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

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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">f</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>assume </b><a NAME="E1:54"/><i><font color="Green" title="E48">A1</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="modelc_1.html#R2" title="MODELC_1:pred.2">is-PreEvaluation-for</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1,<font color="Maroon" title="c3">Kai</font>
 ;<br/>

<a NAME="E2:54"/>
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="Olive" title="b1">H</font> is <a href="modelc_1.html#V2" title="MODELC_1:attr.2">atomic</a> implies <font color="Maroon" title="c4">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="c3">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="c4">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="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">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="c4">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="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">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="c4">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="c4">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="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">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="c4">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="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">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="c4">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="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">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="c4">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="54_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">H</font> be   <a href="modelc_1.html#NM1" title="MODELC_1:NM.1">CTL-formula</a>;<br/>

<b>assume </b><a NAME="E1:54_1"/><i><font color="Green" title="E49">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<a NAME="E2:54_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">H</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="txt" href="modelc_1.html#E1:54_1"><i><font color="Green" title="E49">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:54_1"/>
( ( <font color="Maroon" title="c5">H</font> is <a href="modelc_1.html#V2" title="MODELC_1:attr.2">atomic</a> implies <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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="c5">H</font> ) &amp; ( <font color="Maroon" title="c5">H</font> is <a href="modelc_1.html#V3" title="MODELC_1:attr.3">negative</a> implies <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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="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">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="c5">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c5">H</font> is <a href="modelc_1.html#V4" title="MODELC_1:attr.4">conjunctive</a> implies <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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="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">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="c5">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">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="c5">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c5">H</font> is <a href="modelc_1.html#V5" title="MODELC_1:attr.5">ExistNext</a> implies <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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="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">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="c5">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c5">H</font> is <a href="modelc_1.html#V6" title="MODELC_1:attr.6">ExistGlobal</a> implies <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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="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">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="c5">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c5">H</font> is <a href="modelc_1.html#V7" title="MODELC_1:attr.7">ExistUntill</a> implies <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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="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">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="c5">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">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="c5">H</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:54"><i><font color="Green" title="E48">A1</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>
<b>hence </b><a NAME="E3:54"/>
<font color="Maroon" title="c4">f</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>
 <b>by </b><i><a class="ref" href="modelc_1.html#D27" target="_self" title="MODELC_1:def.27">Def27</a></i>;<br/>


</div>
