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

<b>let </b><font color="Maroon" title="c1">v</font> be   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s2</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="c1">v</font>  st <font color="Olive" title="b1">s2</font> <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Olive" title="b2">s1</font> holds <br/>the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <font color="Olive" title="b2">s1</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="Olive" title="b1">s2</font></span><br/><b>let </b><font color="Maroon" title="c2">s2</font>, <font color="Maroon" title="c3">s1</font> be  <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="c1">v</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">s2</font> <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon" title="c3">s1</font> implies the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <font color="Maroon" title="c3">s1</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="c2">s2</font> )</span><br/>





<b>assume </b><a NAME="E1:92"/>
<font color="Maroon" title="c2">s2</font> <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon" title="c3">s1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <font color="Maroon" title="c3">s1</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="c2">s2</font></span><br/>

<b>then consider </b><font color="Maroon" title="c4">L</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E3:92"/><i><font color="Green" title="E81">A1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">L</font> &amp; <font color="Maroon" title="c4">L</font> <a href="modelc_3.html#R5" title="MODELC_3:pred.5">is_Finseq_for</a> <font color="Maroon" title="c1">v</font> )
 <b>and </b><br/><a NAME="E4:92"/><i><font color="Green" title="E82">A2</font></i>: 
( <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c3">s1</font> &amp; <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s2</font> )
 <b>by </b><i><a class="ref" href="modelc_3.html#D20" target="_self" title="MODELC_3:def.20">Def215</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">N1</font> =  <a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c3">s1</font>;<br/>
<b>set </b><font color="Maroon" title="c6">n</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">L</font>;<br/>
<a NAME="E5:92"/><i><font color="Green" title="E83">A3</font></i>: 
 <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c3">s1</font>
 
<b>by </b><i><a class="ref" href="modelc_3.html#D16" target="_self" title="MODELC_3:def.16">defCastNode01</a>, <a class="txt" href="#E4:92"><i><font color="Green" title="E82">A2</font></i></a></i>;<br/>
<a NAME="E6:92"/><i><font color="Green" title="E84">A4</font></i>: 
 <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">L</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s2</font>
 
<b>by </b><i><a class="ref" href="modelc_3.html#D16" target="_self" title="MODELC_3:def.16">defCastNode01</a>, <a class="txt" href="#E4:92"><i><font color="Green" title="E82">A2</font></i></a></i>;<br/>
<a NAME="E7:92"/><i><font color="Green" title="E85">A5</font></i>: 
the <a href="modelc_3.html#U2" title="MODELC_3:sel.2">LTLnew</a> of <font color="Maroon" title="c2">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_3.html#K7" title="MODELC_3:func.7">{}</a> <font color="Maroon" title="c1">v</font>
 
<b>by </b><i><a class="ref" href="modelc_3.html#D11" target="_self" title="MODELC_3:def.11">Defelementary</a></i>;<br/>
<a NAME="E8:92"/>
the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <font color="Maroon" title="c3">s1</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="c2">s2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">x</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="c3">s1</font> or <font color="Maroon" title="c7">x</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="c2">s2</font> )</span><br/>

<b>assume </b><a NAME="E1:92_1"/><i><font color="Green" title="E86">B1</font></i>: 
<font color="Maroon" title="c7">x</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="c3">s1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</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="c2">s2</font></span><br/>

<a NAME="E2:92_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K1" title="MODELC_3:func.1">Subformulae</a> <font color="Maroon" title="c1">v</font>
 
<b>by </b><i><a class="txt" href="#E1:92_1"><i><font color="Green" title="E86">B1</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</font> as   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> <b>by </b><i><a class="ref" href="modelc_2.html#T1" title="MODELC_2:th.1">MODELC_2:1</a></i>;<br/>
<a NAME="E4:92_1"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">L</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">L</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="c4">L</font> )
 
