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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>



<a NAME="E1:44"/><i><font color="Green">E27</font></i>: 
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="qc_lang3.html#T14">QC_LANG3:14</a></i>;<br/>
<a NAME="E2:44"/><i><font color="Green">E28</font></i>: 
 <a href="cqc_sim1.html#K11" target="_self">NBI</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="cqc_sim1.html#K11" target="_self">NBI</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></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="E1:44_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K11" target="_self">NBI</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then 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="E3:44_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( 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="Maroon">c<sub>4</sub></font> <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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:44_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:44_1_1"/>
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="cqc_sim1.html#E1:44"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E3:44_1_1"/>
not  <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">c<sub>5</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="txt" href="cqc_sim1.html#E3:44_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="cqc_sim1.html#E1:44_1_1"><i><font color="Green">E30</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:44_1"/>
<font color="Maroon">c<sub>3</sub></font> <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="txt" href="cqc_sim1.html#E3:44_1"><i><font color="Green">E29</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:44"/>
 <a href="cqc_sim1.html#K11" target="_self">NBI</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="cqc_sim1.html#K11" target="_self">NBI</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></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="E1:44_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K11" target="_self">NBI</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then 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="E3:44_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( 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="Maroon">c<sub>4</sub></font> <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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:44_2_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:44_2_1"/>
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="cqc_sim1.html#E1:44"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E3:44_2_1"/>
not  <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">c<sub>5</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>2</sub></font>
 <b>by </b><i><a class="txt" href="cqc_sim1.html#E3:44_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="cqc_sim1.html#E1:44_2_1"><i><font color="Green">E30</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:44_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K11" target="_self">NBI</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cqc_sim1.html#E3:44_2"><i><font color="Green">E29</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:44"/><b>then </b><i><font color="Green">E29</font></i>: 
(  <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>  <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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> &amp;  <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>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="cqc_sim1.html#E2:44"><i><font color="Green">E28</font></i></a>, <a class="ref" href="cqc_sim1.html#T19" target="_self">Th19</a></i>;<br/>
<b>thus </b><a NAME="E5:44"/>
(  <a href="cqc_sim1.html#K12" target="_self">index</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="cqc_sim1.html#K12" target="_self">index</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="cqc_sim1.html#K12" target="_self">index</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="cqc_sim1.html#K12" target="_self">index</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="cqc_sim1.html#E4:44"><i><font color="Green">E29</font></i></a></i>;<br/>


</div>
