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

<b>let </b><font color="Maroon" title="c1">r</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">W</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="modelc_2.html#K9" title="MODELC_2:func.9">LTL_WFF</a>  holds <br/> ( <font color="Maroon" title="c1">r</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_2.html#K51" title="MODELC_2:func.51">'X'</a> <font color="Olive" title="b1">W</font> iff  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">r</font>,1 <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a> <font color="Olive" title="b1">W</font> )</span><br/><b>let </b><font color="Maroon" title="c2">W</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="modelc_2.html#K9" title="MODELC_2:func.9">LTL_WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">r</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_2.html#K51" title="MODELC_2:func.51">'X'</a> <font color="Maroon" title="c2">W</font> iff  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">r</font>,1 <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a> <font color="Maroon" title="c2">W</font> )</span><br/>



<a NAME="E1:182"/><i><font color="Green" title="E145">A1</font></i>: 
( <font color="Maroon" title="c1">r</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_2.html#K51" title="MODELC_2:func.51">'X'</a> <font color="Maroon" title="c2">W</font> implies  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">r</font>,1 <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a> <font color="Maroon" title="c2">W</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="182_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:182_1"/><i><font color="Green" title="E146">B1</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_2.html#K51" title="MODELC_2:func.51">'X'</a> <font color="Maroon" title="c2">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">r</font>,1 <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a> <font color="Maroon" title="c2">W</font></span><br/>

<a NAME="E2:182_1"/>
for <font color="Olive" title="b1">H</font> being  <a href="modelc_2.html#NM2" title="MODELC_2:NM.2">LTL-formula</a>  st <font color="Olive" title="b1">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">r</font>,1 <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Olive" title="b1">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="182_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">H</font> be   <a href="modelc_2.html#NM2" title="MODELC_2:NM.2">LTL-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font> implies  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">r</font>,1 <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c3">H</font> )</span><br/>

<b>assume </b><a NAME="E1:182_1_1"/><i><font color="Green" title="E147">C1</font></i>: 
<font color="Maroon" title="c3">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">r</font>,1 <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c3">H</font></span><br/>

<b>set </b><font color="Maroon" title="c4">u</font> =  <a href="modelc_2.html#K6" title="MODELC_2:func.6">'X'</a> <font color="Maroon" title="c3">H</font>;<br/>
<a NAME="E2:182_1_1"/>
 <a href="modelc_2.html#K6" title="MODELC_2:func.6">'X'</a> <font color="Maroon" title="c3">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_2.html#K51" title="MODELC_2:func.51">'X'</a> <font color="Maroon" title="c2">W</font>
 
<b>by </b><i><a class="txt" href="modelc_2.html#E1:182_1_1"><i><font color="Green" title="E147">C1</font></i></a></i>;<br/>
<a NAME="E3:182_1_1"/><b>then </b>
<font color="Maroon" title="c1">r</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> <font color="Maroon" title="c3">H</font>
 
<b>by </b><i><a class="txt" href="modelc_2.html#E1:182_1"><i><font color="Green" title="E146">B1</font></i></a>, <a class="ref" href="modelc_2.html#D66" target="_self" title="MODELC_2:def.66">Def66</a></i>;<br/>
<b>hence </b><a NAME="E4:182_1_1"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">r</font>,1 <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c3">H</font>
 <b>by </b><i><a class="ref" href="modelc_2.html#T67" target="_self" title="MODELC_2:th.67">Th22</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="E3:182_1"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">r</font>,1 <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a> <font color="Maroon" title="c2">W</font>
 <b>by </b><i><a class="ref" href="modelc_2.html#D66" target="_self" title="MODELC_2:def.66">Def66</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>
<a NAME="E2:182"/>
(  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">r</font>,1 <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a> <font color="Maroon" title="c2">W</font> implies <font color="Maroon" title="c1">r</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_2.html#K51" title="MODELC_2:func.51">'X'</a> <font color="Maroon" title="c2">W</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="182_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:182_2"/><i><font color="Green" title="E146">B1</font></i>: 
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">r</font>,1 <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a> <font color="Maroon" title="c2">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">r</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_2.html#K51" title="MODELC_2:func.51">'X'</a> <font color="Maroon" title="c2">W</font></span><br/>

