_1"/><b>then </b>
<font color="Maroon">C</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T14">FINSET_1:14</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a><b> such that </b><br/><a NAME="E9:36_9_1"/><i><font color="Green">E182</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">l</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="E11:36_9_1"/><i><font color="Green">E183</font></i>: 
 <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="qc_lang3.html#T36">QC_LANG3:36</a></i>;<br/>
<a NAME="E12:36_9_1"/>
not  <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="goedelcp.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E13:36_9_1"/><b>then </b>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_9_1_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="E1:36_9_1_1"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</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="E3:36_9_1_1"/><i><font color="Green">E185</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; not  <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="txt" href="goedelcp.html#E1:36"><i><font color="Green">E54</font></i></a></i>;<br/><b>thus </b><a NAME="E4:36_9_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="goedelcp.html#E3:36"><i><font color="Green">E55</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">j</font> =  <a href="henmodel.html#K1">min*</a> <font color="Maroon">A</font>;<br/>
<a NAME="E17:36_9_1"/>
 <a href="henmodel.html#K1">min*</a> <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="henmodel.html#D1">HENMODEL: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="E19:36_9_1"/><i><font color="Green">E186</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <font color="Maroon">A</font> &amp; not  <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 ;<br/>
<a NAME="E20:36_9_1"/><i><font color="Green">E187</font></i>: 
( not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="mcart_1.html#K2">`2</a>  &amp; not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="txt" href="goedelcp.html#E1:36"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E21:36_9_1"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">CX</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="hidden.html#R2">in</a> 0 </span>}</span> 
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_9_1_2"><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="E1:36_9_1_2"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="hidden.html#R2">in</a> 0 </span>}</span> 
 ;<br/><b>consider </b><font color="Maroon">n</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="E3:36_9_1_2"/><i><font color="Green">E190</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> 0 )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:36_9_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="goedelcp.html#E8:36"><i><font color="Green">E58</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E23:36_9_1"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="hidden.html#R2">in</a> 0 </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E24:36_9_1"/><b>then </b><i><font color="Green">E191</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">CX</font>
 
<b>by </b><i><a class="txt" href="goedelcp.html#E5:36"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E25:36_9_1"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,0</span>)</span> <a href="substut2.html#K4">.</a> <span class="p2">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,0</span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p3">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p4">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">CX</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"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E26:36_9_1"/><b>then </b><i><font color="Green">E192</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">CX</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"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,0</span>)</span> <a href="substut2.html#K4">.</a> <span class="p4">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,0</span>)</span>,<span class="p4">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p5">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p0">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">CX</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"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 1 as   <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/>
<a NAME="E28:36_9_1"/><i><font color="Green">E193</font></i>: 
 <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="mcart_1.html#K2">`2</a> 
 
<b>by </b><i><a class="ref" href="goedelcp.html#T1" target="_self">Th1</a>, <a class="txt" href="goedelcp.html#E8:36"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E29:36_9_1"/><b>then </b>
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="txt" href="goedelcp.html#E3:36"><i><font color="Green">E55</font></i></a></i>;<br/>
<a NAME="E30:36_9_1"/><b>then </b>
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="goedelcp.html#E3:36"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E31:36_9_1"/>
<font color="Maroon">S<sub>3</sub></font>[0]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:36_9"/><i><font color="Green">E194</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="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_9_2"><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="E1:36_9_2"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/>
<a NAME="E3:36_9_2"/><i><font color="Green">E196</font></i>: 
 ex <font color="Olive">K</font>, <font color="Olive">L</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  &amp; <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p4">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p5">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p0">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">L</font> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">K</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p0">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Olive">L</font> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">K</font> = <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K2">`2</a> ;<br/>
<b>reconsider </b><font color="Maroon">K</font> = <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">L</font> = <font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/>
<b>set </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p2">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p3">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p4">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E5:36_9_2"/><i><font color="Green">E197</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p2">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p3">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p4">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p4">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p5">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p0">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">B</font> = <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  ;<br/>
