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

<span class="kw">let </span><font color="Maroon" title="c1">w</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v</font> being  <a href="modelc_3.html#V5" title="MODELC_3:attr.5">neg-inner-most</a>  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a>  st <font color="Maroon" title="c1">w</font> <a href="modelc_3.html#R9" title="MODELC_3:pred.9">is-accepted-by</a>  <a href="modelc_3.html#K34" title="MODELC_3:func.34">BAutomaton</a> <font color="Olive" title="b1">v</font> holds <br/><font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Olive" title="b1">v</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">v</font> be   <a href="modelc_3.html#V5" title="MODELC_3:attr.5">neg-inner-most</a>  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">w</font> <a href="modelc_3.html#R9" title="MODELC_3:pred.9">is-accepted-by</a>  <a href="modelc_3.html#K34" title="MODELC_3:func.34">BAutomaton</a> <font color="Maroon" title="c2">v</font> implies <font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">v</font> )</span><br/>



<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c1">w</font>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<span class="kw">assume </span><a NAME="E1:147"/>
<font color="Maroon" title="c1">w</font> <a href="modelc_3.html#R9" title="MODELC_3:pred.9">is-accepted-by</a>  <a href="modelc_3.html#K34" title="MODELC_3:func.34">BAutomaton</a> <font color="Maroon" title="c2">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">v</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">run</font> being   <a href="nat_1.html#NM2" title="NAT_1:NM.2">sequence</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:147"/><span class="lab"><font color="Green" title="E117">A1</font></span>: 
<font color="Maroon" title="c3">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K31" title="MODELC_3:func.31">InitS_LTL</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">and </span><br/><a NAME="E4:147"/><span class="lab"><font color="Green" title="E118">A2</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K30" title="MODELC_3:func.30">Tran_LTL</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">and </span><br/><a NAME="E5:147"/><span class="lab"><font color="Green" title="E119">A3</font></span>: 
for <font color="Olive" title="b1">FSet</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">FSet</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K33" title="MODELC_3:func.33">FinalS_LTL</a> <font color="Maroon" title="c2">v</font> holds <br/><span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon" title="c3">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">FSet</font> </span>}</span>  is   <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#D39" target="_self" title="MODELC_3:def.39">Def39</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>  <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font> =  <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>,<font color="Maroon" title="c2">v</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">Run01</font> = <font color="Maroon">H<sub>2</sub></font>(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1);<br/>
<span class="kw">set </span><font color="Maroon" title="c5">Run00</font> = <font color="Maroon">H<sub>2</sub></font>( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> );<br/>
<a NAME="E6:147"/><span class="lab"><font color="Green" title="E120">A4</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font>) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K29" title="MODELC_3:func.29">Label_</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">i</font>) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">i</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K29" title="MODELC_3:func.29">Label_</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">z</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">i</font>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E1:147_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">i</font>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K30" title="MODELC_3:func.30">Tran_LTL</a> <font color="Maroon" title="c2">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:147"><span class="lab"><font color="Green" title="E118">A2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font></span>)</span>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> ,<span class="p2">(<span class="default"><a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E3:147_1"/><span class="lab"><font color="Green" title="E121">A5</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">i</font>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 <span class="kw">and </span><br/><a NAME="E4:147_1"/><span class="lab"><font color="Green" title="E122">A6</font></span>: 
 ex <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">s1</font> being  <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font> ex <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">s</font>,<font color="Olive" title="b3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b2">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">s1</font> <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K29" title="MODELC_3:func.29">Label_</a> <font color="Olive" title="b2">s1</font> )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">s</font>, <font color="Maroon" title="c10">s1</font> being   <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font>, <font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:147_1"/><span class="lab"><font color="Green" title="E123">A7</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">s</font>,<font color="Maroon" title="c11">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c10">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:147_1"/><span class="lab"><font color="Green" title="E124">A8</font></span>: 
( <font color="Maroon" title="c10">s1</font> <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon" title="c9">s</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K29" title="MODELC_3:func.29">Label_</a> <font color="Maroon" title="c10">s1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:147_1"><span class="lab"><font color="Green" title="E122">A6</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E125">A9</font></span>: <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) = 
 <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <font color="Maroon" title="c10">s1</font>,<font color="Maroon" title="c2">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:147_1"><span class="lab"><font color="Green" title="E121">A5</font></span></a>, <a class="txt" href="#E6:147_1"><span class="lab"><font color="Green" title="E123">A7</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>
<br/>.= 
<font color="Maroon" title="c10">s1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#D16" target="_self" title="MODELC_3:def.16">Def16</a></span>
;<br/>
<a NAME="E9:147_1"/><span class="lab"><font color="Green" title="E126">A10</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">s</font>,<font color="Maroon" title="c11">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">i</font>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:147_1"><span class="lab"><font color="Green" title="E121">A5</font></span></a>, <a class="txt" href="#E6:147_1"><span class="lab"><font color="Green" title="E123">A7</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<span class="kw">then </span><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">i</font>) = 
 <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <font color="Maroon" title="c9">s</font>,<font color="Maroon" title="c2">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>
<br/>.= 
<font color="Maroon" title="c9">s</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#D16" target="_self" title="MODELC_3:def.16">Def16</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E11:147_1"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">i</font>) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">i</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K29" title="MODELC_3:func.29">Label_</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:147_1"><span class="lab"><font color="Green" title="E124">A8</font></span></a>, <a class="txt" href="#E9:147_1"><span class="lab"><font color="Green" title="E126">A10</font></span></a>, <a class="txt" href="#E8:147_1"><span class="lab"><font color="Green" title="E125">A9</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:147"/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">A11</font></span>: 
<font color="Maroon">H<sub>2</sub></font>(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon">H<sub>2</sub></font>( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> )
 
;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">F</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a>  st <font color="Olive" title="b2">F</font> <a href="modelc_2.html#R2" title="MODELC_2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">v</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; <font color="Olive" title="b2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Olive" title="b1">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Olive" title="b2">F</font>;<br/>
<a NAME="E8:147"/><span class="lab"><font color="Green" title="E122">A12</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">i</font> = <font color="Maroon" title="c6">i</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E2:147_2"/>
<font color="Maroon" title="c3">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font>
 
;<br/>
<a NAME="E3:147_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="modelc_3.html#K22" title="MODELC_3:func.22">LTLNodes</a> <font color="Maroon" title="c2">v</font> st <br/>( <font color="Maroon" title="c3">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> is   <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E4:147_2"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#D16" target="_self" title="MODELC_3:def.16">Def16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:147"/><span class="lab"><font color="Green" title="E123">A13</font></span>: 
for <font color="Olive" title="b1">FSet</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">FSet</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K33" title="MODELC_3:func.33">FinalS_LTL</a> <font color="Maroon" title="c2">v</font> holds <br/> not <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">FSet</font> </span>}</span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">FSet</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">FSet</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K33" title="MODELC_3:func.33">FinalS_LTL</a> <font color="Maroon" title="c2">v</font> implies  not <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">FSet</font> </span>}</span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon" title="c3">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">FSet</font> </span>}</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">Y</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">FSet</font> </span>}</span> ;<br/>
<a NAME="E1:147_3"/><span class="lab"><font color="Green" title="E124">A14</font></span>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon" title="c3">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">FSet</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">FSet</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon" title="c3">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">FSet</font> </span>}</span>  or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">FSet</font> </span>}</span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:147_3_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon" title="c3">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">FSet</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">FSet</font> </span>}</span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">k</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E3:147_3_1"/><span class="lab"><font color="Green" title="E125">A15</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">and </span><br/><a NAME="E4:147_3_1"/><span class="lab"><font color="Green" title="E126">A16</font></span>: 
<font color="Maroon" title="c3">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">FSet</font>
 ;<br/>
<a NAME="E5:147_3_1"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c10">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">FSet</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:147"><span class="lab"><font color="Green" title="E122">A12</font></span></a>, <a class="txt" href="#E4:147_3_1"><span class="lab"><font color="Green" title="E126">A16</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:147_3_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">FSet</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:147_3_1"><span class="lab"><font color="Green" title="E125">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E2:147_3"/>
<font color="Maroon" title="c6">FSet</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K33" title="MODELC_3:func.33">FinalS_LTL</a> <font color="Maroon" title="c2">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">FSet</font> </span>}</span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/>

