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

<b>let </b><font color="Maroon" title="c1">V</font> be    <a href="modelc_1.html#L2" title="MODELC_1:struct.2">CTLModelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">Kai</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modelc_1.html#K19" title="MODELC_1:func.19">atomic_WFF</a> ,the <a href="modelc_1.html#U6" title="MODELC_1:sel.6">BasicAssign</a> of <font color="Maroon" title="c1">V</font>;<br/>



<b>set </b><font color="Maroon" title="c3">M</font> =  <a href="modelc_1.html#K23" title="MODELC_1:func.23">EvalFamily</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>;<br/>
<a NAME="E1:63"/>
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K23" title="MODELC_1:func.23">EvalFamily</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font> holds <br/><font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:63_1"/><i><font color="Green" title="E56">A1</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K23" title="MODELC_1:func.23">EvalFamily</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:63_1"/><i><font color="Green" title="E57">A2</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K21" title="MODELC_1:func.21">EvalSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:63_1"><i><font color="Green" title="E56">A1</font></i></a>, <a class="ref" href="modelc_1.html#D31" target="_self" title="MODELC_1:def.31">Def31</a></i>;<br/>
<b>thus </b><a NAME="E4:63_1"/>
<font color="Maroon" title="c4">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:63_1"><i><font color="Green" title="E57">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c4">Choice</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:63"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">Choice</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K23" title="MODELC_1:func.23">EvalFamily</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>
 <b>and </b><br/><a NAME="E4:63"/><i><font color="Green" title="E56">A3</font></i>: 
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K23" title="MODELC_1:func.23">EvalFamily</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font> holds <br/><font color="Maroon" title="c4">Choice</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font>
 <b>by </b><i><a class="ref" href="wellord2.html#T28" title="WELLORD2:th.28">WELLORD2:28</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c4">Choice</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K21" title="MODELC_1:func.21">EvalSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<span class="p2">(<span class="default"><a href="modelc_1.html#K2" title="MODELC_1:func.2">k_nat</a> $1</span>)</span></span>)</span>;<br/>
<a NAME="E5:63"/><i><font color="Green" title="E57">A4</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> ,the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">Assignations</a> of <font color="Maroon" title="c1">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:63_2"/><i><font color="Green" title="E58">A5</font></i>: 
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>

<b>set </b><font color="Maroon" title="c6">Y</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">n</font>);<br/>
<a NAME="E2:63_2"/><i><font color="Green" title="E59">A6</font></i>: 
 <a href="modelc_1.html#K2" title="MODELC_1:func.2">k_nat</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E1:63_2"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="modelc_1.html#D2" target="_self" title="MODELC_1:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">n</font> = <font color="Maroon" title="c5">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="modelc_1.html#E1:63_2"><i><font color="Green" title="E58">A5</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">Z</font> =  <a href="modelc_1.html#K21" title="MODELC_1:func.21">EvalSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<font color="Maroon" title="c7">n</font>;<br/>
<a NAME="E4:63_2"/>
 <a href="modelc_1.html#K21" title="MODELC_1:func.21">EvalSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K23" title="MODELC_1:func.23">EvalFamily</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E63"><i><font color="Green" title="E54">Lm30</font></i></a></i>;<br/>
<a NAME="E5:63_2"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K21" title="MODELC_1:func.21">EvalSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<font color="Maroon" title="c7">n</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E4:63"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="modelc_1.html#E2:63_2"><i><font color="Green" title="E59">A6</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> ,the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">Assignations</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E7:63_2"/><i><font color="Green" title="E60">A7</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h</font>
