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

<span class="kw">let </span><font color="Maroon" title="c1">w</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">v</font> being  <a href="modelc_3.html#V5" title="MODELC_3:attr.5">neg-inner-most</a>  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a>  st <font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Olive" title="b1">v</font> holds <br/><font color="Maroon" title="c1">w</font> <a href="modelc_3.html#R9" title="MODELC_3:pred.9">is-accepted-by</a>  <a href="modelc_3.html#K34" title="MODELC_3:func.34">BAutomaton</a> <font color="Olive" title="b1">v</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">v</font> be   <a href="modelc_3.html#V5" title="MODELC_3:attr.5">neg-inner-most</a>  <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="c1">w</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">v</font> implies <font color="Maroon" title="c1">w</font> <a href="modelc_3.html#R9" title="MODELC_3:pred.9">is-accepted-by</a>  <a href="modelc_3.html#K34" title="MODELC_3:func.34">BAutomaton</a> <font color="Maroon" title="c2">v</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">B</font> =  <a href="modelc_3.html#K34" title="MODELC_3:func.34">BAutomaton</a> <font color="Maroon" title="c2">v</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">LS</font> =  <a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">LT</font> =  <a href="modelc_3.html#K30" title="MODELC_3:func.30">Tran_LTL</a> <font color="Maroon" title="c2">v</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">IS</font> =  <a href="modelc_3.html#K31" title="MODELC_3:func.31">InitS_LTL</a> <font color="Maroon" title="c2">v</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">FS</font> =  <a href="modelc_3.html#K33" title="MODELC_3:func.33">FinalS_LTL</a> <font color="Maroon" title="c2">v</font>;<br/>
<span class="kw">assume </span><a NAME="E1:158"/><span class="lab"><font color="Green" title="E127">A1</font></span>: 
<font color="Maroon" title="c1">w</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c2">v</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">w</font> <a href="modelc_3.html#R9" title="MODELC_3:pred.9">is-accepted-by</a>  <a href="modelc_3.html#K34" title="MODELC_3:func.34">BAutomaton</a> <font color="Maroon" title="c2">v</font></span><br/>

<a NAME="E2:158"/>
 ex <font color="Olive" title="b1">run</font> being  <a href="nat_1.html#NM2" title="NAT_1:NM.2">sequence</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font></span>)</span> st <br/>( <font color="Olive" title="b1">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K31" title="MODELC_3:func.31">InitS_LTL</a> <font color="Maroon" title="c2">v</font> &amp; ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c1">w</font>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K30" title="MODELC_3:func.30">Tran_LTL</a> <font color="Maroon" title="c2">v</font> &amp; ( for <font color="Olive" title="b3">FSet</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">FSet</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K33" title="MODELC_3:func.33">FinalS_LTL</a> <font color="Maroon" title="c2">v</font> holds <br/><span class="p1">{<span class="default"> <font color="Olive" title="b4">k</font> where <font color="Olive" title="b4">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">FSet</font> </span>}</span>  is   <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ) ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="158_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c8">chf</font> being    <a href="orders_1.html#M1" title="ORDERS_1:mode.1">Choice_Function</a> of  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K1" title="MODELC_3:func.1">Subformulae</a> <font color="Maroon" title="c2">v</font></span>)</span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font> = <span class="p1">(<span class="default"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c8">chf</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K2" title="MODELC_1:func.2">k_nat</a> $1</span>)</span>;<br/>
<a NAME="E2:158_1"/><span class="lab"><font color="Green" title="E128">A2</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font>
 
;<br/>
<a NAME="E3:158_1"/>
 ex <font color="Olive" title="b1">run</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font></span>)</span> st <br/>for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Olive" title="b1">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">x</font>)
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S2" title="FUNCT_2:sch.2">FUNCT_2:sch 2</a>(<a class="txt" href="#E2:158_1"><span class="lab"><font color="Green" title="E128">A2</font></span></a>);<br/></span>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">run</font> being   <a href="nat_1.html#NM2" title="NAT_1:NM.2">sequence</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:158_1"/><span class="lab"><font color="Green" title="E129">A3</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon" title="c9">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 ;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>  <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="c2">v</font> =  <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>,<font color="Maroon" title="c2">v</font>;<br/>
<a NAME="E6:158_1"/><span class="lab"><font color="Green" title="E130">A4</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( <font color="Maroon" title="c9">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is   <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="c2">v</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font>) is   <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="c2">v</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="158_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">n</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"> ( <font color="Maroon" title="c9">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> is   <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="c2">v</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c10">n</font>) is   <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="c2">v</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">n</font> = <font color="Maroon" title="c10">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">Rn</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">n</font>);<br/>