<b>reconsider </b><font color="Maroon">C</font> = <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <a href="qc_lang1.html#K2">bound_QC-variables</a>  ;<br/>
<a NAME="E8:36_9_2"/>
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#T20">CQC_SIM1:20</a></i>;<br/>
<a NAME="E9:36_9_2"/><b>then </b>
(  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span> is <a href="finset_1.html#V1">finite</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:36_9_2"/><b>then </b>
<font color="Maroon">C</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T14">FINSET_1:14</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a><b> such that </b><br/><a NAME="E12:36_9_2"/><i><font color="Green">E198</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">l</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="E14:36_9_2"/><i><font color="Green">E199</font></i>: 
 <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="qc_lang3.html#T36">QC_LANG3:36</a></i>;<br/>
<a NAME="E15:36_9_2"/>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="goedelcp.html#E3:36"><i><font color="Green">E55</font></i></a>, <a class="txt" href="goedelcp.html#E5:36"><i><font color="Green">E56</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_9_2_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="E1:36_9_2_1"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</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="E3:36_9_2_1"/><i><font color="Green">E201</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; not  <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:36_9_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="goedelcp.html#E8:36"><i><font color="Green">E58</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">j</font> =  <a href="henmodel.html#K1">min*</a> <font color="Maroon">A</font>;<br/>
<a NAME="E19:36_9_2"/>
 <a href="henmodel.html#K1">min*</a> <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="henmodel.html#D1">HENMODEL: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="E21:36_9_2"/><i><font color="Green">E202</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <font color="Maroon">A</font> &amp; not  <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 ;<br/>
<a NAME="E22:36_9_2"/><i><font color="Green">E203</font></i>: 
( not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E23:36_9_2"/>
<span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p4">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p5">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p0">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="goedelcp.html#T2" target="_self">Th2</a>, <a class="txt" href="goedelcp.html#E1:36"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/>
<a NAME="E24:36_9_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p5">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p0">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<a NAME="E25:36_9_2"/><b>then </b>
 <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p5">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p0">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<a NAME="E26:36_9_2"/><b>then </b><i><font color="Green">E204</font></i>: 
 <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="txt" href="goedelcp.html#E1:36"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E27:36_9_2"/><b>then </b>
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="goedelcp.html#E8:36"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E28:36_9_2"/><b>then </b>
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="goedelcp.html#E8:36"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E29:36_9_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:36_9"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="goedelcp.html#D2" target="_self">Def2</a>, <a class="ref" href="goedelcp.html#T1" target="_self">Th1</a>);<br/></i>
<b>hence </b><a NAME="E4:36_9"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> (  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  &amp; not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> </span>}</span> </span>)</span> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> is <a href="henmodel.html#V1">Consistent</a>;<br/>
<a NAME="E22:36"/><i><font color="Green">E205</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_10"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_10_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="E1:36_10_1"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> 0 </span>}</span> 
 ;<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="E3:36_10_1"/><i><font color="Green">E207</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> 0 )
 <b>by </b><i><a class="ref" href="goedelcp.html#T1" target="_self">Th1</a></i>;<br/><b>thus </b><a NAME="E4:36_10_1"/>
contradiction
 <b>by </b><i><a class="ref" href="goedelcp.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:36_10"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> 0 </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E3:36_10"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">CX</font> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:36_10"/>
<font color="Maroon">S<sub>4</sub></font>[0]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E23:36"/><i><font color="Green">E208</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="Maroon">S<sub>4</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_11"><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="E1:36_11"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_11"/><i><font color="Green">E210</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="cqc_lang.html#K7">CQC-WFF</a>  &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="qc_lang1.html#K2">bound_QC-variables</a>  &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  <b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_11_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_11_1"/>
not <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="henmodel.html#V1">Consistent</a>
 ;<br/><a NAME="E2:36_11_1"/><b>then </b>
( <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="henmodel.html#R1">|-</a>  <a href="cqc_lang.html#K9">VERUM</a>  &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="henmodel.html#R1">|-</a>  <a href="cqc_lang.html#K10">'not'</a> <a href="cqc_lang.html#K9">VERUM</a>  )
 <b>by </b><i><a class="ref" href="henmodel.html#T6">HENMODEL:6</a></i>;<br/><a NAME="E3:36_11_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r</font></span><a href="domain_1.html#K6">}</a></span> <a href="henmodel.html#R1">|-</a>  <a href="cqc_lang.html#K10">'not'</a> <a href="cqc_lang.html#K9">VERUM</a> 
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">f1</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="E5:36_11_1"/><i><font color="Green">E211</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> &amp;  <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</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">r</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"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="henmodel.html#T8">HENMODEL:8</a></i>;<br/><a NAME="E6:36_11_1"/><i><font color="Green">E212</font></i>: 