<span class="kw">hence </span><a NAME="E3:147_3"/>
not <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">FSet</font> </span>}</span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:147"><span class="lab"><font color="Green" title="E119">A3</font></span></a>, <a class="txt" href="#E1:147_3"><span class="lab"><font color="Green" title="E124">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:147"/><span class="lab"><font color="Green" title="E124">A17</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:147_4"/><span class="lab"><font color="Green" title="E125">A18</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:147_4"/><span class="lab"><font color="Green" title="E126">A19</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">F</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a>  st <font color="Olive" title="b2">F</font> <a href="modelc_2.html#R2" title="MODELC_2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">v</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Olive" title="b2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Olive" title="b1">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Olive" title="b2">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a>  st <font color="Olive" title="b1">F</font> <a href="modelc_2.html#R2" title="MODELC_2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">v</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Olive" title="b1">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Olive" title="b1">F</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">F</font> be   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">F</font> <a href="modelc_2.html#R2" title="MODELC_2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">v</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) implies  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:147_4_1"/><span class="lab"><font color="Green" title="E127">A20</font></span>: 
<font color="Maroon" title="c8">F</font> <a href="modelc_2.html#R2" title="MODELC_2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">and </span><br/><a NAME="E2:147_4_1"/><span class="lab"><font color="Green" title="E128">A21</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">and </span><br/><a NAME="E3:147_4_1"/><span class="lab"><font color="Green" title="E129">A22</font></span>: 
<font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c9">zeta</font> =  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="147_4_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:147_4_1_1_1"/>
( <font color="Maroon" title="c8">F</font> is  <a href="modelc_3.html#V6" title="MODELC_3:attr.6">Sub_atomic</a>  or <font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V4" title="MODELC_2:attr.4">conjunctive</a>  or <font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V5" title="MODELC_2:attr.5">disjunctive</a>  or <font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V6" title="MODELC_2:attr.6">next</a>  or <font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V7" title="MODELC_2:attr.7">Until</a>  or <font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V8" title="MODELC_2:attr.8">Release</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1"><span class="lab"><font color="Green" title="E127">A20</font></span></a>, <a class="ref" href="modelc_3.html#T63" target="_self" title="MODELC_3:th.63">Th63</a>, <a class="ref" href="modelc_3.html#T65" target="_self" title="MODELC_3:th.65">Th65</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="147_4_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:147_4_1_1_1_1"/><span class="lab"><font color="Green" title="E130">A23</font></span>: 
<font color="Maroon" title="c8">F</font> is  <a href="modelc_3.html#V6" title="MODELC_3:attr.6">Sub_atomic</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c10">Gi9</font> =  <a href="modelc_2.html#K30" title="MODELC_2:func.30">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c1">w</font>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span>,<font color="Maroon" title="c7">i</font>;<br/><span class="kw">set </span><font color="Maroon" title="c11">Gi</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">i</font>);<br/><a NAME="E2:147_4_1_1_1_1"/>
 <a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="modelc_2.html#K30" title="MODELC_2:func.30">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c1">w</font>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span>,<font color="Maroon" title="c7">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T81" title="MODELC_2:th.81">MODELC_2:81</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A24</font></span>: <span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c1">w</font>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#D43" title="MODELC_2:def.43">MODELC_2:def 43</a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">i</font>)