 <b>and </b><br/><a NAME="E8:63_2"/>
<font color="Maroon" title="c9">h</font> <a href="modelc_1.html#R2" title="MODELC_1:pred.2">is-PreEvaluation-for</a> <font color="Maroon" title="c7">n</font>,<font color="Maroon" title="c2">Kai</font>
 ;<br/>
<b>thus </b><a NAME="E9:63_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">n</font>) is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> ,the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">Assignations</a> of <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E7:63_2"><i><font color="Green" title="E60">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:63"/><i><font color="Green" title="E58">A8</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> ,the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">Assignations</a> of <font color="Maroon" title="c1">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:63_3"/><i><font color="Green" title="E59">A9</font></i>: 
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>

<a NAME="E2:63_3"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">n</font>) is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> ,the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">Assignations</a> of <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E5:63"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="modelc_1.html#E1:63_3"><i><font color="Green" title="E59">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:63_3"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> ,the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">Assignations</a> of <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c5">f1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> ,the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">Assignations</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E8:63"/><i><font color="Green" title="E59">A10</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2" title="FUNCT_2:sch.2">FUNCT_2:sch 2</a>(<a class="txt" href="modelc_1.html#E6:63"><i><font color="Green" title="E58">A8</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c6">v0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">Assignations</a> of <font color="Maroon" title="c1">V</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><a href="modelc_1.html#K22" title="MODELC_1:func.22">CastEval</a> <font color="Maroon" title="c1">V</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="modelc_1.html#K18" title="MODELC_1:func.18">CastCTLformula</a> $1</span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c6">v0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<a NAME="E10:63"/><i><font color="Green" title="E60">A11</font></i>: 
for <font color="Olive" title="b1">H</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a>  holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">H</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">Assignations</a> of <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> ,the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">Assignations</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E12:63"/><i><font color="Green" title="E61">A12</font></i>: 
for <font color="Olive" title="b1">H</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a>  holds <br/><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">H</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2" title="FUNCT_2:sch.2">FUNCT_2:sch 2</a>(<a class="txt" href="modelc_1.html#E10:63"><i><font color="Green" title="E60">A11</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c7">f</font>
;<br/>

<a NAME="E13:63"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c7">f</font> <a href="modelc_1.html#R2" title="MODELC_1:pred.2">is-PreEvaluation-for</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">Kai</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_4"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><font color="Maroon" title="c7">f</font> <a href="modelc_1.html#R2" title="MODELC_1:pred.2">is-PreEvaluation-for</a> $1,<font color="Maroon" title="c2">Kai</font>;<br/>
<a NAME="E1:63_4"/><i><font color="Green" title="E62">A13</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:63_4_1"/>