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</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> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></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/><div><i><font color="Green">E213</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_11_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_11_1_1"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:36_11_1_1"/><b>then </b><i><font color="Green">E215</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,0</span>)</span> <a href="substut2.html#K4">.</a> <span class="p2">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,0</span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p3">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p4">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">CX</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"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><b>reconsider </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,0</span>)</span> <a href="substut2.html#K4">.</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,0</span>)</span>,<span class="p1">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p2">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p3">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">CX</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"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/><a NAME="E4:36_11_1_1"/><i><font color="Green">E216</font></i>: 
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</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> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></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"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <font color="Maroon">s</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="goedelcp.html#E3:36"><i><font color="Green">E55</font></i></a>, <a class="ref" href="calcul_1.html#T51">CALCUL_1:51</a></i>;<br/><a NAME="E5:36_11_1_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">f1</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">r</font></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="E6:36_11_1_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">f1</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">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 ;<br/><a NAME="E7:36_11_1_1"/><b>then </b>
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="calcul_1.html#K1">Ant</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p5"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</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> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></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="calcul_1.html#K2">Suc</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p5"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></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> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="goedelcp.html#E1:36"><i><font color="Green">E54</font></i></a>, </i>;<br/><a NAME="E8:36_11_1_1"/><b>then </b>
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</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> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></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="calcul_1.html#K2">Suc</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p5"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></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> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="calcul_1.html#T5">CALCUL_1:5</a></i>;<br/><a NAME="E9:36_11_1_1"/><b>then </b>
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</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> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></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"><font color="Maroon">r</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"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="calcul_1.html#T5">CALCUL_1:5</a></i>;<br/><a NAME="E10:36_11_1_1"/><b>then </b><i><font color="Green">E217</font></i>: 
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</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> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></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> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i>, <a class="txt" href="goedelcp.html#E8:36"><i><font color="Green">E58</font></i></a>, , <a class="ref" href="calcul_2.html#T24">CALCUL_2:24</a></i>;<br/><a NAME="E11:36_11_1_1"/>
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</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">s</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">s</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="E12:36_11_1_1"/><b>then </b><i><font color="Green">E218</font></i>: 
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</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">s</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <font color="Maroon">s</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="calcul_1.html#T52">CALCUL_1:52</a></i>;<br/><a NAME="E13:36_11_1_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">f1</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">r</font></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="E14:36_11_1_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">f1</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">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 ;<br/><a NAME="E15:36_11_1_1"/><b>then </b>
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="calcul_1.html#K1">Ant</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p5"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></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="calcul_1.html#K2">Suc</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p5"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></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> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="goedelcp.html#E1:36"><i><font color="Green">E54</font></i></a>, </i>;<br/><a NAME="E16:36_11_1_1"/><b>then </b>
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></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="calcul_1.html#K2">Suc</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p5"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></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> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="calcul_1.html#T5">CALCUL_1:5</a></i>;<br/><a NAME="E17:36_11_1_1"/><b>then </b>
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></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"><font color="Maroon">r</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"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="calcul_1.html#T5">CALCUL_1:5</a></i>;<br/><a NAME="E18:36_11_1_1"/><b>then </b><i><font color="Green">E219</font></i>: 