;<br/><a NAME="E4:147_4_1_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">i</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K29" title="MODELC_3:func.29">Label_</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147"><span class="lab"><font color="Green" title="E120">A4</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">X</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="modelc_2.html#K15" title="MODELC_2:func.15">atomic_LTL</a> <span class="kw"> such that </span><br/><a NAME="E6:147_4_1_1_1_1"/><span class="lab"><font color="Green" title="E132">A25</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">X</font>
 <span class="kw">and </span><br/><a NAME="E7:147_4_1_1_1_1"/><span class="lab"><font color="Green" title="E133">A26</font></span>: 
 <a href="modelc_3.html#K27" title="MODELC_3:func.27">atomic_LTL</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">X</font>
 <span class="kw">and </span><br/><a NAME="E8:147_4_1_1_1_1"/><span class="lab"><font color="Green" title="E134">A27</font></span>: 
 <a href="modelc_3.html#K28" title="MODELC_3:func.28">Neg_atomic_LTL</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c12">X</font>
 ;<br/><a NAME="E9:147_4_1_1_1_1"/><span class="lab"><font color="Green" title="E135">A28</font></span>: 
<span class="p1">(<span class="default"><a href="modelc_3.html#K28" title="MODELC_3:func.28">Neg_atomic_LTL</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:147_4_1_1_1_1"><span class="lab"><font color="Green" title="E134">A27</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1_1_1_1_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="147_4_1_1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:147_4_1_1_1_1_2_1"/>
( <font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V2" title="MODELC_2:attr.2">atomic</a>  or ( <font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V3" title="MODELC_2:attr.3">negative</a>  &amp;  <a href="modelc_2.html#K10" title="MODELC_2:func.10">the_argument_of</a> <font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V2" title="MODELC_2:attr.2">atomic</a>  ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1_1_1_1"><span class="lab"><font color="Green" title="E130">A23</font></span></a>, <a class="ref" href="modelc_3.html#T64" target="_self" title="MODELC_3:th.64">Th64</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="147_4_1_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:147_4_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E136">A29</font></span>: 
<font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V2" title="MODELC_2:attr.2">atomic</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font></span><br/><div class="add"><a NAME="E2:147_4_1_1_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K27" title="MODELC_3:func.27">atomic_LTL</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:147_4_1"><span class="lab"><font color="Green" title="E129">A22</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:147_4_1_1_1_1_2_1_1"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147_4_1_1_1_1"><span class="lab"><font color="Green" title="E132">A25</font></span></a>, <a class="txt" href="#E7:147_4_1_1_1_1"><span class="lab"><font color="Green" title="E133">A26</font></span></a>, <a class="txt" href="#E3:147_4_1_1_1_1"><span class="lab"><font color="Green" title="E131">A24</font></span></a>, <a class="txt" href="#E1:147_4_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E136">A29</font></span></a>, <a class="ref" href="modelc_2.html#T63" title="MODELC_2:th.63">MODELC_2:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="147_4_1_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:147_4_1_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E136">A30</font></span>: 
( <font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V3" title="MODELC_2:attr.3">negative</a>  &amp;  <a href="modelc_2.html#K10" title="MODELC_2:func.10">the_argument_of</a> <font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V2" title="MODELC_2:attr.2">atomic</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c13">Fa</font> =  <a href="modelc_2.html#K10" title="MODELC_2:func.10">the_argument_of</a> <font color="Maroon" title="c8">F</font>;<br/><a NAME="E2:147_4_1_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E137">A31</font></span>: 
<font color="Maroon" title="c8">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K3" title="MODELC_2:func.3">'not'</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K10" title="MODELC_2:func.10">the_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E136">A30</font></span></a>, <a class="ref" href="modelc_2.html#T4" title="MODELC_2:th.4">MODELC_2:4</a></span>;<br/><a NAME="E3:147_4_1_1_1_1_2_1_2"/><span class="kw">then </span>
(  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font> iff  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#NR9" title="MODELC_2:NR.9">|/=</a>  <a href="modelc_2.html#K10" title="MODELC_2:func.10">the_argument_of</a> <font color="Maroon" title="c8">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T64" title="MODELC_2:th.64">MODELC_2:64</a></span>;<br/><a NAME="E4:147_4_1_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E138">A32</font></span>: 
(  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font> iff not  <a href="modelc_2.html#K10" title="MODELC_2:func.10">the_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">i</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:147_4_1_1_1_1"><span class="lab"><font color="Green" title="E131">A24</font></span></a>, <a class="txt" href="#E1:147_4_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E136">A30</font></span></a>, <a class="ref" href="modelc_2.html#T63" title="MODELC_2:th.63">MODELC_2:63</a></span>;<br/><a NAME="E5:147_4_1_1_1_1_2_1_2"/>
 <a href="modelc_2.html#K10" title="MODELC_2:func.10">the_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K28" title="MODELC_3:func.28">Neg_atomic_LTL</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:147_4_1"><span class="lab"><font color="Green" title="E129">A22</font></span></a>, <a class="txt" href="#E1:147_4_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E136">A30</font></span></a>, <a class="txt" href="#E2:147_4_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E137">A31</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:147_4_1_1_1_1_2_1_2"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147_4_1_1_1_1"><span class="lab"><font color="Green" title="E132">A25</font></span></a>, <a class="txt" href="#E9:147_4_1_1_1_1"><span class="lab"><font color="Green" title="E135">A28</font></span></a>, <a class="txt" href="#E4:147_4_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E138">A32</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E11:147_4_1_1_1_1"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="147_4_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:147_4_1_1_1_2"/><span class="lab"><font color="Green" title="E130">A33</font></span>: 
( <font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V4" title="MODELC_2:attr.4">conjunctive</a>  or <font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V5" title="MODELC_2:attr.5">disjunctive</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c10">h1</font> =  <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font>;<br/><a NAME="E2:147_4_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:147_4_1"><span class="lab"><font color="Green" title="E128">A21</font></span></a>, <a class="txt" href="#E1:147_4_1_1_1_2"><span class="lab"><font color="Green" title="E130">A33</font></span></a>, <a class="ref" href="modelc_2.html#T11" title="MODELC_2:th.11">MODELC_2:11</a></span>;<br/><a NAME="E3:147_4_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A34</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">Runi1</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1);<br/><span class="kw">set </span><font color="Maroon" title="c12">Runi</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font>);<br/><a NAME="E4:147_4_1_1_1_2"/><span class="lab"><font color="Green" title="E132">A35</font></span>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147"><span class="lab"><font color="Green" title="E120">A4</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c13">h2</font> =  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font>;<br/><a NAME="E5:147_4_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:147_4_1"><span class="lab"><font color="Green" title="E128">A21</font></span></a>, <a class="txt" href="#E1:147_4_1_1_1_2"><span class="lab"><font color="Green" title="E130">A33</font></span></a>, <a class="ref" href="modelc_2.html#T11" title="MODELC_2:th.11">MODELC_2:11</a></span>;<br/><a NAME="E6:147_4_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A36</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">Runi1</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) as   <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:147_4_1_1_1_2"><span class="lab"><font color="Green" title="E132">A35</font></span></a>, <a class="ref" href="modelc_3.html#D20" target="_self" title="MODELC_3:def.20">Def20</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">Runi</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font>) as   <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:147_4_1_1_1_2"><span class="lab"><font color="Green" title="E132">A35</font></span></a>, <a class="ref" href="modelc_3.html#D20" target="_self" title="MODELC_3:def.20">Def20</a></span>;<br/><a NAME="E9:147_4_1_1_1_2"/><span class="lab"><font color="Green" title="E134">A37</font></span>: 
( <font color="Maroon" title="c14">Runi1</font> <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon" title="c15">Runi</font> &amp; <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c14">Runi1</font> implies ( ( <font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V4" title="MODELC_2:attr.4">conjunctive</a>  implies (  <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c14">Runi1</font> &amp;  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c14">Runi1</font> ) ) &amp; (  not <font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V5" title="MODELC_2:attr.5">disjunctive</a>  or  <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c14">Runi1</font> or  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c14">Runi1</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#T41" target="_self" title="MODELC_3:th.41">Th41</a></span>;<br/><a NAME="E10:147_4_1_1_1_2"/><span class="lab"><font color="Green" title="E135">A38</font></span>: 
(  <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="modelc_2.html#R2" title="MODELC_2:pred.2">is_subformula_of</a> <font color="Maroon" title="c8">F</font> &amp;  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="modelc_2.html#R2" title="MODELC_2:pred.2">is_subformula_of</a> <font color="Maroon" title="c8">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1_1_1_2"><span class="lab"><font color="Green" title="E130">A33</font></span></a>, <a class="ref" href="modelc_2.html#T31" title="MODELC_2:th.31">MODELC_2:31</a></span>;<br/><a NAME="E11:147_4_1_1_1_2"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1_1_1_2_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="147_4_1_1_1_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:147_4_1_1_1_2_1_1_1"/>
( <font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V4" title="MODELC_2:attr.4">conjunctive</a>  or <font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V5" title="MODELC_2:attr.5">disjunctive</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1_1_1_2"><span class="lab"><font color="Green" title="E130">A33</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="147_4_1_1_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:147_4_1_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E136">A39</font></span>: 
<font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V4" title="MODELC_2:attr.4">conjunctive</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font></span><br/><div class="add"><a NAME="E2:147_4_1_1_1_2_1_1_1_1"/><span class="kw">then </span>
(  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font> &amp;  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147"><span class="lab"><font color="Green" title="E120">A4</font></span></a>, <a class="txt" href="#E1:147_4"><span class="lab"><font color="Green" title="E125">A18</font></span></a>, <a class="txt" href="#E1:147_4_1"><span class="lab"><font color="Green" title="E127">A20</font></span></a>, <a class="txt" href="#E3:147_4_1"><span class="lab"><font color="Green" title="E129">A22</font></span></a>, <a class="txt" href="#E10:147_4_1_1_1_2"><span class="lab"><font color="Green" title="E135">A38</font></span></a>, <a class="txt" href="#E3:147_4_1_1_1_2"><span class="lab"><font color="Green" title="E131">A34</font></span></a>, <a class="txt" href="#E6:147_4_1_1_1_2"><span class="lab"><font color="Green" title="E133">A36</font></span></a>, <a class="txt" href="#E9:147_4_1_1_1_2"><span class="lab"><font color="Green" title="E134">A37</font></span></a>, <a class="ref" href="modelc_2.html#T35" title="MODELC_2:th.35">MODELC_2:35</a></span>;<br/><a NAME="E3:147_4_1_1_1_2_1_1_1_1"/><span class="kw">then </span>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="modelc_2.html#K4" title="MODELC_2:func.4">'&amp;'</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T65" title="MODELC_2:th.65">MODELC_2:65</a></span>;<br/><span class="kw">hence </span><a NAME="E4:147_4_1_1_1_2_1_1_1_1"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E136">A39</font></span></a>, <a class="ref" href="modelc_2.html#T6" title="MODELC_2:th.6">MODELC_2:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="147_4_1_1_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:147_4_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E136">A40</font></span>: 
<font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V5" title="MODELC_2:attr.5">disjunctive</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font></span><br/><div class="add"><a NAME="E2:147_4_1_1_1_2_1_1_1_2"/><span class="kw">then </span>
(  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font> or  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147"><span class="lab"><font color="Green" title="E120">A4</font></span></a>, <a class="txt" href="#E1:147_4"><span class="lab"><font color="Green" title="E125">A18</font></span></a>, <a class="txt" href="#E1:147_4_1"><span class="lab"><font color="Green" title="E127">A20</font></span></a>, <a class="txt" href="#E3:147_4_1"><span class="lab"><font color="Green" title="E129">A22</font></span></a>, <a class="txt" href="#E10:147_4_1_1_1_2"><span class="lab"><font color="Green" title="E135">A38</font></span></a>, <a class="txt" href="#E3:147_4_1_1_1_2"><span class="lab"><font color="Green" title="E131">A34</font></span></a>, <a class="txt" href="#E6:147_4_1_1_1_2"><span class="lab"><font color="Green" title="E133">A36</font></span></a>, <a class="txt" href="#E9:147_4_1_1_1_2"><span class="lab"><font color="Green" title="E134">A37</font></span></a>, <a class="ref" href="modelc_2.html#T35" title="MODELC_2:th.35">MODELC_2:35</a></span>;<br/><a NAME="E3:147_4_1_1_1_2_1_1_1_2"/><span class="kw">then </span>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="modelc_2.html#K5" title="MODELC_2:func.5">'or'</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T66" title="MODELC_2:th.66">MODELC_2:66</a></span>;<br/><span class="kw">hence </span><a NAME="E4:147_4_1_1_1_2_1_1_1_2"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E136">A40</font></span></a>, <a class="ref" href="modelc_2.html#T7" title="MODELC_2:th.7">MODELC_2:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:147_4_1_1_1_2_1"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E12:147_4_1_1_1_2"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="147_4_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:147_4_1_1_1_3"/><span class="lab"><font color="Green" title="E130">A41</font></span>: 
<font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V6" title="MODELC_2:attr.6">next</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c10">i1</font> = <font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><span class="kw">set </span><font color="Maroon" title="c11">Runi1</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1);<br/><span class="kw">set </span><font color="Maroon" title="c12">Runi2</font> = <font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1);<br/><a NAME="E2:147_4_1_1_1_3"/>
<font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147"><span class="lab"><font color="Green" title="E120">A4</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">Runi2</font> = <font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1) as   <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#D20" target="_self" title="MODELC_3:def.20">Def20</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c14">Runi</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font>);<br/><a NAME="E4:147_4_1_1_1_3"/><span class="lab"><font color="Green" title="E131">A42</font></span>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147"><span class="lab"><font color="Green" title="E120">A4</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c15">h</font> =  <a href="modelc_2.html#K10" title="MODELC_2:func.10">the_argument_of</a> <font color="Maroon" title="c8">F</font>;<br/><a NAME="E5:147_4_1_1_1_3"/><span class="lab"><font color="Green" title="E132">A43</font></span>: 
 <a href="modelc_2.html#K10" title="MODELC_2:func.10">the_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="modelc_2.html#R2" title="MODELC_2:pred.2">is_subformula_of</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1_1_1_3"><span class="lab"><font color="Green" title="E130">A41</font></span></a>, <a class="ref" href="modelc_2.html#T30" title="MODELC_2:th.30">MODELC_2:30</a></span>;<br/><a NAME="E6:147_4_1_1_1_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K10" title="MODELC_2:func.10">the_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:147_4_1"><span class="lab"><font color="Green" title="E128">A21</font></span></a>, <a class="txt" href="#E1:147_4_1_1_1_3"><span class="lab"><font color="Green" title="E130">A41</font></span></a>, <a class="ref" href="modelc_2.html#T10" title="MODELC_2:th.10">MODELC_2:10</a></span>;<br/><a NAME="E7:147_4_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A44</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K10" title="MODELC_2:func.10">the_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">Runi1</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) as   <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:147_4_1_1_1_3"><span class="lab"><font color="Green" title="E131">A42</font></span></a>, <a class="ref" href="modelc_3.html#D20" target="_self" title="MODELC_3:def.20">Def20</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">Runi</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font>) as   <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:147_4_1_1_1_3"><span class="lab"><font color="Green" title="E131">A42</font></span></a>, <a class="ref" href="modelc_3.html#D20" target="_self" title="MODELC_3:def.20">Def20</a></span>;<br/><a NAME="E10:147_4_1_1_1_3"/><span class="lab"><font color="Green" title="E134">A45</font></span>: 
( <font color="Maroon" title="c16">Runi1</font> <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon" title="c17">Runi</font> &amp; <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c16">Runi1</font> &amp; <font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V6" title="MODELC_2:attr.6">next</a>  implies  <a href="modelc_2.html#K10" title="MODELC_2:func.10">the_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <font color="Maroon" title="c16">Runi1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#T41" target="_self" title="MODELC_3:th.41">Th41</a></span>;<br/><a NAME="E11:147_4_1_1_1_3"/>
the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <font color="Maroon" title="c16">Runi1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c13">Runi2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147"><span class="lab"><font color="Green" title="E120">A4</font></span></a>, <a class="ref" href="modelc_3.html#T37" target="_self" title="MODELC_3:th.37">Th37</a></span>;<br/><a NAME="E12:147_4_1_1_1_3"/><span class="kw">then </span>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K10" title="MODELC_2:func.10">the_argument_of</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147"><span class="lab"><font color="Green" title="E120">A4</font></span></a>, <a class="txt" href="#E1:147_4"><span class="lab"><font color="Green" title="E125">A18</font></span></a>, <a class="txt" href="#E1:147_4_1"><span class="lab"><font color="Green" title="E127">A20</font></span></a>, <a class="txt" href="#E3:147_4_1"><span class="lab"><font color="Green" title="E129">A22</font></span></a>, <a class="txt" href="#E1:147_4_1_1_1_3"><span class="lab"><font color="Green" title="E130">A41</font></span></a>, <a class="txt" href="#E5:147_4_1_1_1_3"><span class="lab"><font color="Green" title="E132">A43</font></span></a>, <a class="txt" href="#E7:147_4_1_1_1_3"><span class="lab"><font color="Green" title="E133">A44</font></span></a>, <a class="txt" href="#E10:147_4_1_1_1_3"><span class="lab"><font color="Green" title="E134">A45</font></span></a>, <a class="ref" href="modelc_2.html#T35" title="MODELC_2:th.35">MODELC_2:35</a></span>;<br/><a NAME="E13:147_4_1_1_1_3"/><span class="kw">then </span>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,1 <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K10" title="MODELC_2:func.10">the_argument_of</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T80" title="MODELC_2:th.80">MODELC_2:80</a></span>;<br/><a NAME="E14:147_4_1_1_1_3"/><span class="kw">then </span>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K6" title="MODELC_2:func.6">'X'</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K10" title="MODELC_2:func.10">the_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T67" title="MODELC_2:th.67">MODELC_2:67</a></span>;<br/><span class="kw">hence </span><a NAME="E15:147_4_1_1_1_3"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1_1_1_3"><span class="lab"><font color="Green" title="E130">A41</font></span></a>, <a class="ref" href="modelc_2.html#T5" title="MODELC_2:th.5">MODELC_2:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="147_4_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:147_4_1_1_1_4"/><span class="lab"><font color="Green" title="E130">A46</font></span>: 
<font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V7" title="MODELC_2:attr.7">Until</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c10">Fin</font> =  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font>;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c3">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> $1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">i</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">FRun</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span> ;<br/><a NAME="E2:147_4_1_1_1_4"/><span class="lab"><font color="Green" title="E131">A47</font></span>: 
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="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1_1_1_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c12">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:147_4_1_1_1_4_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c12">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c13">y</font> = <span class="p1">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">i</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">y</font> = <span class="p1">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">i</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E3:147_4_1_1_1_4_1"/>
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c12">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E4:147_4_1_1_1_4_1"/>
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c12">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c12">runQ</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> ,<span class="p1">(<span class="default"><a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:147_4_1_1_1_4"/><span class="lab"><font color="Green" title="E132">A48</font></span>: 
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="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon" title="c12">runQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">x</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S2" title="FUNCT_2:sch.2">FUNCT_2:sch 2</a>(<a class="txt" href="#E2:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E131">A47</font></span></a>);<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c13">runQ</font> = <font color="Maroon" title="c12">runQ</font> as   <a href="nat_1.html#NM2" title="NAT_1:NM.2">sequence</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font></span>)</span> ;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>4</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>  <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font> =  <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">runQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>,<font color="Maroon" title="c2">v</font>;<br/><a NAME="E6:147_4_1_1_1_4"/><span class="lab"><font color="Green" title="E133">A49</font></span>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">i</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1_1_1_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c14">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c14">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">i</font>)</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c15">m</font> = <font color="Maroon" title="c14">m</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c15">m</font>) = 
 <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c15">m</font>),<font color="Maroon" title="c2">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E132">A48</font></span></a></span>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c15">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">i</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#D1" title="MODELC_2:def.1">MODELC_2:def 1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:147_4_1_1_1_4_2"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c14">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c14">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">i</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E7:147_4_1_1_1_4"/><span class="lab"><font color="Green" title="E134">A50</font></span>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">m</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1_1_1_4_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c14">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c14">m</font>)</span><br/>