<a NAME="E2:158_1_1"/>
<font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">N</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="modelc_3.html#K22" title="MODELC_3:func.22">LTLNodes</a> <font color="Maroon" title="c2">v</font><span class="kw"> such that </span><br/><a NAME="E4:158_1_1"/><span class="lab"><font color="Green" title="E131">A5</font></span>: 
<font color="Maroon" title="c13">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font>
 <span class="kw">and </span><br/><a NAME="E5:158_1_1"/><span class="lab"><font color="Green" title="E132">A6</font></span>: 
<font color="Maroon" title="c13">N</font> is   <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="c2">v</font>
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">N</font> = <font color="Maroon" title="c13">N</font> as   <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="c2">v</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:158_1_1"><span class="lab"><font color="Green" title="E132">A6</font></span></a></span>;<br/>
the <a href="modelc_3.html#U2" title="MODELC_3:sel.2">LTLnew</a> of <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">n</font>) = 
the <a href="modelc_3.html#U2" title="MODELC_3:sel.2">LTLnew</a> of <font color="Maroon" title="c14">N</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:158_1_1"><span class="lab"><font color="Green" title="E131">A5</font></span></a>, <a class="ref" href="modelc_3.html#D16" target="_self" title="MODELC_3:def.16">Def16</a></span>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#D11" target="_self" title="MODELC_3:def.11">Def11</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:158_1_1"/>
( <font color="Maroon" title="c9">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> is   <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="c2">v</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c10">n</font>) is   <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="c2">v</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:158_1_1"><span class="lab"><font color="Green" title="E131">A5</font></span></a>, <a class="txt" href="#E5:158_1_1"><span class="lab"><font color="Green" title="E132">A6</font></span></a>, <a class="ref" href="modelc_3.html#D11" target="_self" title="MODELC_3:def.11">Def11</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="E7:158_1"/><span class="lab"><font color="Green" title="E131">A7</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c9">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c8">chf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="158_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">n</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"> <font color="Maroon" title="c9">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c8">chf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">n</font> = <font color="Maroon" title="c10">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E2:158_1_2"/>
<font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:158_1"><span class="lab"><font color="Green" title="E129">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:158_1_2"/>
<font color="Maroon" title="c9">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c8">chf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D2" title="MODELC_1:def.2">MODELC_1:def 2</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="E8:158_1"/><span class="lab"><font color="Green" title="E132">A8</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font>) &amp;  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Olive" title="b1">n</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_3.html#K14" title="MODELC_3:func.14">*</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="158_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">n</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"> ( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c10">n</font>) &amp;  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c10">n</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_3.html#K14" title="MODELC_3:func.14">*</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c11">Rn</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c10">n</font>);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">Rn</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c10">n</font>) as   <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="c2">v</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:158_1"><span class="lab"><font color="Green" title="E130">A4</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">n1</font> = <font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">w1</font> =  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c10">n</font>;<br/>
<a NAME="E2:158_1_3"/><span class="lab"><font color="Green" title="E133">A9</font></span>: 
<font color="Maroon" title="c9">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c8">chf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:158_1"><span class="lab"><font color="Green" title="E131">A7</font></span></a></span>;<br/>
<a NAME="E3:158_1_3"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c10">n</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"><span class="p2">(<span class="default"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c8">chf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<font color="Maroon" title="c2">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:158_1"><span class="lab"><font color="Green" title="E131">A7</font></span></a></span>;<br/>
<a NAME="E4:158_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">A10</font></span>: 
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c10">n</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_3.html#K14" title="MODELC_3:func.14">*</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K12" title="MODELC_3:func.12">'X'</a> <font color="Maroon" title="c12">Rn</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:158"><span class="lab"><font color="Green" title="E127">A1</font></span></a>, <a class="ref" href="modelc_3.html#T73" target="_self" title="MODELC_3:th.73">Th73</a></span>;<br/>
<font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c8">chf</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:158_1"><span class="lab"><font color="Green" title="E131">A7</font></span></a></span>