for <font color="Olive" title="b1">H</font> being  <a href="modelc_1.html#NM1" title="MODELC_1:NM.1">CTL-formula</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>( ( <font color="Olive" title="b1">H</font> is <a href="modelc_1.html#V2" title="MODELC_1:attr.2">atomic</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Kai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> ) &amp; ( <font color="Olive" title="b1">H</font> is <a href="modelc_1.html#V3" title="MODELC_1:attr.3">negative</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U8" title="MODELC_1:sel.8">Not</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Olive" title="b1">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive" title="b1">H</font> is <a href="modelc_1.html#V4" title="MODELC_1:attr.4">conjunctive</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U7" title="MODELC_1:sel.7">And</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Olive" title="b1">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Olive" title="b1">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive" title="b1">H</font> is <a href="modelc_1.html#V5" title="MODELC_1:attr.5">ExistNext</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U9" title="MODELC_1:sel.9">EneXt</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Olive" title="b1">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive" title="b1">H</font> is <a href="modelc_1.html#V6" title="MODELC_1:attr.6">ExistGlobal</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U10" title="MODELC_1:sel.10">EGlobal</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Olive" title="b1">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive" title="b1">H</font> is <a href="modelc_1.html#V7" title="MODELC_1:attr.7">ExistUntill</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U11" title="MODELC_1:sel.11">EUntill</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Olive" title="b1">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Olive" title="b1">H</font></span>)</span></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="modelc_1.html#E32"><i><font color="Green" title="E25">Lm10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:63_4_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <b>by </b><i><a class="ref" href="modelc_1.html#D27" target="_self" title="MODELC_1:def.27">Def27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:63_4"/><i><font color="Green" title="E63">A14</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:63_4_2"/><i><font color="Green" title="E64">A15</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>]
 ;<br/>

<a NAME="E2:63_4_2"/>
for <font color="Olive" title="b1">H</font> being  <a href="modelc_1.html#NM1" title="MODELC_1:NM.1">CTL-formula</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/>( ( <font color="Olive" title="b1">H</font> is <a href="modelc_1.html#V2" title="MODELC_1:attr.2">atomic</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Kai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> ) &amp; ( <font color="Olive" title="b1">H</font> is <a href="modelc_1.html#V3" title="MODELC_1:attr.3">negative</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U8" title="MODELC_1:sel.8">Not</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Olive" title="b1">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive" title="b1">H</font> is <a href="modelc_1.html#V4" title="MODELC_1:attr.4">conjunctive</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U7" title="MODELC_1:sel.7">And</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Olive" title="b1">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Olive" title="b1">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive" title="b1">H</font> is <a href="modelc_1.html#V5" title="MODELC_1:attr.5">ExistNext</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U9" title="MODELC_1:sel.9">EneXt</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Olive" title="b1">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive" title="b1">H</font> is <a href="modelc_1.html#V6" title="MODELC_1:attr.6">ExistGlobal</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U10" title="MODELC_1:sel.10">EGlobal</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Olive" title="b1">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive" title="b1">H</font> is <a href="modelc_1.html#V7" title="MODELC_1:attr.7">ExistUntill</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U11" title="MODELC_1:sel.11">EUntill</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Olive" title="b1">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Olive" title="b1">H</font></span>)</span></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_4_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">H</font> be   <a href="modelc_1.html#NM1" title="MODELC_1:NM.1">CTL-formula</a>;<br/>

<b>assume </b><a NAME="E1:63_4_2_1"/><i><font color="Green" title="E65">A16</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_4_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_4_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_4_2_1_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="modelc_1.html#E1:63_4_2_1"><i><font color="Green" title="E65">A16</font></i></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_4_2_1_1_1_1"><b>case </b></a><a NAME="E1:63_4_2_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:63_4_2_1_1_1_1"/>