<span class="kw">set </span><font color="Maroon" title="c15">m1</font> = <font color="Maroon" title="c14">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">i</font>;<br/>
<span class="lab"><font color="Green" title="E135">A51</font></span>: <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c14">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) = 
<font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c14">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">i</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E133">A49</font></span></a></span>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c14">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)

;<br/>
<a NAME="E2:147_4_1_1_1_4_3"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c14">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c14">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">i</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E133">A49</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:147_4_1_1_1_4_3"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c14">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c14">m</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147"><span class="lab"><font color="Green" title="E120">A4</font></span></a>, <a class="txt" href="#E1:147_4_1_1_1_4_3"><span class="lab"><font color="Green" title="E135">A51</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">set </span><font color="Maroon" title="c14">FRunQ</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span> ;<br/><a NAME="E8:147_4_1_1_1_4"/><span class="lab"><font color="Green" title="E135">A52</font></span>: 
 <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K33" title="MODELC_3:func.33">FinalS_LTL</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1"><span class="lab"><font color="Green" title="E127">A20</font></span></a>, <a class="txt" href="#E1:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E130">A46</font></span></a></span>;<br/><a NAME="E9:147_4_1_1_1_4"/><span class="lab"><font color="Green" title="E136">A53</font></span>: 
not <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1_1_1_4_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c15">FRun2</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">FRun1</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span> ;<br/>
<a NAME="E1:147_4_1_1_1_4_4"/><span class="lab"><font color="Green" title="E137">A54</font></span>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1_1_1_4_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span>  or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:147_4_1_1_1_4_4_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">k</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E3:147_4_1_1_1_4_4_1"/><span class="lab"><font color="Green" title="E138">A55</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">k</font>
 <span class="kw">and </span><br/><a NAME="E4:147_4_1_1_1_4_4_1"/><span class="lab"><font color="Green" title="E139">A56</font></span>: 
<font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">and </span><br/><a NAME="E5:147_4_1_1_1_4_4_1"/>
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span> 
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1_1_1_4_4_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="147_4_1_1_1_4_4_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:147_4_1_1_1_4_4_1_1_1"/>
( <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c18">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="147_4_1_1_1_4_4_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:147_4_1_1_1_4_4_1_1_1_1"/>
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:147_4_1_1_1_4_4_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:147_4_1_1_1_4_4_1_1_1_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:147_4_1_1_1_4_4_1"><span class="lab"><font color="Green" title="E138">A55</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="147_4_1_1_1_4_4_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:147_4_1_1_1_4_4_1_1_1_2"/>
<font color="Maroon" title="c18">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:147_4_1_1_1_4_4_1_1_1_2"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">k</font>
 ;<br/><a NAME="E3:147_4_1_1_1_4_4_1_1_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T19" title="NAT_1:th.19">NAT_1:19</a></span>;<br/><a NAME="E4:147_4_1_1_1_4_4_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:147_4_1_1_1_4_4_1"><span class="lab"><font color="Green" title="E139">A56</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><span class="kw">hence </span><a NAME="E5:147_4_1_1_1_4_4_1_1_1_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:147_4_1_1_1_4_4_1"><span class="lab"><font color="Green" title="E138">A55</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:147_4_1_1_1_4_4_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:147_4_1_1_1_4_4"/><span class="lab"><font color="Green" title="E138">A57</font></span>: 
( <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  implies <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1_1_1_4_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">deffunc </span><font color="Maroon">H<sub>5</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> $1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">i</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">fun</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E2:147_4_1_1_1_4_4_2"/><span class="lab"><font color="Green" title="E139">A58</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c17">fun</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  &amp; ( 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> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  holds <br/><font color="Maroon" title="c17">fun</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></span>
<a NAME="E3:147_4_1_1_1_4_4_2"/><span class="lab"><font color="Green" title="E140">A59</font></span>: 
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> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : ( <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">k</font> where <font color="Olive" title="b3">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b3">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span>  holds <br/><span class="p1">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1_1_1_4_4_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span>  implies <span class="p1">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> <font color="Maroon" title="c18">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:147_4_1_1_1_4_4_2_1"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> <font color="Maroon" title="c18">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">k</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E3:147_4_1_1_1_4_4_2_1"/><span class="lab"><font color="Green" title="E141">A60</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">k</font>
 <span class="kw">and </span><br/><a NAME="E4:147_4_1_1_1_4_4_2_1"/><span class="lab"><font color="Green" title="E142">A61</font></span>: 
<font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">k</font>
 <span class="kw">and </span><br/><a NAME="E5:147_4_1_1_1_4_4_2_1"/><span class="lab"><font color="Green" title="E143">A62</font></span>: 
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span> 
 ;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">k2</font> = <font color="Maroon" title="c19">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">i</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">k2</font> = <font color="Maroon" title="c19">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">i</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:147_4_1_1_1_4_4_2_1"><span class="lab"><font color="Green" title="E142">A61</font></span></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></span>;<br/>
<span class="lab"><font color="Green" title="E144">A63</font></span>: <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c21">k2</font>) = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c21">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">i</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E133">A49</font></span></a></span>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c19">k</font>)

