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<b>let </b><font color="Maroon" title="c1">H</font> be   <a href="modelc_1.html#NM1" title="MODELC_1:NM.1">CTL-formula</a>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1"/>
( <font color="Maroon" title="c1">H</font> is <a href="modelc_1.html#V2" title="MODELC_1:attr.2">atomic</a> or <font color="Maroon" title="c1">H</font> is <a href="modelc_1.html#V3" title="MODELC_1:attr.3">negative</a> or <font color="Maroon" title="c1">H</font> is <a href="modelc_1.html#V4" title="MODELC_1:attr.4">conjunctive</a> or <font color="Maroon" title="c1">H</font> is <a href="modelc_1.html#V5" title="MODELC_1:attr.5">ExistNext</a> or <font color="Maroon" title="c1">H</font> is <a href="modelc_1.html#V6" title="MODELC_1:attr.6">ExistGlobal</a> or <font color="Maroon" title="c1">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="29_1_1"><b>suppose </b></a><a NAME="E1:29_1_1"/>
<font color="Maroon" title="c1">H</font> is <a href="modelc_1.html#V2" title="MODELC_1:attr.2">atomic</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c2">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="E3:29_1_1"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K7" title="MODELC_1:func.7">atom.</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="modelc_1.html#D14" target="_self" title="MODELC_1:def.14">Def14</a></i>;<br/><b>thus </b><a NAME="E4:29_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:29_1_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2"><b>suppose </b></a><a NAME="E1:29_1_2"/>
<font color="Maroon" title="c1">H</font> is <a href="modelc_1.html#V3" title="MODELC_1:attr.3">negative</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c2">H1</font> being   <a href="modelc_1.html#NM1" title="MODELC_1:NM.1">CTL-formula</a><b> such that </b><br/><a NAME="E3:29_1_2"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K8" title="MODELC_1:func.8">'not'</a> <font color="Maroon" title="c2">H1</font>
 <b>by </b><i><a class="ref" href="modelc_1.html#D15" target="_self" title="MODELC_1:def.15">Def15</a></i>;<br/><a NAME="E4:29_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">H1</font></span>)</span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:29_1_2"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="finseq_5.html#T8" title="FINSEQ_5:th.8">FINSEQ_5:8</a></i>;<br/><b>hence </b><a NAME="E5:29_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_3"><b>suppose </b></a><a NAME="E1:29_1_3"/>
<font color="Maroon" title="c1">H</font> is <a href="modelc_1.html#V4" title="MODELC_1:attr.4">conjunctive</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">G</font> being   <a href="modelc_1.html#NM1" title="MODELC_1:NM.1">CTL-formula</a><b> such that </b><br/><a NAME="E3:29_1_3"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="modelc_1.html#K9" title="MODELC_1:func.9">'&amp;'</a> <font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="ref" href="modelc_1.html#D16" target="_self" title="MODELC_1:def.16">Def16</a></i>;<br/><a NAME="E4:29_1_3"/><i><font color="Green" title="E27">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:29_1_3"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="modelc_1.html#E13"><i><font color="Green" title="E7">Lm2</font></i></a></i>;<br/><a NAME="E5:29_1_3"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font></span>)</span> &amp; 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G</font></span>)</span> )
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><b>hence </b><a NAME="E6:29_1_3"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E4:29_1_3"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_4"><b>suppose </b></a><a NAME="E1:29_1_4"/>
<font color="Maroon" title="c1">H</font> is <a href="modelc_1.html#V5" title="MODELC_1:attr.5">ExistNext</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c2">H1</font> being   <a href="modelc_1.html#NM1" title="MODELC_1:NM.1">CTL-formula</a><b> such that </b><br/><a NAME="E3:29_1_4"/><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K10" title="MODELC_1:func.10">EX</a> <font color="Maroon" title="c2">H1</font>
 <b>by </b><i><a class="ref" href="modelc_1.html#D17" target="_self" title="MODELC_1:def.17">Def17</a></i>;<br/><a NAME="E4:29_1_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">H1</font></span>)</span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:29_1_4"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="finseq_5.html#T8" title="FINSEQ_5:th.8">FINSEQ_5:8</a></i>;<br/><b>hence </b><a NAME="E5:29_1_4"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_5"><b>suppose </b></a><a NAME="E1:29_1_5"/>
<font color="Maroon" title="c1">H</font> is <a href="modelc_1.html#V6" title="MODELC_1:attr.6">ExistGlobal</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c2">H1</font> being   <a href="modelc_1.html#NM1" title="MODELC_1:NM.1">CTL-formula</a><b> such that </b><br/><a NAME="E3:29_1_5"/><i><font color="Green" title="E26">A6</font></i>: 
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K11" title="MODELC_1:func.11">EG</a> <font color="Maroon" title="c2">H1</font>
 <b>by </b><i><a class="ref" href="modelc_1.html#D18" target="_self" title="MODELC_1:def.18">Def18</a></i>;<br/><a NAME="E4:29_1_5"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">H1</font></span>)</span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:29_1_5"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="finseq_5.html#T8" title="FINSEQ_5:th.8">FINSEQ_5:8</a></i>;<br/><b>hence </b><a NAME="E5:29_1_5"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_6"><b>suppose </b></a><a NAME="E1:29_1_6"/>
<font color="Maroon" title="c1">H</font> is <a href="modelc_1.html#V7" title="MODELC_1:attr.7">ExistUntill</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">G</font> being   <a href="modelc_1.html#NM1" title="MODELC_1:NM.1">CTL-formula</a><b> such that </b><br/><a NAME="E3:29_1_6"/><i><font color="Green" title="E26">A7</font></i>: 
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="modelc_1.html#K12" title="MODELC_1:func.12">EU</a> <font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="ref" href="modelc_1.html#D19" target="_self" title="MODELC_1:def.19">Def19</a></i>;<br/><a NAME="E4:29_1_6"/><i><font color="Green" title="E27">A8</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:29_1_6"><i><font color="Green" title="E26">A7</font></i></a>, <a class="txt" href="modelc_1.html#E13"><i><font color="Green" title="E7">Lm2</font></i></a></i>;<br/><a NAME="E5:29_1_6"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font></span>)</span> &amp; 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G</font></span>)</span> )
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><b>hence </b><a NAME="E6:29_1_6"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E4:29_1_6"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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