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</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">s</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"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i>, <a class="txt" href="goedelcp.html#E8:36"><i><font color="Green">E58</font></i></a>, , <a class="ref" href="calcul_2.html#T24">CALCUL_2:24</a></i>;<br/><b>set </b><font color="Maroon">y</font> =  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">CX</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"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span>;<br/><b>reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0 as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/><b>set </b><font color="Maroon">C</font> =  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">CX</font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/><a NAME="E20:36_11_1_1"/>
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">r1</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#T20">CQC_SIM1:20</a></i>;<br/><a NAME="E21:36_11_1_1"/><b>then </b>
 <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r1</font></span><a href="domain_1.html#K6">}</a></span> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i/>;<br/><a NAME="E22:36_11_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <font color="Maroon">CX</font></span>)</span> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="ref" href="finset_1.html#T14">FINSET_1:14</a></i>;<br/><a NAME="E23:36_11_1_1"/><b>then </b>
 <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">CX</font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a><b> such that </b><br/><a NAME="E25:36_11_1_1"/><i><font color="Green">E221</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">CX</font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">l</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="E27:36_11_1_1"/><i><font color="Green">E222</font></i>: 
 <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="qc_lang3.html#T36">QC_LANG3:36</a></i>;<br/><a NAME="E28:36_11_1_1"/>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">CX</font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> 
 <b>by </b><i>, </i>;<br/><b>then reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">CX</font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_11_1_1_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="E1:36_11_1_1_1"/><i><font color="Green">E223</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</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="E3:36_11_1_1_1"/><i><font color="Green">E224</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; not  <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">CX</font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:36_11_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">CX</font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>set </b><font color="Maroon">j</font> =  <a href="henmodel.html#K1">min*</a> <font color="Maroon">A</font>;<br/><a NAME="E32:36_11_1_1"/>
 <a href="henmodel.html#K1">min*</a> <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="henmodel.html#D1">HENMODEL: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="E34:36_11_1_1"/><i><font color="Green">E225</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <font color="Maroon">A</font> &amp; not  <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">CX</font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 ;<br/><a NAME="E35:36_11_1_1"/>
not  <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <font color="Maroon">CX</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E36:36_11_1_1"/><b>then </b><i><font color="Green">E226</font></i>: 
( not  <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <font color="Maroon">CX</font> &amp; not  <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r1</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E37:36_11_1_1"/><i><font color="Green">E227</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">CX</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="hidden.html#R2">in</a> 0 </span>}</span> 
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_11_1_1_2"><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="E1:36_11_1_1_2"/><i><font color="Green">E228</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="hidden.html#R2">in</a> 0 </span>}</span> 
 ;<br/><b>consider </b><font color="Maroon">n</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="E3:36_11_1_1_2"/><i><font color="Green">E229</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> 0 )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:36_11_1_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E39:36_11_1_1"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="hidden.html#R2">in</a> 0 </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E40:36_11_1_1"/><b>then </b>
 <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <font color="Maroon">CX</font>
 <b>by </b><i><a class="txt" href="goedelcp.html#E1:36"><i><font color="Green">E54</font></i></a>, , , </i>;<br/><a NAME="E41:36_11_1_1"/><b>then </b>
not  <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E42:36_11_1_1"/><b>then </b><i><font color="Green">E230</font></i>: 
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="calcul_1.html#K3">still_not-bound_in</a> <font color="Maroon">f1</font>
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">r2</font> =  <a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,0 as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/><b>reconsider </b><font color="Maroon">y2</font> =  <a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,0 as   <a href="qc_lang1.html#NM3">bound_QC-variable</a> ;<br/><a NAME="E45:36_11_1_1"/>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K1" target="_self">ExCl</a> 
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>then consider </b><font color="Maroon">y1</font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a>, <font color="Maroon">s1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E47:36_11_1_1"/><i><font color="Green">E233</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">y1</font>,<font color="Maroon">s1</font>
 <b>by </b><i/>;<br/><a NAME="E48:36_11_1_1"/><i><font color="Green">E234</font></i>: 
( <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="goedelcp.html#K3" target="_self">Ex-the_scope_of</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a>  <a href="goedelcp.html#K3" target="_self">Ex-the_scope_of</a> <font color="Maroon">r1</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E49:36_11_1_1"/><i><font color="Green">E235</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a>  <a href="goedelcp.html#K2" target="_self">Ex-bound_in</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a>  <a href="goedelcp.html#K2" target="_self">Ex-bound_in</a> <font color="Maroon">r1</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E50:36_11_1_1"/><i><font color="Green">E236</font></i>: 