;<br/>
<a NAME="E8:147_4_1_1_1_4_4_2_1"/>
(  ex <font color="Olive" title="b1">k1</font> being   <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>  st <br/>( <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k1</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">k1</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> ) &amp; <span class="p1">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> <font color="Maroon" title="c18">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">k2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:147_4_1_1_1_4_4_2_1"><span class="lab"><font color="Green" title="E141">A60</font></span></a>, <a class="txt" href="#E5:147_4_1_1_1_4_4_2_1"><span class="lab"><font color="Green" title="E143">A62</font></span></a>, <a class="ref" href="modelc_2.html#D1" title="MODELC_2:def.1">MODELC_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:147_4_1_1_1_4_4_2_1"/>
<span class="p1">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> <font color="Maroon" title="c18">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:147_4_1_1_1_4_4_2_1"><span class="lab"><font color="Green" title="E144">A63</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:147_4_1_1_1_4_4_2"/><span class="lab"><font color="Green" title="E141">A64</font></span>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : ( <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">k</font> where <font color="Olive" title="b3">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b3">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span>  holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c17">fun</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">fun</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1_1_1_4_4_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span>  implies  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c17">fun</font> &amp; <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">fun</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:147_4_1_1_1_4_4_2_2"/><span class="lab"><font color="Green" title="E142">A65</font></span>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c17">fun</font> &amp; <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">fun</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c19">k</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E3:147_4_1_1_1_4_4_2_2"/><span class="lab"><font color="Green" title="E143">A66</font></span>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">k</font>
 <span class="kw">and </span><br/><a NAME="E4:147_4_1_1_1_4_4_2_2"/><span class="lab"><font color="Green" title="E144">A67</font></span>: 
<font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">k</font>
 <span class="kw">and </span><br/><a NAME="E5:147_4_1_1_1_4_4_2_2"/>
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1_1_1_4_4_2_2"><span class="lab"><font color="Green" title="E142">A65</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">x</font> = <span class="p1">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> <font color="Maroon" title="c18">y</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">i</font>;<br/>
<a NAME="E6:147_4_1_1_1_4_4_2_2"/><span class="lab"><font color="Green" title="E145">A68</font></span>: 
<span class="p1">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> <font color="Maroon" title="c18">y</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c17">fun</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:147_4_1_1_1_4_4_2"><span class="lab"><font color="Green" title="E140">A59</font></span></a>, <a class="txt" href="#E2:147_4_1_1_1_4_4_2"><span class="lab"><font color="Green" title="E139">A58</font></span></a>, <a class="txt" href="#E1:147_4_1_1_1_4_4_2_2"><span class="lab"><font color="Green" title="E142">A65</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">k1</font> = <font color="Maroon" title="c19">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">i</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">k1</font> = <font color="Maroon" title="c19">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">i</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:147_4_1_1_1_4_4_2_2"><span class="lab"><font color="Green" title="E144">A67</font></span></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></span>;<br/>
<a NAME="E8:147_4_1_1_1_4_4_2_2"/><span class="lab"><font color="Green" title="E146">A69</font></span>: 
<span class="p1">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> <font color="Maroon" title="c18">y</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:147_4_1_1_1_4_4_2_2"><span class="lab"><font color="Green" title="E143">A66</font></span></a>, <a class="ref" href="modelc_2.html#D1" title="MODELC_2:def.1">MODELC_2:def 1</a></span>;<br/>
<font color="Maroon" title="c17">fun</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> <font color="Maroon" title="c18">y</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">i</font></span>)</span> = 
<font color="Maroon">H<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="modelc_2.html#K1" title="MODELC_2:func.1">CastNat</a> <font color="Maroon" title="c18">y</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">i</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:147_4_1_1_1_4_4_2"><span class="lab"><font color="Green" title="E140">A59</font></span></a>, <a class="txt" href="#E2:147_4_1_1_1_4_4_2"><span class="lab"><font color="Green" title="E139">A58</font></span></a>, <a class="txt" href="#E1:147_4_1_1_1_4_4_2_2"><span class="lab"><font color="Green" title="E142">A65</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c22">k1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:147_4_1_1_1_4_4_2_2"><span class="lab"><font color="Green" title="E146">A69</font></span></a>, <a class="ref" href="modelc_2.html#D1" title="MODELC_2:def.1">MODELC_2:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c18">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:147_4_1_1_1_4_4_2_2"><span class="lab"><font color="Green" title="E143">A66</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:147_4_1_1_1_4_4_2_2"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c17">fun</font> &amp; <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">fun</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147_4_1_1_1_4_4_2_2"><span class="lab"><font color="Green" title="E145">A68</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E5:147_4_1_1_1_4_4_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/>

<a NAME="E6:147_4_1_1_1_4_4_2"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">fun</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:147_4_1_1_1_4_4_2"><span class="lab"><font color="Green" title="E139">A58</font></span></a>, <a class="ref" href="finset_1.html#T26" title="FINSET_1:th.26">FINSET_1:26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:147_4_1_1_1_4_4_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:147_4_1_1_1_4_4_2"><span class="lab"><font color="Green" title="E141">A64</font></span></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a>, <a class="ref" href="funct_1.html#T19" title="FUNCT_1:th.19">FUNCT_1:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:147_4_1_1_1_4_4"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1_1_1_4_4_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:147_4_1_1_1_4_4_3"/><span class="lab"><font color="Green" title="E139">A70</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span> </span><br/>