<br/>.= 
 <a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c10">n</font></span>)</span>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c8">chf</font>,<font color="Maroon" title="c12">Rn</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:158_1_3"><span class="lab"><font color="Green" title="E133">A9</font></span></a>, <a class="ref" href="modelc_3.html#D50" target="_self" title="MODELC_3:def.50">Def50</a></span>
;<br/>
<a NAME="E6:158_1_3"/><span class="kw">then </span>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_3.html#K36" title="MODELC_3:func.36">chosen_next</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c10">n</font></span>)</span>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c8">chf</font>,<font color="Maroon" title="c12">Rn</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#D16" target="_self" title="MODELC_3:def.16">Def16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:158_1_3"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c10">n</font>) &amp;  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c10">n</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_3.html#K14" title="MODELC_3:func.14">*</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:158_1_3"><span class="lab"><font color="Green" title="E134">A10</font></span></a>, <a class="ref" href="modelc_3.html#T69" target="_self" title="MODELC_3:th.69">Th69</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="E9:158_1"/><span class="lab"><font color="Green" title="E133">A11</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c1">w</font>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K29" title="MODELC_3:func.29">Label_</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="158_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">n</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"> <span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c1">w</font>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K29" title="MODELC_3:func.29">Label_</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">n</font> = <font color="Maroon" title="c10">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">Rn</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">n</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c13">Rn1</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1);<br/>
<span class="kw">set </span><font color="Maroon" title="c14">w1</font> =  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c11">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">X</font> = <span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c1">w</font>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">X</font> = <span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c1">w</font>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="modelc_2.html#K15" title="MODELC_2:func.15">atomic_LTL</a>  ;<br/>
<a NAME="E3:158_1_4"/>
 <a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c11">n</font></span>)</span>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="modelc_2.html#K30" title="MODELC_2:func.30">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c1">w</font>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span>,<font color="Maroon" title="c11">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T81" title="MODELC_2:th.81">MODELC_2:81</a></span>;<br/>
<a NAME="E4:158_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">A12</font></span>: 
<span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c11">n</font></span>)</span>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c1">w</font>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#D43" title="MODELC_2:def.43">MODELC_2:def 43</a></span>;<br/>
<a NAME="E5:158_1_4"/><span class="lab"><font color="Green" title="E135">A13</font></span>: 
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c11">n</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_3.html#K14" title="MODELC_3:func.14">*</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:158_1"><span class="lab"><font color="Green" title="E132">A8</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E136">A14</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="158_1_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:158_1_4_1"/>
not  <a href="modelc_3.html#K28" title="MODELC_3:func.28">Neg_atomic_LTL</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c16">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:158_1_4_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="modelc_3.html#K28" title="MODELC_3:func.28">Neg_atomic_LTL</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c16">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:158_1_4_1"/><span class="lab"><font color="Green" title="E137">A15</font></span>: 
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K28" title="MODELC_3:func.28">Neg_atomic_LTL</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c16">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><a NAME="E5:158_1_4_1"/>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K28" title="MODELC_3:func.28">Neg_atomic_LTL</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:158_1_4_1"><span class="lab"><font color="Green" title="E137">A15</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">x</font> being   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a><span class="kw"> such that </span><br/><a NAME="E7:158_1_4_1"/><span class="lab"><font color="Green" title="E138">A16</font></span>: 
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font> &amp; <font color="Maroon" title="c18">x</font> is  <a href="modelc_2.html#V2" title="MODELC_2:attr.2">atomic</a>  )
 <span class="kw">and </span><br/><a NAME="E8:158_1_4_1"/><span class="lab"><font color="Green" title="E139">A17</font></span>: 
 <a href="modelc_2.html#K3" title="MODELC_2:func.3">'not'</a> <font color="Maroon" title="c18">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">H<sub>2</sub></font>(<font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1)
 ;<br/><a NAME="E9:158_1_4_1"/>