<b>by </b><i><a class="txt" href="#E7:92"><i><font color="Green" title="E85">A5</font></i></a>, <a class="txt" href="#E1:92_1"><i><font color="Green" title="E86">B1</font></i></a>, <a class="txt" href="#E4:92"><i><font color="Green" title="E82">A2</font></i></a>, <a class="txt" href="#E3:92"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:92_1"/><i><font color="Green" title="E87">B3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">L</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U2" title="MODELC_3:sel.2">LTLnew</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>,<font color="Maroon" title="c1">v</font></span>)</span> &amp; not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="modelc_3.html#U2" title="MODELC_3:sel.2">LTLnew</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">v</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E3:92"><i><font color="Green" title="E81">A1</font></i></a>, <a class="txt" href="#E1:92_1"><i><font color="Green" title="E86">B1</font></i></a>, <a class="txt" href="#E5:92"><i><font color="Green" title="E83">A3</font></i></a>, <a class="txt" href="#E7:92"><i><font color="Green" title="E85">A5</font></i></a>, <a class="txt" href="#E6:92"><i><font color="Green" title="E84">A4</font></i></a>, <a class="ref" href="modelc_3.html#T29" target="_self" title="MODELC_3:th.29">ThSucc05</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">N1</font>, <font color="Maroon" title="c11">N2</font> being  <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="c1">v</font><b> such that </b><br/><a NAME="E8:92_1"/><i><font color="Green" title="E88">B301</font></i>: 
( <font color="Maroon" title="c10">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Maroon" title="c11">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c11">N2</font> <a href="modelc_3.html#R4" title="MODELC_3:pred.4">is_succ_of</a> <font color="Maroon" title="c10">N1</font> )
 <b>by </b><i><a class="txt" href="#E6:92_1"><i><font color="Green" title="E87">B3</font></i></a>, <a class="txt" href="#E3:92"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="modelc_3.html#D19" target="_self" title="MODELC_3:def.19">DefFinseq</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">m1</font> = <font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<a NAME="E9:92_1"/><i><font color="Green" title="E89">B302</font></i>: 
<font color="Maroon" title="c10">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>,<font color="Maroon" title="c1">v</font>
 
<b>by </b><i><a class="txt" href="#E8:92_1"><i><font color="Green" title="E88">B301</font></i></a>, <a class="ref" href="modelc_3.html#D16" target="_self" title="MODELC_3:def.16">defCastNode01</a></i>;<br/>
<a NAME="E10:92_1"/><i><font color="Green" title="E90">B303</font></i>: 
<font color="Maroon" title="c11">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">v</font>
 
<b>by </b><i><a class="txt" href="#E8:92_1"><i><font color="Green" title="E88">B301</font></i></a>, <a class="ref" href="modelc_3.html#D16" target="_self" title="MODELC_3:def.16">defCastNode01</a></i>;<br/>
<a NAME="E11:92_1"/><b>then </b><i><font color="Green" title="E91">B4</font></i>: 
<font color="Maroon" title="c8">x</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="c11">N2</font>
 
<b>by </b><i><a class="txt" href="#E6:92_1"><i><font color="Green" title="E87">B3</font></i></a>, <a class="txt" href="#E8:92_1"><i><font color="Green" title="E88">B301</font></i></a>, <a class="txt" href="#E9:92_1"><i><font color="Green" title="E89">B302</font></i></a>, <a class="ref" href="modelc_3.html#T30" target="_self" title="MODELC_3:th.30">ThSucc06</a>, <a class="ref" href="modelc_3.html#T32" target="_self" title="MODELC_3:th.32">ThSucc08</a></i>;<br/>
<a NAME="E12:92_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">L</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">L</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="c4">L</font> )
 
<b>by </b><i><a class="txt" href="#E6:92_1"><i><font color="Green" title="E87">B3</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E13:92_1"/><b>then </b>
the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c11">N2</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 <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">L</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">v</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:92"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="modelc_3.html#T31" target="_self" title="MODELC_3:th.31">ThSucc07</a>, <a class="txt" href="#E10:92_1"><i><font color="Green" title="E90">B303</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:92_1"/>
<font color="Maroon" title="c7">x</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="c2">s2</font>
 <b>by </b><i><a class="txt" href="#E6:92"><i><font color="Green" title="E84">A4</font></i></a>, <a class="txt" href="#E11:92_1"><i><font color="Green" title="E91">B4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:92"/>
the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <font color="Maroon" title="c3">s1</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="c2">s2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