( ( <font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V2" title="MODELC_1:attr.2">atomic</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Kai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> ) &amp; ( <font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V3" title="MODELC_1:attr.3">negative</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U8" title="MODELC_1:sel.8">Not</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V4" title="MODELC_1:attr.4">conjunctive</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U7" title="MODELC_1:sel.7">And</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V5" title="MODELC_1:attr.5">ExistNext</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U9" title="MODELC_1:sel.9">EneXt</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V6" title="MODELC_1:attr.6">ExistGlobal</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U10" title="MODELC_1:sel.10">EGlobal</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V7" title="MODELC_1:attr.7">ExistUntill</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U11" title="MODELC_1:sel.11">EUntill</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:63_4_2"><i><font color="Green" title="E64">A15</font></i></a>, <a class="ref" href="modelc_1.html#D27" target="_self" title="MODELC_1:def.27">Def27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_4_2_1_1_1_2"><b>case </b></a><a NAME="E1:63_4_2_1_1_1_2"/><i><font color="Green" title="E66">A17</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:63_4_2_1_1_1_2"/><i><font color="Green" title="E67">A18</font></i>: 
<font color="Maroon" title="c9">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> 
 <b>by </b><i><a class="ref" href="modelc_1.html#T1" target="_self" title="MODELC_1:th.1">Th1</a></i>;<br/><b>then </b><font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K18" title="MODELC_1:func.18">CastCTLformula</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font></span>)</span>
<b>by </b><i><a class="ref" href="modelc_1.html#D24" target="_self" title="MODELC_1:def.24">Def24</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>( <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font>)
<b>by </b><i><a class="txt" href="modelc_1.html#E8:63"><i><font color="Green" title="E59">A10</font></i></a></i>
;<br/><a NAME="E4:63_4_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E68">A19</font></i>: 
 <a href="modelc_1.html#K22" title="MODELC_1:func.22">CastEval</a> <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="modelc_1.html#K18" title="MODELC_1:func.18">CastCTLformula</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c6">v0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>( <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font>)
 <b>by </b><i><a class="ref" href="modelc_1.html#D30" target="_self" title="MODELC_1:def.30">Def30</a></i>;<br/><b>set </b><font color="Maroon" title="c10">f2</font> = <font color="Maroon">H<sub>1</sub></font>( <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font>);<br/><b>reconsider </b><font color="Maroon" title="c11">f2</font> = <font color="Maroon">H<sub>1</sub></font>( <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font>) as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> ,the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">Assignations</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="modelc_1.html#E4:63_4_2_1_1_1_2"><i><font color="Green" title="E68">A19</font></i></a></i>;<br/><a NAME="E6:63_4_2_1_1_1_2"/><i><font color="Green" title="E69">A20</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E12:63"><i><font color="Green" title="E61">A12</font></i></a>, <a class="txt" href="modelc_1.html#E2:63_4_2_1_1_1_2"><i><font color="Green" title="E67">A18</font></i></a>, <a class="txt" href="modelc_1.html#E4:63_4_2_1_1_1_2"><i><font color="Green" title="E68">A19</font></i></a></i>;<br/><a NAME="E7:63_4_2_1_1_1_2"/><i><font color="Green" title="E70">A21</font></i>: 
<font color="Maroon" title="c11">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Choice</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K21" title="MODELC_1:func.21">EvalSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="modelc_1.html#D2" target="_self" title="MODELC_1:def.2">Def2</a></i>;<br/><a NAME="E8:63_4_2_1_1_1_2"/>
 <a href="modelc_1.html#K21" title="MODELC_1:func.21">EvalSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K23" title="MODELC_1:func.23">EvalFamily</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E63"><i><font color="Green" title="E54">Lm30</font></i></a></i>;<br/><a NAME="E9:63_4_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">Choice</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K21" title="MODELC_1:func.21">EvalSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K21" title="MODELC_1:func.21">EvalSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E4:63"><i><font color="Green" title="E56">A3</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> ,the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">Assignations</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E11:63_4_2_1_1_1_2"/><i><font color="Green" title="E71">A22</font></i>: 