<a NAME="E2:182_2"/><i><font color="Green" title="E147">B2</font></i>: 
for <font color="Olive" title="b1">u</font> being  <a href="modelc_2.html#NM2" title="MODELC_2:NM.2">LTL-formula</a>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font> holds <br/><font color="Maroon" title="c1">r</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> <font color="Olive" title="b1">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="182_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">u</font> be   <a href="modelc_2.html#NM2" title="MODELC_2:NM.2">LTL-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font> implies <font color="Maroon" title="c1">r</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> <font color="Maroon" title="c3">u</font> )</span><br/>

<b>assume </b><a NAME="E1:182_2_1"/><i><font color="Green" title="E148">C1</font></i>: 
<font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">r</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> <font color="Maroon" title="c3">u</font></span><br/>

<a NAME="E2:182_2_1"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">r</font>,1 <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c3">u</font>
 
<b>by </b><i><a class="txt" href="modelc_2.html#E1:182_2"><i><font color="Green" title="E146">B1</font></i></a>, <a class="ref" href="modelc_2.html#D66" target="_self" title="MODELC_2:def.66">Def66</a>, <a class="txt" href="modelc_2.html#E1:182_2_1"><i><font color="Green" title="E148">C1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:182_2_1"/>
<font color="Maroon" title="c1">r</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> <font color="Maroon" title="c3">u</font>
 <b>by </b><i><a class="ref" href="modelc_2.html#T67" target="_self" title="MODELC_2:th.67">Th22</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>
<a NAME="E3:182_2"/>
for <font color="Olive" title="b1">H</font> being  <a href="modelc_2.html#NM2" title="MODELC_2:NM.2">LTL-formula</a>  st <font color="Olive" title="b1">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_2.html#K51" title="MODELC_2:func.51">'X'</a> <font color="Maroon" title="c2">W</font> holds <br/><font color="Maroon" title="c1">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Olive" title="b1">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="182_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">H</font> be   <a href="modelc_2.html#NM2" title="MODELC_2:NM.2">LTL-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_2.html#K51" title="MODELC_2:func.51">'X'</a> <font color="Maroon" title="c2">W</font> implies <font color="Maroon" title="c1">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c3">H</font> )</span><br/>

<b>assume </b><a NAME="E1:182_2_2"/>
<font color="Maroon" title="c3">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_2.html#K51" title="MODELC_2:func.51">'X'</a> <font color="Maroon" title="c2">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c3">H</font></span><br/>

<b>then consider </b><font color="Maroon" title="c4">x</font> being   <a href="modelc_2.html#NM2" title="MODELC_2:NM.2">LTL-formula</a><b> such that </b><br/><a NAME="E3:182_2_2"/><i><font color="Green" title="E148">C1</font></i>: 
<font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <b>and </b><br/><a NAME="E4:182_2_2"/><i><font color="Green" title="E149">C2</font></i>: 
 ex <font color="Olive" title="b1">u</font> being  <a href="modelc_2.html#NM2" title="MODELC_2:NM.2">LTL-formula</a> st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K6" title="MODELC_2:func.6">'X'</a> <font color="Olive" title="b1">u</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c5">u</font> being   <a href="modelc_2.html#NM2" title="MODELC_2:NM.2">LTL-formula</a><b> such that </b><br/><a NAME="E6:182_2_2"/><i><font color="Green" title="E150">C3</font></i>: 
( <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K6" title="MODELC_2:func.6">'X'</a> <font color="Maroon" title="c5">u</font> )
 <b>by </b><i><a class="txt" href="modelc_2.html#E4:182_2_2"><i><font color="Green" title="E149">C2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:182_2_2"/>
<font color="Maroon" title="c1">r</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c3">H</font>
 <b>by </b><i><a class="txt" href="modelc_2.html#E3:182_2_2"><i><font color="Green" title="E148">C1</font></i></a>, <a class="txt" href="modelc_2.html#E2:182_2"><i><font color="Green" title="E147">B2</font></i></a>, <a class="txt" href="modelc_2.html#E6:182_2_2"><i><font color="Green" title="E150">C3</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="E4:182_2"/>
<font color="Maroon" title="c1">r</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_2.html#K51" title="MODELC_2:func.51">'X'</a> <font color="Maroon" title="c2">W</font>
 <b>by </b><i><a class="ref" href="modelc_2.html#D66" target="_self" title="MODELC_2:def.66">Def66</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="E3:182"/>
( <font color="Maroon" title="c1">r</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_2.html#K51" title="MODELC_2:func.51">'X'</a> <font color="Maroon" title="c2">W</font> iff  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">r</font>,1 <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a> <font color="Maroon" title="c2">W</font> )
 <b>by </b><i><a class="txt" href="modelc_2.html#E1:182"><i><font color="Green" title="E145">A1</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>