 <a href="modelc_2.html#K3" title="MODELC_2:func.3">'not'</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K14" title="MODELC_3:func.14">*</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:158_1_4_1"><span class="lab"><font color="Green" title="E139">A17</font></span></a>, <a class="txt" href="modelc_3.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/><a NAME="E10:158_1_4_1"/><span class="kw">then </span>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c11">n</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K3" title="MODELC_2:func.3">'not'</a> <font color="Maroon" title="c18">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:158_1_4"><span class="lab"><font color="Green" title="E135">A13</font></span></a>, <a class="ref" href="modelc_2.html#D66" title="MODELC_2:def.66">MODELC_2:def 66</a></span>;<br/><a NAME="E11:158_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">A18</font></span>: 
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c11">n</font> <a href="modelc_2.html#NR9" title="MODELC_2:NR.9">|/=</a> <font color="Maroon" title="c18">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#T64" title="MODELC_2:th.64">MODELC_2:64</a></span>;<br/><a NAME="E12:158_1_4_1"/>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:158_1_4_1"><span class="lab"><font color="Green" title="E137">A15</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E13:158_1_4_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:158_1_4"><span class="lab"><font color="Green" title="E134">A12</font></span></a>, <a class="txt" href="#E7:158_1_4_1"><span class="lab"><font color="Green" title="E138">A16</font></span></a>, <a class="txt" href="#E11:158_1_4_1"><span class="lab"><font color="Green" title="E140">A18</font></span></a>, <a class="ref" href="modelc_2.html#T63" title="MODELC_2:th.63">MODELC_2:63</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="E7:158_1_4"/>
 <a href="modelc_3.html#K27" title="MODELC_3:func.27">atomic_LTL</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="158_1_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K27" title="MODELC_3:func.27">atomic_LTL</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1) or <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:158_1_4_2"/>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K27" title="MODELC_3:func.27">atomic_LTL</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1)
 ; <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="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">X</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">x</font> being   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a><span class="kw"> such that </span><br/><a NAME="E3:158_1_4_2"/><span class="lab"><font color="Green" title="E137">A19</font></span>: 
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font> &amp; <font color="Maroon" title="c18">x</font> is  <a href="modelc_2.html#V2" title="MODELC_2:attr.2">atomic</a>  )
 <span class="kw">and </span><br/><a NAME="E4:158_1_4_2"/><span class="lab"><font color="Green" title="E138">A20</font></span>: 
<font color="Maroon" title="c18">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">H<sub>2</sub></font>(<font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1)
 ;<br/>
<a NAME="E5:158_1_4_2"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K14" title="MODELC_3:func.14">*</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:158_1_4_2"><span class="lab"><font color="Green" title="E138">A20</font></span></a>, <a class="txt" href="modelc_3.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<a NAME="E6:158_1_4_2"/><span class="kw">then </span>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c11">n</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c18">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:158_1_4"><span class="lab"><font color="Green" title="E135">A13</font></span></a>, <a class="ref" href="modelc_2.html#D66" title="MODELC_2:def.66">MODELC_2:def 66</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:158_1_4_2"/>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:158_1_4"><span class="lab"><font color="Green" title="E134">A12</font></span></a>, <a class="txt" href="#E3:158_1_4_2"><span class="lab"><font color="Green" title="E137">A19</font></span></a>, <a class="ref" href="modelc_2.html#T63" title="MODELC_2:th.63">MODELC_2:63</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="E8:158_1_4"/>
<span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c1">w</font>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K29" title="MODELC_3:func.29">Label_</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:158_1_4"><span class="lab"><font color="Green" title="E136">A14</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>
<a NAME="E10:158_1"/><span class="lab"><font color="Green" title="E134">A21</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c1">w</font>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K30" title="MODELC_3:func.30">Tran_LTL</a> <font color="Maroon" title="c2">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="158_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">n</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"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c1">w</font>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K30" title="MODELC_3:func.30">Tran_LTL</a> <font color="Maroon" title="c2">v</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">n</font> = <font color="Maroon" title="c10">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">R</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">n</font>);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">R</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">n</font>) as   <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="c2">v</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:158_1"><span class="lab"><font color="Green" title="E130">A4</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">n1</font> = <font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">r</font> = <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">r1</font> = <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">R1</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1);<br/>
<span class="kw">set </span><font color="Maroon" title="c18">gA</font> = <span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c1">w</font>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">y</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c1">w</font>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">R1</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1) as   <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="c2">v</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:158_1"><span class="lab"><font color="Green" title="E130">A4</font></span></a></span>;<br/>