<font color="Maroon" title="c11">f2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c12">h</font>
 <b>and </b><br/><a NAME="E12:63_4_2_1_1_1_2"/><i><font color="Green" title="E72">A23</font></i>: 
<font color="Maroon" title="c12">h</font> <a href="modelc_1.html#R2" title="MODELC_1:pred.2">is-PreEvaluation-for</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font>,<font color="Maroon" title="c2">Kai</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E7:63_4_2_1_1_1_2"><i><font color="Green" title="E70">A21</font></i></a></i>;<br/><a NAME="E13:63_4_2_1_1_1_2"/><i><font color="Green" title="E73">A24</font></i>: 
<font color="Maroon" title="c11">f2</font> <a href="modelc_1.html#R2" title="MODELC_1:pred.2">is-PreEvaluation-for</a> <font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c2">Kai</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:63_4_2_1_1_1_2"><i><font color="Green" title="E66">A17</font></i></a>, <a class="txt" href="modelc_1.html#E11:63_4_2_1_1_1_2"><i><font color="Green" title="E71">A22</font></i></a>, <a class="txt" href="modelc_1.html#E12:63_4_2_1_1_1_2"><i><font color="Green" title="E72">A23</font></i></a>, <a class="txt" href="modelc_1.html#E55"><i><font color="Green" title="E47">Lm25</font></i></a></i>;<br/><a NAME="E14:63_4_2_1_1_1_2"/><i><font color="Green" title="E74">A25</font></i>: 
( <font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V3" title="MODELC_1:attr.3">negative</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U8" title="MODELC_1:sel.8">Not</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_4_2_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:63_4_2_1_1_1_2_2"/><i><font color="Green" title="E75">A26</font></i>: 
<font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V3" title="MODELC_1:attr.3">negative</a>
 ;<br/>

<a NAME="E2:63_4_2_1_1_1_2_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E47"><i><font color="Green" title="E40">Lm22</font></i></a></i>;<br/>
<a NAME="E3:63_4_2_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E76">A27</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E1:63_4_2_1_1_1_2"><i><font color="Green" title="E66">A17</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> = 
the <a href="modelc_1.html#U8" title="MODELC_1:sel.8">Not</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E6:63_4_2_1_1_1_2"><i><font color="Green" title="E69">A20</font></i></a>, <a class="txt" href="modelc_1.html#E11:63_4_2_1_1_1_2"><i><font color="Green" title="E71">A22</font></i></a>, <a class="txt" href="modelc_1.html#E12:63_4_2_1_1_1_2"><i><font color="Green" title="E72">A23</font></i></a>, <a class="txt" href="modelc_1.html#E1:63_4_2_1_1_1_2_2"><i><font color="Green" title="E75">A26</font></i></a>, <a class="ref" href="modelc_1.html#D27" target="_self" title="MODELC_1:def.27">Def27</a></i>
<br/>.= 
the <a href="modelc_1.html#U8" title="MODELC_1:sel.8">Not</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E1:63_4_2"><i><font color="Green" title="E64">A15</font></i></a>, <a class="txt" href="modelc_1.html#E13:63_4_2_1_1_1_2"><i><font color="Green" title="E73">A24</font></i></a>, <a class="txt" href="modelc_1.html#E3:63_4_2_1_1_1_2_2"><i><font color="Green" title="E76">A27</font></i></a>, <a class="txt" href="modelc_1.html#E57"><i><font color="Green" title="E49">Lm27</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:63_4_2_1_1_1_2_2"/>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U8" title="MODELC_1:sel.8">Not</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E15:63_4_2_1_1_1_2"/><i><font color="Green" title="E75">A28</font></i>: 
( <font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V5" title="MODELC_1:attr.5">ExistNext</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U9" title="MODELC_1:sel.9">EneXt</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_4_2_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:63_4_2_1_1_1_2_3"/><i><font color="Green" title="E76">A29</font></i>: 
<font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V5" title="MODELC_1:attr.5">ExistNext</a>
 ;<br/>

<a NAME="E2:63_4_2_1_1_1_2_3"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E47"><i><font color="Green" title="E40">Lm22</font></i></a></i>;<br/>
<a NAME="E3:63_4_2_1_1_1_2_3"/><b>then </b><i><font color="Green" title="E77">A30</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E1:63_4_2_1_1_1_2"><i><font color="Green" title="E66">A17</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> = 
