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

<b>let </b><font color="Maroon" title="c1">n</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>let </b><font color="Maroon" title="c2">V</font> be    <a href="modelc_1.html#L2" title="MODELC_1:struct.2">CTLModelStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">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="c2">V</font>;<br/><b>let </b><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">f2</font> be   <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="c2">V</font>;<br/>









<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="c4">f1</font> <a href="modelc_1.html#R2" title="MODELC_1:pred.2">is-PreEvaluation-for</a> $1,<font color="Maroon" title="c3">Kai</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="modelc_1.html#R2" title="MODELC_1:pred.2">is-PreEvaluation-for</a> $1,<font color="Maroon" title="c3">Kai</font> implies 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> $1 holds <br/><font color="Maroon" title="c4">f1</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="c5">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> );<br/>
<a NAME="E1:56"/><i><font color="Green" title="E50">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="modelc_1.html#E32"><i><font color="Green" title="E25">Lm10</font></i></a></i>;<br/>
<a NAME="E2:56"/><i><font color="Green" title="E51">A2</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#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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="E1:56_1"/><i><font color="Green" title="E52">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:56_1"/><i><font color="Green" title="E53">A4</font></i>: 
<font color="Maroon" title="c4">f1</font> <a href="modelc_1.html#R2" title="MODELC_1:pred.2">is-PreEvaluation-for</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c3">Kai</font>
 <b>and </b><br/><a NAME="E3:56_1"/><i><font color="Green" title="E54">A5</font></i>: 
<font color="Maroon" title="c5">f2</font> <a href="modelc_1.html#R2" title="MODELC_1:pred.2">is-PreEvaluation-for</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c3">Kai</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c7">H</font> be   <a href="modelc_1.html#NM1" title="MODELC_1:NM.1">CTL-formula</a>;<br/>

<b>assume </b><a NAME="E4:56_1"/><i><font color="Green" title="E55">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1_1"/>
( <font color="Maroon" title="c7">H</font> is <a href="modelc_1.html#V2" title="MODELC_1:attr.2">atomic</a> or <font color="Maroon" title="c7">H</font> is <a href="modelc_1.html#V3" title="MODELC_1:attr.3">negative</a> or <font color="Maroon" title="c7">H</font> is <a href="modelc_1.html#V5" title="MODELC_1:attr.5">ExistNext</a> or <font color="Maroon" title="c7">H</font> is <a href="modelc_1.html#V6" title="MODELC_1:attr.6">ExistGlobal</a> or <font color="Maroon" title="c7">H</font> is <a href="modelc_1.html#V4" title="MODELC_1:attr.4">conjunctive</a> or <font color="Maroon" title="c7">H</font> is <a href="modelc_1.html#V7" title="MODELC_1:attr.7">ExistUntill</a> )
 </span><b>by </b><i><a class="ref" href="modelc_1.html#T2" target="_self" title="MODELC_1:th.2">Th2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1"><b>suppose </b></a><a NAME="E1:56_1_1_1"/><i><font color="Green" title="E56">A7</font></i>: 
<font color="Maroon" title="c7">H</font> is <a href="modelc_1.html#V2" title="MODELC_1:attr.2">atomic</a>
 ;<br/><div class="add"><a NAME="E2:56_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Kai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">H</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E4:56_1"><i><font color="Green" title="E55">A6</font></i></a>, <a class="txt" href="modelc_1.html#E2:56_1"><i><font color="Green" title="E53">A4</font></i></a>, <a class="ref" href="modelc_1.html#D27" target="_self" title="MODELC_1:def.27">Def27</a></i>;<br/><b>hence </b><a NAME="E3:56_1_1_1"/>
<font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">H</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E4:56_1"><i><font color="Green" title="E55">A6</font></i></a>, <a class="txt" href="modelc_1.html#E1:56_1_1_1"><i><font color="Green" title="E56">A7</font></i></a>, <a class="txt" href="modelc_1.html#E3:56_1"><i><font color="Green" title="E54">A5</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="56_1_1_2"><b>suppose </b></a><a NAME="E1:56_1_1_2"/><i><font color="Green" title="E56">A8</font></i>: 
<font color="Maroon" title="c7">H</font> is <a href="modelc_1.html#V3" title="MODELC_1:attr.3">negative</a>
 ;<br/><div class="add"><a NAME="E2:56_1_1_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="c7">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="c7">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="E3:56_1_1_2"/><b>then </b><i><font color="Green" title="E57">A9</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="c7">H</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="txt" href="modelc_1.html#E4:56_1"><i><font color="Green" title="E55">A6</font></i></a></i>;<br/><font color="Maroon" title="c5">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">H</font> = 
