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

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

<b>thus </b><a NAME="E1:41"/>
(  <a href="cqc_sim1.html#K12" target="_self">index</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V6">closed</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1"/>
 <a href="cqc_sim1.html#K12" target="_self">index</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:41_1"/><b>then </b>
 <a href="cqc_sim1.html#K10" target="_self">min</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K11" target="_self">NBI</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<a NAME="E3:41_1"/><b>then </b>
0 <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K11" target="_self">NBI</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E4:41_1"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/> not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
;<br/>
<a NAME="E5:41_1"/><b>then </b><i><font color="Green">E24</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/> not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font> ) )
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1_1"/><i><font color="Green">E25</font></i>: 
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="qc_lang1.html#NM3">bound_QC-variable</a> <b>by </b><i><a class="txt" href="cqc_sim1.html#E1:41_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:41_1_1"/><i><font color="Green">E26</font></i>: 
 <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="qc_lang3.html#T36">QC_LANG3:36</a></i>;<br/><a NAME="E6:41_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><b>hence </b><a NAME="E7:41_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="cqc_sim1.html#E5:41_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="cqc_sim1.html#E1:41_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="cqc_sim1.html#E5:41_1_1"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:41_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V6">closed</a>
 <b>by </b><i><a class="ref" href="qc_lang1.html#D30">QC_LANG1:def 30</a></i>;<br/>


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

<b>assume </b><a NAME="E2:41"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V6">closed</a>
 ;<br/>

<a NAME="E3:41"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/> not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D30">QC_LANG1:def 30</a></i>;<br/>
<a NAME="E4:41"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/> not  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
;<br/>
<a NAME="E5:41"/><b>then </b>
0 <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K11" target="_self">NBI</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E6:41"/><b>then </b>
 <a href="cqc_sim1.html#K10" target="_self">min</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K11" target="_self">NBI</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E7:41"/><b>then </b>
 <a href="cqc_sim1.html#K10" target="_self">min</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K11" target="_self">NBI</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<b>hence </b><a NAME="E8:41"/>
 <a href="cqc_sim1.html#K12" target="_self">index</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div>
