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

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E1:181"/><span class="lab"><font color="Green" title="E145">A1</font></span>: 
(  <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="181_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:181_1"/><span class="lab"><font color="Green" title="E146">A2</font></span>: 
 <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:181_1"/><span class="lab"><font color="Green" title="E147">A3</font></span>: 
for <font color="Olive" title="b1">u</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">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="181_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">u</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="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/>

<span class="kw">assume </span><a NAME="E1:181_1_1"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:181_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">r</font>,1 <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c3">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:181_1"><span class="lab"><font color="Green" title="E146">A2</font></span></a>, <a class="ref" href="modelc_2.html#D66" target="_self" title="MODELC_2:def.66">Def66</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:181_1_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T67" target="_self" title="MODELC_2:th.67">Th67</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="E3:181_1"/>
for <font color="Olive" title="b1">H</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">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="181_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">H</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="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/>

<span class="kw">assume </span><a NAME="E1:181_1_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<a NAME="E2:181_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <br/>( <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp;  ex <font color="Olive" title="b2">u</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <br/>( <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font> &amp; <font color="Olive" title="b1">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="b2">u</font> ) )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:181_1_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:181_1"><span class="lab"><font color="Green" title="E147">A3</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="E4:181_1"/>
<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 class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#D66" target="_self" title="MODELC_2:def.66">Def66</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="E2:181"/>
( <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="181_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:181_2"/><span class="lab"><font color="Green" title="E146">A4</font></span>: 
<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:()"><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">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="E2:181_2"/>
for <font color="Olive" title="b1">H</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">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="181_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">H</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="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/>

<span class="kw">set </span><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/>
<span class="kw">assume </span><a NAME="E1:181_2_1"/>
<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:()"><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">r</font>,1 <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c3">H</font></span><br/>

<a NAME="E2:181_2_1"/><span class="kw">then </span>
 <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>
 
;<br/>
<a NAME="E3:181_2_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:181_2"><span class="lab"><font color="Green" title="E146">A4</font></span></a>, <a class="ref" href="modelc_2.html#D66" target="_self" title="MODELC_2:def.66">Def66</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:181_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">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T67" target="_self" title="MODELC_2:th.67">Th67</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="E3:181_2"/>
 <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 class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#D66" target="_self" title="MODELC_2:def.66">Def66</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="E3:181"/>
( <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:181"><span class="lab"><font color="Green" title="E145">A1</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>