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</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"><font color="Maroon">s1</font> <a href="substut2.html#K4">.</a> <font color="Maroon">y1</font>,<span class="p4">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p5">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">A</font></span>)</span></span>)</span></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> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i>, , <a class="ref" href="goedelcp.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E51:36_11_1_1"/>
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">r1</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E52:36_11_1_1"/><b>then </b>
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="calcul_1.html#K3">still_not-bound_in</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r1</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="calcul_1.html#T60">CALCUL_1:60</a></i>;<br/><a NAME="E53:36_11_1_1"/><b>then </b>
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3">still_not-bound_in</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3">still_not-bound_in</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r1</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E54:36_11_1_1"/><b>then </b><i><font color="Green">E237</font></i>: 
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="calcul_1.html#K3">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">f1</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">r1</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="calcul_1.html#T59">CALCUL_1:59</a></i>;<br/><a NAME="E55:36_11_1_1"/>
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span>
 <b>by </b><i><a class="ref" href="qc_lang3.html#T7">QC_LANG3:7</a>, <a class="ref" href="qc_lang3.html#T11">QC_LANG3:11</a></i>;<br/><a NAME="E56:36_11_1_1"/><b>then </b>
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="calcul_1.html#K3">still_not-bound_in</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> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="calcul_1.html#T60">CALCUL_1:60</a></i>;<br/><a NAME="E57:36_11_1_1"/><b>then </b>
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3">still_not-bound_in</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r1</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3">still_not-bound_in</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> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="goedelcp.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E58:36_11_1_1"/><b>then </b>
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="calcul_1.html#K3">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r1</font></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> <a href="cqc_lang.html#K9">VERUM</a> </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#T59">CALCUL_1:59</a></i>;<br/><a NAME="E59:36_11_1_1"/><b>then </b><i><font color="Green">E238</font></i>: 
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</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">r1</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"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="goedelcp.html#T4" target="_self">Th4</a>, <a class="ref" href="calcul_1.html#T61">CALCUL_1:61</a></i>;<br/><a NAME="E60:36_11_1_1"/>
 <a href="calcul_1.html#R4">|-</a> <font color="Maroon">f1</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> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></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="E61:36_11_1_1"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="henmodel.html#R1">|-</a>  <a href="cqc_lang.html#K10">'not'</a> <a href="cqc_lang.html#K9">VERUM</a> 
 <b>by </b><i><a class="txt" href="goedelcp.html#E1:36"><i><font color="Green">E54</font></i></a>, <a class="ref" href="henmodel.html#D2">HENMODEL:def 2</a></i>;<br/><b>hence </b><a NAME="E62:36_11_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="goedelcp.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_11_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_11_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><b>then consider </b><font color="Maroon">k1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:36_11_1_2"/><i><font color="Green">E240</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">k1</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="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:36_11_1_2"/><i><font color="Green">E241</font></i>: 
 ex <font color="Olive">K</font>, <font color="Olive">L</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  &amp; <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p4">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p5">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p0">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">L</font> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">K</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p0">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Olive">L</font> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> )
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">K</font> = <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> ;<br/><b>set </b><font color="Maroon">r1</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">L</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/><b>set </b><font color="Maroon">p1</font> = <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p2">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p3">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p4">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span>;<br/><a NAME="E6:36_11_1_2"/><i><font color="Green">E242</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p2">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p3">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p4">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p4">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p5">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p0">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="txt" href="goedelcp.html#E8:36"><i><font color="Green">E58</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><b>reconsider </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p2">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p3">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/><a NAME="E8:36_11_1_2"/><i><font color="Green">E243</font></i>: 
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</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> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></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"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p4">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p5">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p0">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="goedelcp.html#E3:36"><i><font color="Green">E55</font></i></a>, , <a class="ref" href="calcul_1.html#T51">CALCUL_1:51</a></i>;<br/><a NAME="E9:36_11_1_2"/>
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">f1</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">r</font></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="E10:36_11_1_2"/><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">f1</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">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 ;<br/><a NAME="E11:36_11_1_2"/><b>then </b>
