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

<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">J</font> be    <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c3">v</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ;<br/>







<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E31">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<b>thus </b><a NAME="E2:33"/>
( <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">v</font> <a href="calcul_1.html#R9" title="CALCUL_1:pred.9">|=</a>  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c4">f</font> implies <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">v</font> <a href="calcul_1.html#R9" title="CALCUL_1:pred.9">|=</a> <font color="Maroon" title="c4">f</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" title="c2">J</font>,<font color="Maroon" title="c3">v</font> <a href="calcul_1.html#R9" title="CALCUL_1:pred.9">|=</a>  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c4">f</font> )
 ;<br/>

<a NAME="E2:33_1"/><b>then </b><i><font color="Green" title="E32">A2</font></i>: 
( <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">v</font> <a href="calcul_1.html#R6" title="CALCUL_1:pred.6">|=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c4">f</font> )
 
<b>by </b><i><a class="ref" href="calcul_1.html#D14" target="_self" title="CALCUL_1:def.14">Def14</a></i>;<br/>
<b>let </b><font color="Maroon" title="c5">p</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="calcul_1.html#D11" target="_self" title="CALCUL_1:def.11">CALCUL_1:def 11</a>,<a class="ref" href="calcul_1.html#D14" target="_self" title="CALCUL_1:def.14">CALCUL_1:def 14</a><br/>

<b>assume </b><a NAME="E3:33_1"/><i><font color="Green" title="E33">A3</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font>
 ;<br/>

<a NAME="E4:33_1"/>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c4">f</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="calcul_1.html#E1:33"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="calcul_1.html#E3:33_1"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="calcul_1.html#T3" target="_self" title="CALCUL_1:th.3">Th3</a></i>;<br/>
<a NAME="E5:33_1"/><b>then </b>
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c4">f</font></span>)</span> or <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c4">f</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:33_1"/>
<font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E2:33_1"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="calcul_1.html#D11" target="_self" title="CALCUL_1:def.11">Def11</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E3:33"/>
( <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">v</font> <a href="calcul_1.html#R9" title="CALCUL_1:pred.9">|=</a> <font color="Maroon" title="c4">f</font> implies ( <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">v</font> <a href="calcul_1.html#R9" title="CALCUL_1:pred.9">|=</a>  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c4">f</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" title="E32">A4</font></i>: 
<font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">v</font> <a href="calcul_1.html#R6" title="CALCUL_1:pred.6">|=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="calcul_1.html#D14" target="_self" title="CALCUL_1:def.14">CALCUL_1:def 14</a><br/>

<b>thus </b><a NAME="E2:33_2"/>
<font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">v</font> <a href="calcul_1.html#R6" title="CALCUL_1:pred.6">|=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c4">f</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="calcul_1.html#D14" target="_self" title="CALCUL_1:def.14">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" title="c5">p</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="calcul_1.html#D11" target="_self" title="CALCUL_1:def.11">CALCUL_1:def 11</a><br/>

<b>assume </b><a NAME="E1:33_2_1"/><i><font color="Green" title="E33">A5</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c4">f</font></span>)</span>
 ;<br/>

<a NAME="E2:33_2_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c4">f</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E3:33_2_1"/><b>then </b>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c4">f</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="calcul_1.html#E1:33_2_1"><i><font color="Green" title="E33">A5</font></i></a></i>;<br/>
<a NAME="E4:33_2_1"/><b>then </b>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="calcul_1.html#E1:33"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="calcul_1.html#T3" target="_self" title="CALCUL_1:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E5:33_2_1"/>
<font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:33_2"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="calcul_1.html#D11" target="_self" title="CALCUL_1:def.11">Def11</a></i>;<br/>


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

<b>thus </b><a NAME="E3:33_2"/>
<font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c4">f</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" title="E33">A6</font></i>: 
 <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="calcul_1.html#E1:33"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="calcul_1.html#D2" target="_self" title="CALCUL_1:def.2">Def2</a></i>;<br/>
<a NAME="E2:33_2_2"/>
( 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> )
 
<b>by </b><i><a class="txt" href="calcul_1.html#E1:33"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:33_2_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E4:33_2_2"/><b>then </b>
 <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="calcul_1.html#E1:33_2_2"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>hence </b><a NAME="E5:33_2_2"/>
<font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:33_2"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="calcul_1.html#D11" target="_self" title="CALCUL_1:def.11">Def11</a></i>;<br/>


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


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


</div>
