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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="valuat_1.html#M1">interpretation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>







<b>assume </b><a NAME="E1:33"/><i><font color="Green">E31</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>thus </b><a NAME="E2:33"/>
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="calcul_1.html#R9" target="_self">|=</a>  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="valuat_1.html#R1">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="calcul_1.html#R9" target="_self">|=</a> <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1"/>
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="calcul_1.html#R9" target="_self">|=</a>  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="valuat_1.html#R1">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E2:33_1"/><b>then </b><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="calcul_1.html#R6" target="_self">|=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="valuat_1.html#R1">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="calcul_1.html#D14" target="_self">Def14</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="calcul_1.html#D11" target="_self">CALCUL_1:def 11</a>,<a class="ref" href="calcul_1.html#D14" target="_self">CALCUL_1:def 14</a><br/>

<b>assume </b><a NAME="E3:33_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E4:33_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="calcul_1.html#E1:33"><i><font color="Green">E31</font></i></a>, <a class="txt" href="calcul_1.html#E3:33_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="calcul_1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E5:33_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:33_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E2:33_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="calcul_1.html#D11" target="_self">Def11</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E3:33"/>
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="calcul_1.html#R9" target="_self">|=</a> <font color="Maroon">c<sub>4</sub></font> implies ( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="calcul_1.html#R9" target="_self">|=</a>  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="valuat_1.html#R1">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">c<sub>4</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="calcul_1.html#R6" target="_self">|=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="calcul_1.html#D14" target="_self">CALCUL_1:def 14</a><br/>

<b>thus </b><a NAME="E2:33_2"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="calcul_1.html#R6" target="_self">|=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="calcul_1.html#D14" target="_self">CALCUL_1:def 14</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_1"><b>proof </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="cqc_lang.html#K7">CQC-WFF</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="calcul_1.html#D11" target="_self">CALCUL_1:def 11</a><br/>

<b>assume </b><a NAME="E1:33_2_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:33_2_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E3:33_2_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="calcul_1.html#E1:33_2_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E4:33_2_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="calcul_1.html#E1:33"><i><font color="Green">E31</font></i></a>, <a class="ref" href="calcul_1.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E5:33_2_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:33_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="calcul_1.html#D11" target="_self">Def11</a></i>;<br/>


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

<b>thus </b><a NAME="E3:33_2"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="valuat_1.html#R1">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">c<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:33_2_2"/><i><font color="Green">E33</font></i>: 
 <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="calcul_1.html#E1:33"><i><font color="Green">E31</font></i></a>, <a class="ref" href="calcul_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E2:33_2_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> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="calcul_1.html#E1:33"><i><font color="Green">E31</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:33_2_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E4:33_2_2"/><b>then </b>
 <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="calcul_1.html#E1:33_2_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>hence </b><a NAME="E5:33_2_2"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="valuat_1.html#R1">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:33_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="calcul_1.html#D11" target="_self">Def11</a></i>;<br/>


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


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


</div>