the <a href="modelc_1.html#U9" title="MODELC_1:sel.9">EneXt</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E6:63_4_2_1_1_1_2"><i><font color="Green" title="E69">A20</font></i></a>, <a class="txt" href="modelc_1.html#E11:63_4_2_1_1_1_2"><i><font color="Green" title="E71">A22</font></i></a>, <a class="txt" href="modelc_1.html#E12:63_4_2_1_1_1_2"><i><font color="Green" title="E72">A23</font></i></a>, <a class="txt" href="modelc_1.html#E1:63_4_2_1_1_1_2_3"><i><font color="Green" title="E76">A29</font></i></a>, <a class="ref" href="modelc_1.html#D27" target="_self" title="MODELC_1:def.27">Def27</a></i>
<br/>.= 
the <a href="modelc_1.html#U9" title="MODELC_1:sel.9">EneXt</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E1:63_4_2"><i><font color="Green" title="E64">A15</font></i></a>, <a class="txt" href="modelc_1.html#E13:63_4_2_1_1_1_2"><i><font color="Green" title="E73">A24</font></i></a>, <a class="txt" href="modelc_1.html#E3:63_4_2_1_1_1_2_3"><i><font color="Green" title="E77">A30</font></i></a>, <a class="txt" href="modelc_1.html#E57"><i><font color="Green" title="E49">Lm27</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:63_4_2_1_1_1_2_3"/>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U9" title="MODELC_1:sel.9">EneXt</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E16:63_4_2_1_1_1_2"/><i><font color="Green" title="E76">A31</font></i>: 
( <font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V6" title="MODELC_1:attr.6">ExistGlobal</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U10" title="MODELC_1:sel.10">EGlobal</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_4_2_1_1_1_2_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:63_4_2_1_1_1_2_4"/><i><font color="Green" title="E77">A32</font></i>: 
<font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V6" title="MODELC_1:attr.6">ExistGlobal</a>
 ;<br/>

<a NAME="E2:63_4_2_1_1_1_2_4"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E47"><i><font color="Green" title="E40">Lm22</font></i></a></i>;<br/>
<a NAME="E3:63_4_2_1_1_1_2_4"/><b>then </b><i><font color="Green" title="E78">A33</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E1:63_4_2_1_1_1_2"><i><font color="Green" title="E66">A17</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> = 
the <a href="modelc_1.html#U10" title="MODELC_1:sel.10">EGlobal</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E6:63_4_2_1_1_1_2"><i><font color="Green" title="E69">A20</font></i></a>, <a class="txt" href="modelc_1.html#E11:63_4_2_1_1_1_2"><i><font color="Green" title="E71">A22</font></i></a>, <a class="txt" href="modelc_1.html#E12:63_4_2_1_1_1_2"><i><font color="Green" title="E72">A23</font></i></a>, <a class="txt" href="modelc_1.html#E1:63_4_2_1_1_1_2_4"><i><font color="Green" title="E77">A32</font></i></a>, <a class="ref" href="modelc_1.html#D27" target="_self" title="MODELC_1:def.27">Def27</a></i>
<br/>.= 
the <a href="modelc_1.html#U10" title="MODELC_1:sel.10">EGlobal</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E1:63_4_2"><i><font color="Green" title="E64">A15</font></i></a>, <a class="txt" href="modelc_1.html#E13:63_4_2_1_1_1_2"><i><font color="Green" title="E73">A24</font></i></a>, <a class="txt" href="modelc_1.html#E3:63_4_2_1_1_1_2_4"><i><font color="Green" title="E78">A33</font></i></a>, <a class="txt" href="modelc_1.html#E57"><i><font color="Green" title="E49">Lm27</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:63_4_2_1_1_1_2_4"/>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U10" title="MODELC_1:sel.10">EGlobal</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E17:63_4_2_1_1_1_2"/><i><font color="Green" title="E77">A34</font></i>: 
( <font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V4" title="MODELC_1:attr.4">conjunctive</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U7" title="MODELC_1:sel.7">And</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_4_2_1_1_1_2_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:63_4_2_1_1_1_2_5"/><i><font color="Green" title="E78">A35</font></i>: 
<font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V4" title="MODELC_1:attr.4">conjunctive</a>
 ;<br/>

<a NAME="E2:63_4_2_1_1_1_2_5"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E48"><i><font color="Green" title="E41">Lm23</font></i></a></i>;<br/>
<a NAME="E3:63_4_2_1_1_1_2_5"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E1:63_4_2_1_1_1_2"><i><font color="Green" title="E66">A17</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E4:63_4_2_1_1_1_2_5"/><b>then </b><i><font color="Green" title="E79">A36</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E1:63_4_2"><i><font color="Green" title="E64">A15</font></i></a>, <a class="txt" href="modelc_1.html#E13:63_4_2_1_1_1_2"><i><font color="Green" title="E73">A24</font></i></a>, <a class="txt" href="modelc_1.html#E57"><i><font color="Green" title="E49">Lm27</font></i></a></i>;<br/>