the <a href="modelc_1.html#U8" title="MODELC_1:sel.8">Not</a> of <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c7">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E1:56_1_1_2"><i><font color="Green" title="E56">A8</font></i></a>, <a class="txt" href="modelc_1.html#E4:56_1"><i><font color="Green" title="E55">A6</font></i></a>, <a class="txt" href="modelc_1.html#E3:56_1"><i><font color="Green" title="E54">A5</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="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</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="c7">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E1:56_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="modelc_1.html#E2:56_1"><i><font color="Green" title="E53">A4</font></i></a>, <a class="txt" href="modelc_1.html#E3:56_1"><i><font color="Green" title="E54">A5</font></i></a>, <a class="txt" href="modelc_1.html#E55"><i><font color="Green" title="E47">Lm25</font></i></a>, <a class="txt" href="modelc_1.html#E3:56_1_1_2"><i><font color="Green" title="E57">A9</font></i></a></i>
;<br/><b>hence </b><a NAME="E5:56_1_1_2"/>
<font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">H</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:56_1_1_2"><i><font color="Green" title="E56">A8</font></i></a>, <a class="txt" href="modelc_1.html#E4:56_1"><i><font color="Green" title="E55">A6</font></i></a>, <a class="txt" href="modelc_1.html#E2:56_1"><i><font color="Green" title="E53">A4</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="56_1_1_3"><b>suppose </b></a><a NAME="E1:56_1_1_3"/><i><font color="Green" title="E56">A10</font></i>: 
<font color="Maroon" title="c7">H</font> is <a href="modelc_1.html#V5" title="MODELC_1:attr.5">ExistNext</a>
 ;<br/><div class="add"><a NAME="E2:56_1_1_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="c7">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="c7">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="E3:56_1_1_3"/><b>then </b><i><font color="Green" title="E57">A11</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="c7">H</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="txt" href="modelc_1.html#E4:56_1"><i><font color="Green" title="E55">A6</font></i></a></i>;<br/><font color="Maroon" title="c5">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">H</font> = 
the <a href="modelc_1.html#U9" title="MODELC_1:sel.9">EneXt</a> of <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c7">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E1:56_1_1_3"><i><font color="Green" title="E56">A10</font></i></a>, <a class="txt" href="modelc_1.html#E4:56_1"><i><font color="Green" title="E55">A6</font></i></a>, <a class="txt" href="modelc_1.html#E3:56_1"><i><font color="Green" title="E54">A5</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="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</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="c7">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E3:56_1_1_3"><i><font color="Green" title="E57">A11</font></i></a>, <a class="txt" href="modelc_1.html#E1:56_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="modelc_1.html#E2:56_1"><i><font color="Green" title="E53">A4</font></i></a>, <a class="txt" href="modelc_1.html#E55"><i><font color="Green" title="E47">Lm25</font></i></a>, <a class="txt" href="modelc_1.html#E3:56_1"><i><font color="Green" title="E54">A5</font></i></a></i>
;<br/><b>hence </b><a NAME="E5:56_1_1_3"/>
<font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">H</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:56_1_1_3"><i><font color="Green" title="E56">A10</font></i></a>, <a class="txt" href="modelc_1.html#E4:56_1"><i><font color="Green" title="E55">A6</font></i></a>, <a class="txt" href="modelc_1.html#E2:56_1"><i><font color="Green" title="E53">A4</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="56_1_1_4"><b>suppose </b></a><a NAME="E1:56_1_1_4"/><i><font color="Green" title="E56">A12</font></i>: 
<font color="Maroon" title="c7">H</font> is <a href="modelc_1.html#V6" title="MODELC_1:attr.6">ExistGlobal</a>
 ;<br/><div class="add"><a NAME="E2:56_1_1_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="c7">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="c7">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="E3:56_1_1_4"/><b>then </b><i><font color="Green" title="E57">A13</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="c7">H</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="txt" href="modelc_1.html#E4:56_1"><i><font color="Green" title="E55">A6</font></i></a></i>;<br/><font color="Maroon" title="c5">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">H</font> = 