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="calcul_1.html#K1">Ant</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p5"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</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> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p0">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></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="calcul_1.html#K2">Suc</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p5"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></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> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="goedelcp.html#E1:36"><i><font color="Green">E54</font></i></a>, </i>;<br/><a NAME="E12:36_11_1_2"/><b>then </b>
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</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> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p0">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></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="calcul_1.html#K2">Suc</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p5"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></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> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="calcul_1.html#T5">CALCUL_1:5</a></i>;<br/><a NAME="E13:36_11_1_2"/><b>then </b>
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</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> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p0">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></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"><font color="Maroon">r</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"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="calcul_1.html#T5">CALCUL_1:5</a></i>;<br/><a NAME="E14:36_11_1_2"/><b>then </b><i><font color="Green">E244</font></i>: 
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</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> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></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> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i>, , <a class="ref" href="calcul_2.html#T24">CALCUL_2:24</a></i>;<br/><a NAME="E15:36_11_1_2"/>
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</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">s</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">s</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="E16:36_11_1_2"/><b>then </b><i><font color="Green">E245</font></i>: 
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</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">s</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p4">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p5">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p0">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="calcul_1.html#T52">CALCUL_1:52</a></i>;<br/><a NAME="E17:36_11_1_2"/>
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">f1</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">r</font></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="E18:36_11_1_2"/><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">f1</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">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 ;<br/><a NAME="E19:36_11_1_2"/><b>then </b>
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="calcul_1.html#K1">Ant</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p5"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></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="calcul_1.html#K2">Suc</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p5"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></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> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="goedelcp.html#E1:36"><i><font color="Green">E54</font></i></a>, </i>;<br/><a NAME="E20:36_11_1_2"/><b>then </b>
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></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="calcul_1.html#K2">Suc</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p5"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></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> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="calcul_1.html#T5">CALCUL_1:5</a></i>;<br/><a NAME="E21:36_11_1_2"/><b>then </b>
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></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"><span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p0">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p0">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="substut2.html#K4">.</a> <span class="p5">(<span class="default"><a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p0">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p0">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></span>)</span></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> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="calcul_1.html#T5">CALCUL_1:5</a></i>;<br/><a NAME="E22:36_11_1_2"/><b>then </b><i><font color="Green">E246</font></i>: 
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</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">s</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"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="calcul_2.html#T24">CALCUL_2:24</a></i>;<br/><b>set </b><font color="Maroon">y</font> =  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span>;<br/><b>set </b><font color="Maroon">j</font> =  <a href="henmodel.html#K1">min*</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> ;<br/><b>reconsider </b><font color="Maroon">r2</font> =  <a href="goedelcp.html#K5" target="_self">the_scope_of</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/><b>reconsider </b><font color="Maroon">y2</font> =  <a href="goedelcp.html#K4" target="_self">bound_in</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="qc_lang1.html#NM3">bound_QC-variable</a> ;<br/><b>reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/><a NAME="E26:36_11_1_2"/>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="goedelcp.html#K1" target="_self">ExCl</a> 
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>then consider </b><font color="Maroon">y1</font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a>, <font color="Maroon">s1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E28:36_11_1_2"/><i><font color="Green">E247</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">y1</font>,<font color="Maroon">s1</font>
 <b>by </b><i/>;<br/><a NAME="E29:36_11_1_2"/><i><font color="Green">E248</font></i>: 
( <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="goedelcp.html#K3" target="_self">Ex-the_scope_of</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a>  <a href="goedelcp.html#K3" target="_self">Ex-the_scope_of</a> <font color="Maroon">r1</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E30:36_11_1_2"/>