<a NAME="E2:147_4_1_1_1_4_4_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">k</font> being   <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>  st <br/>( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">x</font> = <font color="Maroon" title="c17">x</font> as    <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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1_1_1_4_4_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="147_4_1_1_1_4_4_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:147_4_1_1_1_4_4_3_1_1"/>
( <font color="Maroon" title="c18">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> or <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="147_4_1_1_1_4_4_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:147_4_1_1_1_4_4_3_1_1_1"/>
<font color="Maroon" title="c18">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span> </span><br/><div class="add"><a NAME="E2:147_4_1_1_1_4_4_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1_1_1_4_4_3"><span class="lab"><font color="Green" title="E139">A70</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:147_4_1_1_1_4_4_3_1_1_1"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="147_4_1_1_1_4_4_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:147_4_1_1_1_4_4_3_1_1_2"/>
<font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span> </span><br/><div class="add"><a NAME="E2:147_4_1_1_1_4_4_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1_1_1_4_4_3"><span class="lab"><font color="Green" title="E139">A70</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:147_4_1_1_1_4_4_3_1_1_2"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:147_4_1_1_1_4_4_3"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:147_4_1_1_1_4_4"/>
not <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:147"><span class="lab"><font color="Green" title="E123">A13</font></span></a>, <a class="txt" href="#E8:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E135">A52</font></span></a>, <a class="txt" href="#E2:147_4_1_1_1_4_4"><span class="lab"><font color="Green" title="E138">A57</font></span></a>, <a class="txt" href="#E1:147_4_1_1_1_4_4"><span class="lab"><font color="Green" title="E137">A54</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">set </span><font color="Maroon" title="c15">h2</font> =  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font>;<br/><span class="kw">set </span><font color="Maroon" title="c16">h1</font> =  <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font>;<br/><a NAME="E10:147_4_1_1_1_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:147_4_1"><span class="lab"><font color="Green" title="E128">A21</font></span></a>, <a class="txt" href="#E1:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E130">A46</font></span></a>, <a class="ref" href="modelc_2.html#T11" title="MODELC_2:th.11">MODELC_2:11</a></span>;<br/><a NAME="E11:147_4_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E137">A71</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E12:147_4_1_1_1_4"/><span class="lab"><font color="Green" title="E138">A72</font></span>: 
( ( for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/>( <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">m</font>) &amp;  <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">m</font>) &amp; not  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">m</font>) ) ) implies <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1_1_1_4_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:147_4_1_1_1_4_5"/><span class="lab"><font color="Green" title="E139">A73</font></span>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/>( <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">m</font>) &amp;  <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">m</font>) &amp; not  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">m</font>) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1_1_1_4_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:147_4_1_1_1_4_5_1"/>
not <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:147_4_1_1_1_4_5_1"/><span class="lab"><font color="Green" title="E140">A74</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span> 
 <span class="kw">and </span><br/><a NAME="E4:147_4_1_1_1_4_5_1"/><span class="lab"><font color="Green" title="E141">A75</font></span>: 
not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c18">k</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E6:147_4_1_1_1_4_5_1"/><span class="lab"><font color="Green" title="E142">A76</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">k</font>
 <span class="kw">and </span><br/><a NAME="E7:147_4_1_1_1_4_5_1"/><span class="lab"><font color="Green" title="E143">A77</font></span>: 
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c18">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:147_4_1_1_1_4_5_1"><span class="lab"><font color="Green" title="E140">A74</font></span></a></span>;<br/><a NAME="E8:147_4_1_1_1_4_5_1"/>
<font color="Maroon" title="c18">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:147_4_1_1_1_4_5_1"><span class="lab"><font color="Green" title="E141">A75</font></span></a>, <a class="txt" href="#E6:147_4_1_1_1_4_5_1"><span class="lab"><font color="Green" title="E142">A76</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E9:147_4_1_1_1_4_5_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">k</font>
 ;<br/><a NAME="E10:147_4_1_1_1_4_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E144">A78</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T19" title="NAT_1:th.19">NAT_1:19</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c19">RQk</font> = <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c18">k</font>);<br/><span class="kw">consider </span><font color="Maroon" title="c20">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font><span class="kw"> such that </span><br/><a NAME="E12:147_4_1_1_1_4_5_1"/><span class="lab"><font color="Green" title="E145">A79</font></span>: 
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c18">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">y</font>
 <span class="kw">and </span><br/><a NAME="E13:147_4_1_1_1_4_5_1"/><span class="lab"><font color="Green" title="E146">A80</font></span>: 
(  not <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <font color="Maroon" title="c20">y</font>,<font color="Maroon" title="c2">v</font></span>)</span> or  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <font color="Maroon" title="c20">y</font>,<font color="Maroon" title="c2">v</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:147_4_1_1_1_4_5_1"><span class="lab"><font color="Green" title="E143">A77</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">y</font> = <font color="Maroon" title="c20">y</font> as   <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:147_4_1_1_1_4_5_1"><span class="lab"><font color="Green" title="E145">A79</font></span></a></span>;<br/><a NAME="E15:147_4_1_1_1_4_5_1"/>
 <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <font color="Maroon" title="c21">y</font>,<font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c18">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:147_4_1_1_1_4_5_1"><span class="lab"><font color="Green" title="E145">A79</font></span></a>, <a class="ref" href="modelc_3.html#D16" target="_self" title="MODELC_3:def.16">Def16</a></span>;<br/><span class="kw">hence </span><a NAME="E16:147_4_1_1_1_4_5_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1_1_1_4_5"><span class="lab"><font color="Green" title="E139">A73</font></span></a>, <a class="txt" href="#E10:147_4_1_1_1_4_5_1"><span class="lab"><font color="Green" title="E144">A78</font></span></a>, <a class="txt" href="#E13:147_4_1_1_1_4_5_1"><span class="lab"><font color="Green" title="E146">A80</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:147_4_1_1_1_4_5"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c2">v</font> </span>}</span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E13:147_4_1_1_1_4"/>
<font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>4</sub></font>(1)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:147_4_1"><span class="lab"><font color="Green" title="E129">A22</font></span></a>, <a class="txt" href="#E6:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E133">A49</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E15:147_4_1_1_1_4"/><span class="lab"><font color="Green" title="E139">A81</font></span>: 
<font color="Maroon" title="c17">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">and </span><br/><a NAME="E16:147_4_1_1_1_4"/><span class="lab"><font color="Green" title="E140">A82</font></span>: 
 <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c17">j</font>)
 <span class="kw">and </span><br/><a NAME="E17:147_4_1_1_1_4"/><span class="lab"><font color="Green" title="E141">A83</font></span>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">j</font> holds <br/>( <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">m</font>) &amp;  <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">m</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E130">A46</font></span></a>, <a class="txt" href="#E7:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E134">A50</font></span></a>, <a class="txt" href="#E9:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E136">A53</font></span></a>, <a class="txt" href="#E12:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E138">A72</font></span></a>, <a class="ref" href="modelc_3.html#T54" target="_self" title="MODELC_3:th.54">Th54</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c18">j0</font> = <font color="Maroon" title="c17">j</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">j0</font> = <font color="Maroon" title="c17">j</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E139">A81</font></span></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c20">j1</font> = <font color="Maroon" title="c19">j0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">i</font>;<br/><a NAME="E19:147_4_1_1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">j0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">i</font>
 ;<br/><a NAME="E20:147_4_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E142">A84</font></span>: 
 <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c19">j0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E133">A49</font></span></a>, <a class="txt" href="#E16:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E140">A82</font></span></a></span>;<br/><a NAME="E21:147_4_1_1_1_4"/><span class="lab"><font color="Green" title="E143">A85</font></span>: 
 <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="modelc_2.html#R2" title="MODELC_2:pred.2">is_subformula_of</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E130">A46</font></span></a>, <a class="ref" href="modelc_2.html#T31" title="MODELC_2:th.31">MODELC_2:31</a></span>;<br/><a NAME="E22:147_4_1_1_1_4"/><span class="lab"><font color="Green" title="E144">A86</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">j0</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<font color="Olive" title="b1">k</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1_1_1_4_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">j0</font> implies  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<font color="Maroon" title="c21">k</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:147_4_1_1_1_4_6"/><span class="lab"><font color="Green" title="E145">A87</font></span>: 
<font color="Maroon" title="c21">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">j0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<font color="Maroon" title="c21">k</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c22">k1</font> = <font color="Maroon" title="c21">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c23">ki</font> = <font color="Maroon" title="c21">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">i</font>;<br/>
<a NAME="E2:147_4_1_1_1_4_6"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">j0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1_1_1_4_6"><span class="lab"><font color="Green" title="E145">A87</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<a NAME="E3:147_4_1_1_1_4_6"/><span class="kw">then </span>
 <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c21">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E141">A83</font></span></a></span>;<br/>
<a NAME="E4:147_4_1_1_1_4_6"/><span class="kw">then </span>
 <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c21">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">i</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E133">A49</font></span></a></span>;<br/>
<a NAME="E5:147_4_1_1_1_4_6"/><span class="kw">then </span>
 <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c21">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)
 
;<br/>
<a NAME="E6:147_4_1_1_1_4_6"/><span class="kw">then </span>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c21">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4"><span class="lab"><font color="Green" title="E125">A18</font></span></a>, <a class="txt" href="#E1:147_4_1"><span class="lab"><font color="Green" title="E127">A20</font></span></a>, <a class="txt" href="#E21:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E143">A85</font></span></a>, <a class="txt" href="#E11:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E137">A71</font></span></a>, <a class="ref" href="modelc_2.html#T35" title="MODELC_2:th.35">MODELC_2:35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:147_4_1_1_1_4_6"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<font color="Maroon" title="c21">k</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T80" title="MODELC_2:th.80">MODELC_2:80</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E23:147_4_1_1_1_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:147_4_1"><span class="lab"><font color="Green" title="E128">A21</font></span></a>, <a class="txt" href="#E1:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E130">A46</font></span></a>, <a class="ref" href="modelc_2.html#T11" title="MODELC_2:th.11">MODELC_2:11</a></span>;<br/><a NAME="E24:147_4_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E145">A88</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E25:147_4_1_1_1_4"/>
 <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="modelc_2.html#R2" title="MODELC_2:pred.2">is_subformula_of</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E130">A46</font></span></a>, <a class="ref" href="modelc_2.html#T31" title="MODELC_2:th.31">MODELC_2:31</a></span>;<br/><a NAME="E26:147_4_1_1_1_4"/><span class="kw">then </span>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4"><span class="lab"><font color="Green" title="E125">A18</font></span></a>, <a class="txt" href="#E1:147_4_1"><span class="lab"><font color="Green" title="E127">A20</font></span></a>, <a class="txt" href="#E24:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E145">A88</font></span></a>, <a class="txt" href="#E20:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E142">A84</font></span></a>, <a class="ref" href="modelc_2.html#T35" title="MODELC_2:th.35">MODELC_2:35</a></span>;<br/><a NAME="E27:147_4_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E146">A89</font></span>: 
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<font color="Maroon" title="c19">j0</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T80" title="MODELC_2:th.80">MODELC_2:80</a></span>;<br/><a NAME="E28:147_4_1_1_1_4"/>
<font color="Maroon" title="c8">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E130">A46</font></span></a>, <a class="ref" href="modelc_2.html#T8" title="MODELC_2:th.8">MODELC_2:8</a></span>;<br/><span class="kw">hence </span><a NAME="E29:147_4_1_1_1_4"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E146">A89</font></span></a>, <a class="txt" href="#E22:147_4_1_1_1_4"><span class="lab"><font color="Green" title="E144">A86</font></span></a>, <a class="ref" href="modelc_2.html#T68" title="MODELC_2:th.68">MODELC_2:68</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="147_4_1_1_1_5"><span class="kw">suppose </span></a><a NAME="E1:147_4_1_1_1_5"/><span class="lab"><font color="Green" title="E130">A90</font></span>: 
<font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V8" title="MODELC_2:attr.8">Release</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c10">h2</font> =  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font>;<br/><a NAME="E2:147_4_1_1_1_5"/><span class="lab"><font color="Green" title="E131">A91</font></span>: 
 <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="modelc_2.html#R2" title="MODELC_2:pred.2">is_subformula_of</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1_1_1_5"><span class="lab"><font color="Green" title="E130">A90</font></span></a>, <a class="ref" href="modelc_2.html#T31" title="MODELC_2:th.31">MODELC_2:31</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">h1</font> =  <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $1 holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<font color="Olive" title="b1">k</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K3" title="MODELC_2:func.3">'not'</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span> ) implies (  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,$1 <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font> &amp; <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1) ) );<br/><a NAME="E3:147_4_1_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_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:147_4_1"><span class="lab"><font color="Green" title="E128">A21</font></span></a>, <a class="txt" href="#E1:147_4_1_1_1_5"><span class="lab"><font color="Green" title="E130">A90</font></span></a>, <a class="ref" href="modelc_2.html#T11" title="MODELC_2:th.11">MODELC_2:11</a></span>;<br/><a NAME="E4:147_4_1_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E132">A92</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:147_4_1_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_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:147_4_1"><span class="lab"><font color="Green" title="E128">A21</font></span></a>, <a class="txt" href="#E1:147_4_1_1_1_5"><span class="lab"><font color="Green" title="E130">A90</font></span></a>, <a class="ref" href="modelc_2.html#T11" title="MODELC_2:th.11">MODELC_2:11</a></span>;<br/><a NAME="E6:147_4_1_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A93</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E7:147_4_1_1_1_5"/><span class="lab"><font color="Green" title="E134">A94</font></span>: 
 <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="modelc_2.html#R2" title="MODELC_2:pred.2">is_subformula_of</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1_1_1_5"><span class="lab"><font color="Green" title="E130">A90</font></span></a>, <a class="ref" href="modelc_2.html#T31" title="MODELC_2:th.31">MODELC_2:31</a></span>;<br/><a NAME="E8:147_4_1_1_1_5"/><span class="lab"><font color="Green" title="E135">A95</font></span>: 
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1_1_1_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">j</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:147_4_1_1_1_5_1"/><span class="lab"><font color="Green" title="E136">A96</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">j</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:147_4_1_1_1_5_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1_1_1_5_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c13">i1</font> = <font color="Maroon" title="c7">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">j</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">Run1</font> = <font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1);<br/>
<span class="kw">set </span><font color="Maroon" title="c15">Run0</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">j</font>);<br/>
<span class="kw">assume </span><a NAME="E1:147_4_1_1_1_5_1_1"/><span class="lab"><font color="Green" title="E137">A97</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<font color="Olive" title="b1">k</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K3" title="MODELC_2:func.3">'not'</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font> &amp; <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) )</span><br/>

