<?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#K25" title="MODELC_2:func.25">Inf_seq</a> <a href="modelc_2.html#K43" title="MODELC_2:func.43">AtomicFamily</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K48" title="MODELC_2:func.48">'X'</a> <font color="Olive" title="b1">W</font> iff  <a href="modelc_2.html#K29" title="MODELC_2:func.29">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="firebrick">::  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#K48" title="MODELC_2:func.48">'X'</a> <font color="Maroon" title="c2">W</font> iff  <a href="modelc_2.html#K29" title="MODELC_2:func.29">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="E1:180"/><span class="lab"><font color="Green" title="E138">A1</font></span>: 
(  <a href="modelc_2.html#K29" title="MODELC_2:func.29">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#K48" title="MODELC_2:func.48">'X'</a> <font color="Maroon" title="c2">W</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="180_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:180_1"/><span class="lab"><font color="Green" title="E139">A2</font></span>: 
 <a href="modelc_2.html#K29" title="MODELC_2:func.29">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="firebrick">::  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#K48" title="MODELC_2:func.48">'X'</a> <font color="Maroon" title="c2">W</font></span><br/>

<a NAME="E2:180_1"/><span class="lab"><font color="Green" title="E140">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="180_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="firebrick">::  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="E1:180_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="firebrick">::  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="E2:180_1_1"/><span class="kw">then </span>
 <a href="modelc_2.html#K29" title="MODELC_2:func.29">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="#E1:180_1"><span class="lab"><font color="Green" title="E139">A2</font></span></a>, <a class="ref" href="modelc_2.html#D64" target="_self" title="MODELC_2:def.64">Def68</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:180_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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:180_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#K48" title="MODELC_2:func.48">'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="180_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="firebrick">::  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#K48" title="MODELC_2:func.48">'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="E1:180_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#K48" title="MODELC_2:func.48">'X'</a> <font color="Maroon" title="c2">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E2:180_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="E3:180_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="#E2:180_1"><span class="lab"><font color="Green" title="E140">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:180_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#K48" title="MODELC_2:func.48">'X'</a> <font color="Maroon" title="c2">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#D64" target="_self" title="MODELC_2:def.64">Def68</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:180"/>
( <font color="Maroon" title="c1">r</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_2.html#K48" title="MODELC_2:func.48">'X'</a> <font color="Maroon" title="c2">W</font> implies  <a href="modelc_2.html#K29" title="MODELC_2:func.29">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="180_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:180_2"/><span class="lab"><font color="Green" title="E139">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#K48" title="MODELC_2:func.48">'X'</a> <font color="Maroon" title="c2">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="modelc_2.html#K29" title="MODELC_2:func.29">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="E2:180_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#K29" title="MODELC_2:func.29">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="180_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="firebrick">::  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#K29" title="MODELC_2:func.29">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="E1:180_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="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="modelc_2.html#K29" title="MODELC_2:func.29">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="E2:180_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#K48" title="MODELC_2:func.48">'X'</a> <font color="Maroon" title="c2">W</font>
 
;<br/>
<a NAME="E3:180_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="#E1:180_2"><span class="lab"><font color="Green" title="E139">A4</font></span></a>, <a class="ref" href="modelc_2.html#D64" target="_self" title="MODELC_2:def.64">Def68</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:180_2_1"/>
 <a href="modelc_2.html#K29" title="MODELC_2:func.29">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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:180_2"/>
 <a href="modelc_2.html#K29" title="MODELC_2:func.29">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#D64" target="_self" title="MODELC_2:def.64">Def68</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:180"/>
( <font color="Maroon" title="c1">r</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_2.html#K48" title="MODELC_2:func.48">'X'</a> <font color="Maroon" title="c2">W</font> iff  <a href="modelc_2.html#K29" title="MODELC_2:func.29">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="#E1:180"><span class="lab"><font color="Green" title="E138">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