the <a href="modelc_1.html#U10" title="MODELC_1:sel.10">EGlobal</a> of <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c7">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E1:56_1_1_4"><i><font color="Green" title="E56">A12</font></i></a>, <a class="txt" href="modelc_1.html#E4:56_1"><i><font color="Green" title="E55">A6</font></i></a>, <a class="txt" href="modelc_1.html#E3:56_1"><i><font color="Green" title="E54">A5</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="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</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="c7">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E3:56_1_1_4"><i><font color="Green" title="E57">A13</font></i></a>, <a class="txt" href="modelc_1.html#E1:56_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="modelc_1.html#E2:56_1"><i><font color="Green" title="E53">A4</font></i></a>, <a class="txt" href="modelc_1.html#E55"><i><font color="Green" title="E47">Lm25</font></i></a>, <a class="txt" href="modelc_1.html#E3:56_1"><i><font color="Green" title="E54">A5</font></i></a></i>
;<br/><b>hence </b><a NAME="E5:56_1_1_4"/>
<font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">H</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:56_1_1_4"><i><font color="Green" title="E56">A12</font></i></a>, <a class="txt" href="modelc_1.html#E4:56_1"><i><font color="Green" title="E55">A6</font></i></a>, <a class="txt" href="modelc_1.html#E2:56_1"><i><font color="Green" title="E53">A4</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="56_1_1_5"><b>suppose </b></a><a NAME="E1:56_1_1_5"/><i><font color="Green" title="E56">A14</font></i>: 
<font color="Maroon" title="c7">H</font> is <a href="modelc_1.html#V4" title="MODELC_1:attr.4">conjunctive</a>
 ;<br/><div class="add"><a NAME="E2:56_1_1_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="c7">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="c7">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="E3:56_1_1_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="c7">H</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="txt" href="modelc_1.html#E4:56_1"><i><font color="Green" title="E55">A6</font></i></a></i>;<br/><a NAME="E4:56_1_1_5"/><b>then </b><i><font color="Green" title="E57">A15</font></i>: 
<font color="Maroon" title="c4">f1</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="c7">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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="c7">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:56_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="modelc_1.html#E2:56_1"><i><font color="Green" title="E53">A4</font></i></a>, <a class="txt" href="modelc_1.html#E55"><i><font color="Green" title="E47">Lm25</font></i></a>, <a class="txt" href="modelc_1.html#E3:56_1"><i><font color="Green" title="E54">A5</font></i></a></i>;<br/><a NAME="E5:56_1_1_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="c7">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="c7">H</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:56_1_1_5"><i><font color="Green" title="E56">A14</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="E6:56_1_1_5"/><b>then </b><i><font color="Green" title="E58">A16</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="c7">H</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="txt" href="modelc_1.html#E4:56_1"><i><font color="Green" title="E55">A6</font></i></a></i>;<br/><font color="Maroon" title="c5">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">H</font> = 
the <a href="modelc_1.html#U7" title="MODELC_1:sel.7">And</a> of <font color="Maroon" title="c2">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c7">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c7">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E1:56_1_1_5"><i><font color="Green" title="E56">A14</font></i></a>, <a class="txt" href="modelc_1.html#E4:56_1"><i><font color="Green" title="E55">A6</font></i></a>, <a class="txt" href="modelc_1.html#E3:56_1"><i><font color="Green" title="E54">A5</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="c2">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</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="c7">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</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="c7">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E4:56_1_1_5"><i><font color="Green" title="E57">A15</font></i></a>, <a class="txt" href="modelc_1.html#E6:56_1_1_5"><i><font color="Green" title="E58">A16</font></i></a>, <a class="txt" href="modelc_1.html#E1:56_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="modelc_1.html#E2:56_1"><i><font color="Green" title="E53">A4</font></i></a>, <a class="txt" href="modelc_1.html#E55"><i><font color="Green" title="E47">Lm25</font></i></a>, <a class="txt" href="modelc_1.html#E3:56_1"><i><font color="Green" title="E54">A5</font></i></a></i>
;<br/><b>hence </b><a NAME="E8:56_1_1_5"/>