<a NAME="E2:147_4_1_1_1_5_1_1"/><span class="lab"><font color="Green" title="E138">A98</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<font color="Olive" title="b1">k</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K3" title="MODELC_2:func.3">'not'</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1_1_1_5_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:147_4_1_1_1_5_1_1_1"/><span class="lab"><font color="Green" title="E139">A99</font></span>: 
<font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c16">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j</font> implies  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<font color="Maroon" title="c16">k</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K3" title="MODELC_2:func.3">'not'</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:147_4_1_1_1_5_1_1_1"/>
<font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<font color="Maroon" title="c16">k</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K3" title="MODELC_2:func.3">'not'</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span></span><br/>

<a NAME="E3:147_4_1_1_1_5_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1_1_1_5_1_1_1"><span class="lab"><font color="Green" title="E139">A99</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:147_4_1_1_1_5_1_1_1"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<font color="Maroon" title="c16">k</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K3" title="MODELC_2:func.3">'not'</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E137">A97</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:147_4_1_1_1_5_1_1"/><span class="lab"><font color="Green" title="E139">A100</font></span>: 
<font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147"><span class="lab"><font color="Green" title="E120">A4</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">Run0</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">j</font>) as   <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#D20" target="_self" title="MODELC_3:def.20">Def20</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">i2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">Run2</font> = <font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1);<br/>
<a NAME="E5:147_4_1_1_1_5_1_1"/><span class="lab"><font color="Green" title="E140">A101</font></span>: 
<font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147"><span class="lab"><font color="Green" title="E120">A4</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">Run2</font> = <font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1) as   <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#D20" target="_self" title="MODELC_3:def.20">Def20</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">Run1</font> = <font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) as   <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:147_4_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E139">A100</font></span></a>, <a class="ref" href="modelc_3.html#D20" target="_self" title="MODELC_3:def.20">Def20</a></span>;<br/>
<a NAME="E8:147_4_1_1_1_5_1_1"/>
<font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E9:147_4_1_1_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E141">A102</font></span>: 
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<font color="Maroon" title="c12">j</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K3" title="MODELC_2:func.3">'not'</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E137">A97</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E142">A103</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1_1_1_5_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:147_4_1_1_1_5_1_1_2"/>
 <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c20">Run1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:147_4_1_1_1_5_1_1_2"/><span class="kw">then </span>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">j</font></span>)</span> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4"><span class="lab"><font color="Green" title="E125">A18</font></span></a>, <a class="txt" href="#E1:147_4_1"><span class="lab"><font color="Green" title="E127">A20</font></span></a>, <a class="txt" href="#E7:147_4_1_1_1_5"><span class="lab"><font color="Green" title="E134">A94</font></span></a>, <a class="txt" href="#E4:147_4_1_1_1_5"><span class="lab"><font color="Green" title="E132">A92</font></span></a>, <a class="ref" href="modelc_2.html#T35" title="MODELC_2:th.35">MODELC_2:35</a></span>;<br/><a NAME="E3:147_4_1_1_1_5_1_1_2"/><span class="kw">then </span>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<font color="Maroon" title="c12">j</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T80" title="MODELC_2:th.80">MODELC_2:80</a></span>;<br/><span class="kw">hence </span><a NAME="E4:147_4_1_1_1_5_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:147_4_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E141">A102</font></span></a>, <a class="ref" href="modelc_2.html#T64" title="MODELC_2:th.64">MODELC_2:64</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E11:147_4_1_1_1_5_1_1"/><span class="lab"><font color="Green" title="E143">A104</font></span>: 
( <font color="Maroon" title="c20">Run1</font> <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon" title="c16">Run0</font> &amp; <font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V8" title="MODELC_2:attr.8">Release</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c20">Run1</font> &amp; not  <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c20">Run1</font> implies (  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c20">Run1</font> &amp; <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <font color="Maroon" title="c20">Run1</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#T39" target="_self" title="MODELC_3:th.39">Th39</a></span>;<br/>
<a NAME="E12:147_4_1_1_1_5_1_1"/><span class="kw">then </span>
 <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c19">Run2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147"><span class="lab"><font color="Green" title="E120">A4</font></span></a>, <a class="txt" href="#E1:147_4_1_1_1_5"><span class="lab"><font color="Green" title="E130">A90</font></span></a>, <a class="txt" href="#E1:147_4_1_1_1_5_1"><span class="lab"><font color="Green" title="E136">A96</font></span></a>, <a class="txt" href="#E5:147_4_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E140">A101</font></span></a>, <a class="txt" href="#E2:147_4_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E138">A98</font></span></a>, <a class="txt" href="#E10:147_4_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E142">A103</font></span></a>, <a class="ref" href="modelc_3.html#T40" target="_self" title="MODELC_3:th.40">Th40</a></span>;<br/>
<a NAME="E13:147_4_1_1_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E144">A105</font></span>: 
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4"><span class="lab"><font color="Green" title="E125">A18</font></span></a>, <a class="txt" href="#E1:147_4_1"><span class="lab"><font color="Green" title="E127">A20</font></span></a>, <a class="txt" href="#E2:147_4_1_1_1_5"><span class="lab"><font color="Green" title="E131">A91</font></span></a>, <a class="txt" href="#E6:147_4_1_1_1_5"><span class="lab"><font color="Green" title="E133">A93</font></span></a>, <a class="ref" href="modelc_2.html#T35" title="MODELC_2:th.35">MODELC_2:35</a></span>;<br/>
<a NAME="E14:147_4_1_1_1_5_1_1"/>
<font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c19">Run2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147"><span class="lab"><font color="Green" title="E120">A4</font></span></a>, <a class="txt" href="#E1:147_4_1_1_1_5"><span class="lab"><font color="Green" title="E130">A90</font></span></a>, <a class="txt" href="#E1:147_4_1_1_1_5_1"><span class="lab"><font color="Green" title="E136">A96</font></span></a>, <a class="txt" href="#E5:147_4_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E140">A101</font></span></a>, <a class="txt" href="#E2:147_4_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E138">A98</font></span></a>, <a class="txt" href="#E10:147_4_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E142">A103</font></span></a>, <a class="txt" href="#E11:147_4_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E143">A104</font></span></a>, <a class="ref" href="modelc_3.html#T40" target="_self" title="MODELC_3:th.40">Th40</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:147_4_1_1_1_5_1_1"/>
(  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font> &amp; <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:147_4_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E144">A105</font></span></a>, <a class="ref" href="modelc_2.html#T80" title="MODELC_2:th.80">MODELC_2:80</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:147_4_1_1_1_5_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E9:147_4_1_1_1_5"/><span class="lab"><font color="Green" title="E136">A106</font></span>: 
<font color="Maroon" title="c8">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="modelc_2.html#K8" title="MODELC_2:func.8">'R'</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1_1_1_5"><span class="lab"><font color="Green" title="E130">A90</font></span></a>, <a class="ref" href="modelc_2.html#T9" title="MODELC_2:th.9">MODELC_2:9</a></span>;<br/><a NAME="E10:147_4_1_1_1_5"/><span class="lab"><font color="Green" title="E137">A107</font></span>: 
( ( for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>] ) implies  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1_1_1_5_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:147_4_1_1_1_5_2"/>
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font></span><br/>