<a NAME="E5:63_4_2_1_1_1_2_5"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E1:63_4_2_1_1_1_2_5"><i><font color="Green" title="E78">A35</font></i></a>, <a class="txt" href="modelc_1.html#E48"><i><font color="Green" title="E41">Lm23</font></i></a></i>;<br/>
<a NAME="E6:63_4_2_1_1_1_2_5"/><b>then </b><i><font color="Green" title="E80">A37</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E1:63_4_2_1_1_1_2"><i><font color="Green" title="E66">A17</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> = 
the <a href="modelc_1.html#U7" title="MODELC_1:sel.7">And</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E6:63_4_2_1_1_1_2"><i><font color="Green" title="E69">A20</font></i></a>, <a class="txt" href="modelc_1.html#E11:63_4_2_1_1_1_2"><i><font color="Green" title="E71">A22</font></i></a>, <a class="txt" href="modelc_1.html#E12:63_4_2_1_1_1_2"><i><font color="Green" title="E72">A23</font></i></a>, <a class="txt" href="modelc_1.html#E1:63_4_2_1_1_1_2_5"><i><font color="Green" title="E78">A35</font></i></a>, <a class="ref" href="modelc_1.html#D27" target="_self" title="MODELC_1:def.27">Def27</a></i>
<br/>.= 
the <a href="modelc_1.html#U7" title="MODELC_1:sel.7">And</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E1:63_4_2"><i><font color="Green" title="E64">A15</font></i></a>, <a class="txt" href="modelc_1.html#E13:63_4_2_1_1_1_2"><i><font color="Green" title="E73">A24</font></i></a>, <a class="txt" href="modelc_1.html#E4:63_4_2_1_1_1_2_5"><i><font color="Green" title="E79">A36</font></i></a>, <a class="txt" href="modelc_1.html#E6:63_4_2_1_1_1_2_5"><i><font color="Green" title="E80">A37</font></i></a>, <a class="txt" href="modelc_1.html#E57"><i><font color="Green" title="E49">Lm27</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E8:63_4_2_1_1_1_2_5"/>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U7" title="MODELC_1:sel.7">And</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E18:63_4_2_1_1_1_2"/>
( <font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V7" title="MODELC_1:attr.7">ExistUntill</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U11" title="MODELC_1:sel.11">EUntill</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_4_2_1_1_1_2_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:63_4_2_1_1_1_2_6"/><i><font color="Green" title="E78">A38</font></i>: 
<font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V7" title="MODELC_1:attr.7">ExistUntill</a>
 ;<br/>

<a NAME="E2:63_4_2_1_1_1_2_6"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E48"><i><font color="Green" title="E41">Lm23</font></i></a></i>;<br/>
<a NAME="E3:63_4_2_1_1_1_2_6"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E1:63_4_2_1_1_1_2"><i><font color="Green" title="E66">A17</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E4:63_4_2_1_1_1_2_6"/><b>then </b><i><font color="Green" title="E79">A39</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E1:63_4_2"><i><font color="Green" title="E64">A15</font></i></a>, <a class="txt" href="modelc_1.html#E13:63_4_2_1_1_1_2"><i><font color="Green" title="E73">A24</font></i></a>, <a class="txt" href="modelc_1.html#E57"><i><font color="Green" title="E49">Lm27</font></i></a></i>;<br/>
<a NAME="E5:63_4_2_1_1_1_2_6"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">H</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E1:63_4_2_1_1_1_2_6"><i><font color="Green" title="E78">A38</font></i></a>, <a class="txt" href="modelc_1.html#E48"><i><font color="Green" title="E41">Lm23</font></i></a></i>;<br/>
<a NAME="E6:63_4_2_1_1_1_2_6"/><b>then </b><i><font color="Green" title="E80">A40</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E1:63_4_2_1_1_1_2"><i><font color="Green" title="E66">A17</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> = 
the <a href="modelc_1.html#U11" title="MODELC_1:sel.11">EUntill</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E6:63_4_2_1_1_1_2"><i><font color="Green" title="E69">A20</font></i></a>, <a class="txt" href="modelc_1.html#E11:63_4_2_1_1_1_2"><i><font color="Green" title="E71">A22</font></i></a>, <a class="txt" href="modelc_1.html#E12:63_4_2_1_1_1_2"><i><font color="Green" title="E72">A23</font></i></a>, <a class="txt" href="modelc_1.html#E1:63_4_2_1_1_1_2_6"><i><font color="Green" title="E78">A38</font></i></a>, <a class="ref" href="modelc_1.html#D27" target="_self" title="MODELC_1:def.27">Def27</a></i>
<br/>.= 
