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

<span class="kw">let </span><font color="Maroon" title="c1">S</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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"> for <font color="Olive" title="b1">BASSIGN</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K32" title="MODELC_1:func.32">ModelSP</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <font color="Maroon" title="c1">S</font></span>)</span></span>)</span><br/> for <font color="Olive" title="b2">t</font> being   <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> <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b3">f</font>, <font color="Olive" title="b4">g</font> being  <a href="modelc_2.html#NM2" title="MODELC_2:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_2.html#K45" title="MODELC_2:func.45">LTLModel</a> <font color="Maroon" title="c1">S</font>,<font color="Olive" title="b1">BASSIGN</font></span>)</span> holds <br/> ( <font color="Olive" title="b2">t</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Olive" title="b3">f</font> <a href="modelc_2.html#K26" title="MODELC_2:func.26">'R'</a> <font color="Olive" title="b4">g</font> iff for <font color="Olive" title="b5">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b6">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b6">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">m</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Olive" title="b2">t</font>,<font color="Olive" title="b6">j</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Olive" title="b3">f</font> ) holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Olive" title="b2">t</font>,<font color="Olive" title="b5">m</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Olive" title="b4">g</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">BASSIGN</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K32" title="MODELC_1:func.32">ModelSP</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <font color="Maroon" title="c1">S</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"> for <font color="Olive" title="b1">t</font> being   <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> <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being  <a href="modelc_2.html#NM2" title="MODELC_2:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_2.html#K45" title="MODELC_2:func.45">LTLModel</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">BASSIGN</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">t</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Olive" title="b2">f</font> <a href="modelc_2.html#K26" title="MODELC_2:func.26">'R'</a> <font color="Olive" title="b3">g</font> iff for <font color="Olive" title="b4">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b5">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">m</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Olive" title="b1">t</font>,<font color="Olive" title="b5">j</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Olive" title="b2">f</font> ) holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Olive" title="b1">t</font>,<font color="Olive" title="b4">m</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Olive" title="b3">g</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">t</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> <font color="Maroon" title="c1">S</font>; <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>, <font color="Olive" title="b2">g</font> being  <a href="modelc_2.html#NM2" title="MODELC_2:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_2.html#K45" title="MODELC_2:func.45">LTLModel</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">BASSIGN</font></span>)</span> holds <br/> ( <font color="Maroon" title="c3">t</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Olive" title="b1">f</font> <a href="modelc_2.html#K26" title="MODELC_2:func.26">'R'</a> <font color="Olive" title="b2">g</font> iff for <font color="Olive" title="b3">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b4">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">m</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b4">j</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Olive" title="b1">f</font> ) holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b3">m</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Olive" title="b2">g</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be   <a href="modelc_2.html#NM2" title="MODELC_2:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_2.html#K45" title="MODELC_2:func.45">LTLModel</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">BASSIGN</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="c3">t</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_2.html#K26" title="MODELC_2:func.26">'R'</a> <font color="Maroon" title="c5">g</font> iff for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b2">j</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c4">f</font> ) holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">m</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c5">g</font> )</span><br/>









<a NAME="E1:156"/><span class="lab"><font color="Green" title="E129">A1</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 ( for <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b2">j</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c4">f</font> ) holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">m</font> <a href="modelc_2.html#NR7" title="MODELC_2:NR.7">|/=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c5">g</font> ) implies for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b2">j</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c4">f</font> ) holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">m</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c5">g</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="156_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:156_1"/><span class="lab"><font color="Green" title="E130">A2</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 ( for <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b2">j</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c4">f</font> ) holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">m</font> <a href="modelc_2.html#NR7" title="MODELC_2:NR.7">|/=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c5">g</font>
 ; <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">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b2">j</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c4">f</font> ) holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">m</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c5">g</font></span><br/>

<a NAME="E2:156_1"/>
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b2">j</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c4">f</font> ) holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">m</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c5">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="156_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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"> ( ( 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="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">j</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c4">f</font> ) implies  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c6">m</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c5">g</font> )</span><br/>