<a NAME="E2:147_4_1_1_1_5_2"/><span class="kw">then </span>
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<font color="Olive" title="b2">k</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K3" title="MODELC_2:func.3">'not'</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span> ) holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<font color="Olive" title="b1">j</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E3:147_4_1_1_1_5_2"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:147_4_1_1_1_5"><span class="lab"><font color="Green" title="E136">A106</font></span></a>, <a class="ref" href="modelc_2.html#T69" title="MODELC_2:th.69">MODELC_2:69</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E11:147_4_1_1_1_5"/><span class="lab"><font color="Green" title="E138">A108</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_1_1_1_5_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c12">Run0</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c13">Run1</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1);<br/>
<a NAME="E1:147_4_1_1_1_5_3"/><span class="lab"><font color="Green" title="E139">A109</font></span>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147"><span class="lab"><font color="Green" title="E120">A4</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">Run1</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) as   <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#D20" target="_self" title="MODELC_3:def.20">Def20</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">Run0</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font>) as   <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4_1_1_1_5_3"><span class="lab"><font color="Green" title="E139">A109</font></span></a>, <a class="ref" href="modelc_3.html#D20" target="_self" title="MODELC_3:def.20">Def20</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:147_4_1_1_1_5_3"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<font color="Olive" title="b1">k</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K3" title="MODELC_2:func.3">'not'</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c8">F</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font> &amp; <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ) )</span><br/>

<a NAME="E5:147_4_1_1_1_5_3"/><span class="lab"><font color="Green" title="E140">A110</font></span>: 
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T79" title="MODELC_2:th.79">MODELC_2:79</a></span>;<br/>
<a NAME="E6:147_4_1_1_1_5_3"/>
( <font color="Maroon" title="c14">Run1</font> <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon" title="c15">Run0</font> &amp; <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c14">Run1</font> &amp; <font color="Maroon" title="c8">F</font> is  <a href="modelc_2.html#V8" title="MODELC_2:attr.8">Release</a>  implies  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c14">Run1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#T41" target="_self" title="MODELC_3:th.41">Th41</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:147_4_1_1_1_5_3"/>
(  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c8">F</font> &amp; <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147"><span class="lab"><font color="Green" title="E120">A4</font></span></a>, <a class="txt" href="#E1:147_4"><span class="lab"><font color="Green" title="E125">A18</font></span></a>, <a class="txt" href="#E1:147_4_1"><span class="lab"><font color="Green" title="E127">A20</font></span></a>, <a class="txt" href="#E3:147_4_1"><span class="lab"><font color="Green" title="E129">A22</font></span></a>, <a class="txt" href="#E1:147_4_1_1_1_5"><span class="lab"><font color="Green" title="E130">A90</font></span></a>, <a class="txt" href="#E2:147_4_1_1_1_5"><span class="lab"><font color="Green" title="E131">A91</font></span></a>, <a class="txt" href="#E6:147_4_1_1_1_5"><span class="lab"><font color="Green" title="E133">A93</font></span></a>, <a class="txt" href="#E5:147_4_1_1_1_5_3"><span class="lab"><font color="Green" title="E140">A110</font></span></a>, <a class="ref" href="modelc_2.html#T35" title="MODELC_2:th.35">MODELC_2:35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E12:147_4_1_1_1_5"/>
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E11:147_4_1_1_1_5"><span class="lab"><font color="Green" title="E138">A108</font></span></a>, <a class="txt" href="#E8:147_4_1_1_1_5"><span class="lab"><font color="Green" title="E135">A95</font></span></a>);<br/></span><span class="kw">hence </span><a NAME="E13:147_4_1_1_1_5"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:147_4_1_1_1_5"><span class="lab"><font color="Green" title="E137">A107</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:147_4_1"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:147_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a>  st <font color="Olive" title="b1">F</font> <a href="modelc_2.html#R2" title="MODELC_2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">v</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Olive" title="b1">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Olive" title="b1">F</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">F</font> be   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">F</font> <a href="modelc_2.html#R2" title="MODELC_2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">v</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) implies  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:147_4_2"/><span class="lab"><font color="Green" title="E127">A111</font></span>: 
<font color="Maroon" title="c8">F</font> <a href="modelc_2.html#R2" title="MODELC_2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">and </span><br/><a NAME="E2:147_4_2"/><span class="lab"><font color="Green" title="E128">A112</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">and </span><br/><a NAME="E3:147_4_2"/><span class="lab"><font color="Green" title="E129">A113</font></span>: 
<font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c9">L</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">L</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_4_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="147_4_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:147_4_2_1_1"/>
( <font color="Maroon" title="c10">L</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> or <font color="Maroon" title="c10">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:147_4_2"><span class="lab"><font color="Green" title="E128">A112</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="147_4_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:147_4_2_1_1_1"/>
<font color="Maroon" title="c10">L</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:147_4_2_1_1_1"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:147_4"><span class="lab"><font color="Green" title="E125">A18</font></span></a>, <a class="txt" href="#E1:147_4_2"><span class="lab"><font color="Green" title="E127">A111</font></span></a>, <a class="txt" href="#E3:147_4_2"><span class="lab"><font color="Green" title="E129">A113</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="147_4_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:147_4_2_1_1_2"/>
<font color="Maroon" title="c10">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:147_4_2_1_1_2"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:147_4"><span class="lab"><font color="Green" title="E126">A19</font></span></a>, <a class="txt" href="#E1:147_4_2"><span class="lab"><font color="Green" title="E127">A111</font></span></a>, <a class="txt" href="#E3:147_4_2"><span class="lab"><font color="Green" title="E129">A113</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:147_4_2"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c7">i</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c8">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:147_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:147"/><span class="lab"><font color="Green" title="E125">A114</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T3" title="MODELC_2:th.3">MODELC_2:3</a></span>;<br/>
<a NAME="E12:147"/><span class="lab"><font color="Green" title="E126">A115</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E11:147"><span class="lab"><font color="Green" title="E125">A114</font></span></a>, <a class="txt" href="#E10:147"><span class="lab"><font color="Green" title="E124">A17</font></span></a>);<br/></span>
<span class="kw">set </span><font color="Maroon" title="c6">n</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">v</font>;<br/>
<a NAME="E13:147"/><span class="lab"><font color="Green" title="E127">A116</font></span>: 
( <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">v</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">Run01</font> = <font color="Maroon">H<sub>2</sub></font>(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1) as   <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:147"><span class="lab"><font color="Green" title="E121">A11</font></span></a>, <a class="ref" href="modelc_3.html#D20" target="_self" title="MODELC_3:def.20">Def20</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">Run00</font> = <font color="Maroon">H<sub>2</sub></font>( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ) as   <a href="modelc_3.html#V1" title="MODELC_3:attr.1">strict</a>   <a href="modelc_3.html#V3" title="MODELC_3:attr.3">elementary</a>   <a href="modelc_3.html#L1" title="MODELC_3:struct.1">LTLnode</a> of <font color="Maroon" title="c2">v</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:147"><span class="lab"><font color="Green" title="E121">A11</font></span></a>, <a class="ref" href="modelc_3.html#D20" target="_self" title="MODELC_3:def.20">Def20</a></span>;<br/>
<a NAME="E16:147"/><span class="lab"><font color="Green" title="E128">A117</font></span>: 
the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <font color="Maroon" title="c8">Run00</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c7">Run01</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:147"><span class="lab"><font color="Green" title="E120">A4</font></span></a>, <a class="ref" href="modelc_3.html#T37" target="_self" title="MODELC_3:th.37">Th37</a></span>;<br/>
<font color="Maroon">H<sub>2</sub></font>( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ) = 
 <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K11" title="MODELC_3:func.11">init</a> <font color="Maroon" title="c2">v</font></span>)</span>,<font color="Maroon" title="c2">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:147"><span class="lab"><font color="Green" title="E117">A1</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>
<br/>.= 
 <a href="modelc_3.html#K11" title="MODELC_3:func.11">init</a> <font color="Maroon" title="c2">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#D16" target="_self" title="MODELC_3:def.16">Def16</a></span>
;<br/>
<a NAME="E18:147"/><span class="kw">then </span>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:147"><span class="lab"><font color="Green" title="E128">A117</font></span></a>, <a class="txt" href="#E12:147"><span class="lab"><font color="Green" title="E126">A115</font></span></a>, <a class="txt" href="#E13:147"><span class="lab"><font color="Green" title="E127">A116</font></span></a>, <a class="ref" href="modelc_2.html#T27" title="MODELC_2:th.27">MODELC_2:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:147"/>
<font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T79" title="MODELC_2:th.79">MODELC_2:79</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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