the <a href="modelc_1.html#U11" title="MODELC_1:sel.11">EUntill</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E1:63_4_2"><i><font color="Green" title="E64">A15</font></i></a>, <a class="txt" href="modelc_1.html#E13:63_4_2_1_1_1_2"><i><font color="Green" title="E73">A24</font></i></a>, <a class="txt" href="modelc_1.html#E4:63_4_2_1_1_1_2_6"><i><font color="Green" title="E79">A39</font></i></a>, <a class="txt" href="modelc_1.html#E6:63_4_2_1_1_1_2_6"><i><font color="Green" title="E80">A40</font></i></a>, <a class="txt" href="modelc_1.html#E57"><i><font color="Green" title="E49">Lm27</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E8:63_4_2_1_1_1_2_6"/>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U11" title="MODELC_1:sel.11">EUntill</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E19:63_4_2_1_1_1_2"/>
( ( <font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V2" title="MODELC_1:attr.2">atomic</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Kai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> ) &amp; ( <font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V3" title="MODELC_1:attr.3">negative</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U8" title="MODELC_1:sel.8">Not</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V4" title="MODELC_1:attr.4">conjunctive</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U7" title="MODELC_1:sel.7">And</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V5" title="MODELC_1:attr.5">ExistNext</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U9" title="MODELC_1:sel.9">EneXt</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V6" title="MODELC_1:attr.6">ExistGlobal</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U10" title="MODELC_1:sel.10">EGlobal</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V7" title="MODELC_1:attr.7">ExistUntill</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U11" title="MODELC_1:sel.11">EUntill</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="modelc_1.html#E6:63_4_2_1_1_1_2"><i><font color="Green" title="E69">A20</font></i></a>, <a class="txt" href="modelc_1.html#E11:63_4_2_1_1_1_2"><i><font color="Green" title="E71">A22</font></i></a>, <a class="txt" href="modelc_1.html#E12:63_4_2_1_1_1_2"><i><font color="Green" title="E72">A23</font></i></a>, <a class="txt" href="modelc_1.html#E14:63_4_2_1_1_1_2"><i><font color="Green" title="E74">A25</font></i></a>, <a class="txt" href="modelc_1.html#E15:63_4_2_1_1_1_2"><i><font color="Green" title="E75">A28</font></i></a>, <a class="txt" href="modelc_1.html#E16:63_4_2_1_1_1_2"><i><font color="Green" title="E76">A31</font></i></a>, <a class="txt" href="modelc_1.html#E17:63_4_2_1_1_1_2"><i><font color="Green" title="E77">A34</font></i></a>, <a class="ref" href="modelc_1.html#D27" target="_self" title="MODELC_1:def.27">Def27</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:63_4_2_1"/>
( ( <font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V2" title="MODELC_1:attr.2">atomic</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Kai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> ) &amp; ( <font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V3" title="MODELC_1:attr.3">negative</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U8" title="MODELC_1:sel.8">Not</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V4" title="MODELC_1:attr.4">conjunctive</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U7" title="MODELC_1:sel.7">And</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V5" title="MODELC_1:attr.5">ExistNext</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U9" title="MODELC_1:sel.9">EneXt</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V6" title="MODELC_1:attr.6">ExistGlobal</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U10" title="MODELC_1:sel.10">EGlobal</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c9">H</font> is <a href="modelc_1.html#V7" title="MODELC_1:attr.7">ExistUntill</a> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="modelc_1.html#U11" title="MODELC_1:sel.11">EUntill</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c9">H</font></span>)</span></span>)</span> ) )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:63_4_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="ref" href="modelc_1.html#D27" target="_self" title="MODELC_1:def.27">Def27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:63_4"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="modelc_1.html#E1:63_4"><i><font color="Green" title="E62">A13</font></i></a>, <a class="txt" href="modelc_1.html#E2:63_4"><i><font color="Green" title="E63">A14</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:63_4"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c7">f</font> <a href="modelc_1.html#R2" title="MODELC_1:pred.2">is-PreEvaluation-for</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">Kai</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:63"/>
<font color="Maroon" title="c7">f</font> <a href="modelc_1.html#R1" title="MODELC_1:pred.1">is-Evaluation-for</a> <font color="Maroon" title="c2">Kai</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E59"><i><font color="Green" title="E51">Lm29</font></i></a></i>;<br/>


</div>