<a NAME="E1:156_1_1"/>
( ( 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="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">j</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c4">f</font> ) implies  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c6">m</font> <a href="modelc_2.html#NR7" title="MODELC_2:NR.7">|/=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c5">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:156_1"><span class="lab"><font color="Green" title="E130">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:156_1_1"/>
( ( 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="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">j</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c4">f</font> ) implies  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c6">m</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c5">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T57" target="_self" title="MODELC_2:th.57">Th57</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="E3:156_1"/>
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b2">j</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c4">f</font> ) holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">m</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c5">g</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="E2:156"/><span class="lab"><font color="Green" title="E130">A3</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 ( for <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b2">j</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c4">f</font> ) holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">m</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c5">g</font> ) implies for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b2">j</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c4">f</font> ) holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">m</font> <a href="modelc_2.html#NR7" title="MODELC_2:NR.7">|/=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c5">g</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="156_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:156_2"/><span class="lab"><font color="Green" title="E131">A4</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 ( for <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b2">j</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c4">f</font> ) holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">m</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c5">g</font>
 ; <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">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b2">j</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c4">f</font> ) holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">m</font> <a href="modelc_2.html#NR7" title="MODELC_2:NR.7">|/=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c5">g</font></span><br/>

<a NAME="E2:156_2"/>
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b2">j</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c4">f</font> ) holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">m</font> <a href="modelc_2.html#NR7" title="MODELC_2:NR.7">|/=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c5">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="156_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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"> ( ( 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="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">j</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c4">f</font> ) implies  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c6">m</font> <a href="modelc_2.html#NR7" title="MODELC_2:NR.7">|/=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c5">g</font> )</span><br/>

<a NAME="E1:156_2_1"/>
( ( 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="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">j</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c4">f</font> ) implies  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c6">m</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c5">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:156_2"><span class="lab"><font color="Green" title="E131">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:156_2_1"/>
( ( 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="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">j</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c4">f</font> ) implies  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c6">m</font> <a href="modelc_2.html#NR7" title="MODELC_2:NR.7">|/=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c5">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T57" target="_self" title="MODELC_2:th.57">Th57</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="E3:156_2"/>
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b2">j</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c4">f</font> ) holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">m</font> <a href="modelc_2.html#NR7" title="MODELC_2:NR.7">|/=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c5">g</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="E3:156"/>
( <font color="Maroon" title="c3">t</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_2.html#K26" title="MODELC_2:func.26">'R'</a> <font color="Maroon" title="c5">g</font> iff <font color="Maroon" title="c3">t</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="modelc_2.html#K25" title="MODELC_2:func.25">'U'</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#D57" target="_self" title="MODELC_2:def.57">Def57</a></span>;<br/>
<a NAME="E4:156"/><span class="kw">then </span>
( <font color="Maroon" title="c3">t</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_2.html#K26" title="MODELC_2:func.26">'R'</a> <font color="Maroon" title="c5">g</font> iff not <font color="Maroon" title="c3">t</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="modelc_2.html#K25" title="MODELC_2:func.25">'U'</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c5">g</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T57" target="_self" title="MODELC_2:th.57">Th57</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:156"/>
( <font color="Maroon" title="c3">t</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_2.html#K26" title="MODELC_2:func.26">'R'</a> <font color="Maroon" title="c5">g</font> iff for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b2">j</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a>  <a href="modelc_2.html#K21" title="MODELC_2:func.21">'not'</a> <font color="Maroon" title="c4">f</font> ) holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c3">t</font>,<font color="Olive" title="b1">m</font> <a href="modelc_2.html#R6" title="MODELC_2:pred.6">|=</a> <font color="Maroon" title="c5">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:156"><span class="lab"><font color="Green" title="E129">A1</font></span></a>, <a class="txt" href="#E2:156"><span class="lab"><font color="Green" title="E130">A3</font></span></a>, <a class="ref" href="modelc_2.html#T60" target="_self" title="MODELC_2:th.60">Th60</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>