<font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">H</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:56_1_1_5"><i><font color="Green" title="E56">A14</font></i></a>, <a class="txt" href="modelc_1.html#E4:56_1"><i><font color="Green" title="E55">A6</font></i></a>, <a class="txt" href="modelc_1.html#E2:56_1"><i><font color="Green" title="E53">A4</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="56_1_1_6"><b>suppose </b></a><a NAME="E1:56_1_1_6"/><i><font color="Green" title="E56">A17</font></i>: 
<font color="Maroon" title="c7">H</font> is <a href="modelc_1.html#V7" title="MODELC_1:attr.7">ExistUntill</a>
 ;<br/><div class="add"><a NAME="E2:56_1_1_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="c7">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="c7">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="E3:56_1_1_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="c7">H</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="txt" href="modelc_1.html#E4:56_1"><i><font color="Green" title="E55">A6</font></i></a></i>;<br/><a NAME="E4:56_1_1_6"/><b>then </b><i><font color="Green" title="E57">A18</font></i>: 
<font color="Maroon" title="c4">f1</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="c7">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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="c7">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:56_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="modelc_1.html#E2:56_1"><i><font color="Green" title="E53">A4</font></i></a>, <a class="txt" href="modelc_1.html#E55"><i><font color="Green" title="E47">Lm25</font></i></a>, <a class="txt" href="modelc_1.html#E3:56_1"><i><font color="Green" title="E54">A5</font></i></a></i>;<br/><a NAME="E5:56_1_1_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="c7">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="c7">H</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:56_1_1_6"><i><font color="Green" title="E56">A17</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="E6:56_1_1_6"/><b>then </b><i><font color="Green" title="E58">A19</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="c7">H</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="txt" href="modelc_1.html#E4:56_1"><i><font color="Green" title="E55">A6</font></i></a></i>;<br/><font color="Maroon" title="c5">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">H</font> = 
the <a href="modelc_1.html#U11" title="MODELC_1:sel.11">EUntill</a> of <font color="Maroon" title="c2">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c7">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c7">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E1:56_1_1_6"><i><font color="Green" title="E56">A17</font></i></a>, <a class="txt" href="modelc_1.html#E4:56_1"><i><font color="Green" title="E55">A6</font></i></a>, <a class="txt" href="modelc_1.html#E3:56_1"><i><font color="Green" title="E54">A5</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="c2">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</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="c7">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</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="c7">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E4:56_1_1_6"><i><font color="Green" title="E57">A18</font></i></a>, <a class="txt" href="modelc_1.html#E6:56_1_1_6"><i><font color="Green" title="E58">A19</font></i></a>, <a class="txt" href="modelc_1.html#E1:56_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="modelc_1.html#E2:56_1"><i><font color="Green" title="E53">A4</font></i></a>, <a class="txt" href="modelc_1.html#E55"><i><font color="Green" title="E47">Lm25</font></i></a>, <a class="txt" href="modelc_1.html#E3:56_1"><i><font color="Green" title="E54">A5</font></i></a></i>
;<br/><b>hence </b><a NAME="E8:56_1_1_6"/>
<font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">H</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:56_1_1_6"><i><font color="Green" title="E56">A17</font></i></a>, <a class="txt" href="modelc_1.html#E4:56_1"><i><font color="Green" title="E55">A6</font></i></a>, <a class="txt" href="modelc_1.html#E2:56_1"><i><font color="Green" title="E53">A4</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>
<a NAME="E3:56"/>
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#E1:56"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="modelc_1.html#E2:56"><i><font color="Green" title="E51">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:56"/>
( <font color="Maroon" title="c4">f1</font> <a href="modelc_1.html#R2" title="MODELC_1:pred.2">is-PreEvaluation-for</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">Kai</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="modelc_1.html#R2" title="MODELC_1:pred.2">is-PreEvaluation-for</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">Kai</font> implies 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="c1">n</font> holds <br/><font color="Maroon" title="c4">f1</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="c5">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> )
 ;<br/>


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