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a>  <a href="goedelcp.html#K2" target="_self">Ex-bound_in</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a>  <a href="goedelcp.html#K2" target="_self">Ex-bound_in</a> <font color="Maroon">r1</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E31:36_11_1_2"/><b>then </b><i><font color="Green">E249</font></i>: 
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</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"><font color="Maroon">s1</font> <a href="substut2.html#K4">.</a> <font color="Maroon">y1</font>,<span class="p4">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p5">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p0">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span></span>)</span></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> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/><a NAME="E33:36_11_1_2"/>
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="goedelcp.html#D1" target="_self">Def1</a>, </i>;<br/><a NAME="E34:36_11_1_2"/><b>then </b>
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <font color="Maroon">Z</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E35:36_11_1_2"/><b>then </b><i><font color="Green">E251</font></i>: 
( not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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="goedelcp.html#K6" target="_self">still_not-bound_in</a> <font color="Maroon">Z</font> &amp; not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r1</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E36:36_11_1_2"/>
 <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="txt" href="goedelcp.html#E1:36"><i><font color="Green">E54</font></i></a>, </i>;<br/><a NAME="E37:36_11_1_2"/><b>then </b>
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E38:36_11_1_2"/><b>then </b><i><font color="Green">E252</font></i>: 
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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="calcul_1.html#K3">still_not-bound_in</a> <font color="Maroon">f1</font>
 <b>by </b><i/>;<br/><a NAME="E39:36_11_1_2"/>
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">r1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E40:36_11_1_2"/><b>then </b>
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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="calcul_1.html#K3">still_not-bound_in</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r1</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="calcul_1.html#T60">CALCUL_1:60</a></i>;<br/><a NAME="E41:36_11_1_2"/><b>then </b>
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3">still_not-bound_in</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3">still_not-bound_in</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r1</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E42:36_11_1_2"/><b>then </b><i><font color="Green">E253</font></i>: 
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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="calcul_1.html#K3">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">f1</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">r1</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="calcul_1.html#T59">CALCUL_1:59</a></i>;<br/><a NAME="E43:36_11_1_2"/>
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span>
 <b>by </b><i><a class="ref" href="qc_lang3.html#T7">QC_LANG3:7</a>, <a class="ref" href="qc_lang3.html#T11">QC_LANG3:11</a></i>;<br/><a NAME="E44:36_11_1_2"/><b>then </b>
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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="calcul_1.html#K3">still_not-bound_in</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> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="calcul_1.html#T60">CALCUL_1:60</a></i>;<br/><a NAME="E45:36_11_1_2"/><b>then </b>
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> </span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3">still_not-bound_in</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r1</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3">still_not-bound_in</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> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E46:36_11_1_2"/><b>then </b>
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <span class="p2">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="goedelcp.html#K6" target="_self">still_not-bound_in</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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="calcul_1.html#K3">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r1</font></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> <a href="cqc_lang.html#K9">VERUM</a> </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#T59">CALCUL_1:59</a></i>;<br/><a NAME="E47:36_11_1_2"/><b>then </b>
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</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">r1</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"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i>, , <a class="ref" href="calcul_1.html#T61">CALCUL_1:61</a></i>;<br/><a NAME="E48:36_11_1_2"/><b>then </b>
 <a href="calcul_1.html#R4">|-</a> <font color="Maroon">f1</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> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></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="E49:36_11_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="henmodel.html#R1">|-</a>  <a href="cqc_lang.html#K10">'not'</a> <a href="cqc_lang.html#K9">VERUM</a> 
 <b>by </b><i><a class="txt" href="goedelcp.html#E1:36"><i><font color="Green">E54</font></i></a>, <a class="ref" href="henmodel.html#D2">HENMODEL:def 2</a></i>;<br/><b>hence </b><a NAME="E50:36_11_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="goedelcp.html#T2" target="_self">Th2</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:36_11_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goedelcp.html#E5:36"><i><font color="Green">E56</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:36_11"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E24:36"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="goedelcp.html#D2" target="_self">Def2</a>, <a class="ref" href="goedelcp.html#T1" target="_self">Th1</a>);<br/></i>
<a NAME="E25:36"/><b>then </b>
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/>( <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> is <a href="henmodel.html#V1">Consistent</a> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">CY</font> = <font color="Maroon">CY</font> as  <a href="henmodel.html#V1">Consistent</a> <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a>  <b>by </b><i>, <a class="ref" href="henmodel.html#T11">HENMODEL:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">CY</font>
;<br/>

<b>thus </b><a NAME="E27:36"/>
( <font color="Maroon">CX</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">CY</font> &amp; <font color="Maroon">CY</font> is <a href="goedelcp.html#V2" target="_self">with_examples</a> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>


</div>