<a NAME="E4:158_1_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c1">w</font>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font></span>)</span>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E5:158_1_5"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c1">w</font>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font></span>)</span>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p2">(<span class="default"><a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E6:158_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E135">A22</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c1">w</font>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font></span>)</span>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> ,<span class="p2">(<span class="default"><a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D3" title="ZFMISC_1:def.3">ZFMISC_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">r1</font> = <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <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="c2">v</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:158_1"><span class="lab"><font color="Green" title="E130">A4</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">r</font> = <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font> as   <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="c2">v</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:158_1"><span class="lab"><font color="Green" title="E130">A4</font></span></a></span>;<br/>
<a NAME="E9:158_1_5"/><span class="lab"><font color="Green" title="E136">A23</font></span>: 
( <font color="Maroon" title="c20">R1</font> <a href="modelc_3.html#R6" title="MODELC_3:pred.6">is_next_of</a> <font color="Maroon" title="c13">R</font> &amp; <span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c1">w</font>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K29" title="MODELC_3:func.29">Label_</a> <font color="Maroon" title="c20">R1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:158_1"><span class="lab"><font color="Green" title="E132">A8</font></span></a>, <a class="txt" href="#E9:158_1"><span class="lab"><font color="Green" title="E133">A11</font></span></a></span>;<br/>
<a NAME="E10:158_1_5"/>
( <font color="Maroon" title="c13">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">r</font> &amp; <font color="Maroon" title="c20">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#D16" target="_self" title="MODELC_3:def.16">Def16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:158_1_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c1">w</font>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K30" title="MODELC_3:func.30">Tran_LTL</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:158_1_5"><span class="lab"><font color="Green" title="E135">A22</font></span></a>, <a class="txt" href="#E9:158_1_5"><span class="lab"><font color="Green" title="E136">A23</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>
<a NAME="E11:158_1"/><span class="lab"><font color="Green" title="E135">A24</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">H</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a>  st <font color="Olive" title="b2">H</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">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) &amp; <font color="Olive" title="b2">H</font> is  <a href="modelc_2.html#V7" title="MODELC_2:attr.7">Until</a>  &amp;  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Olive" title="b1">n</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Olive" title="b2">H</font> holds <br/> <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Olive" title="b2">H</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">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="158_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">n</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">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> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) &amp; <font color="Olive" title="b1">H</font> is  <a href="modelc_2.html#V7" title="MODELC_2:attr.7">Until</a>  &amp;  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c10">n</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Olive" title="b1">H</font> holds <br/> <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Olive" title="b1">H</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">H<sub>2</sub></font>(<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)</span><br/><span class="kw">let </span><font color="Maroon" title="c11">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="c11">H</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">H<sub>2</sub></font>(<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) &amp; <font color="Maroon" title="c11">H</font> is  <a href="modelc_2.html#V7" title="MODELC_2:attr.7">Until</a>  &amp;  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c10">n</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c11">H</font> implies  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c11">H</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">H<sub>2</sub></font>(<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c12">n1</font> = <font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<a NAME="E1:158_1_6"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <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"><span class="p2">(<span class="default"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c8">chf</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:158_1"><span class="lab"><font color="Green" title="E131">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:158_1_6"/>
( <font color="Maroon" title="c11">H</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">H<sub>2</sub></font>(<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) &amp; <font color="Maroon" title="c11">H</font> is  <a href="modelc_2.html#V7" title="MODELC_2:attr.7">Until</a>  &amp;  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c10">n</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c11">H</font> implies  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c11">H</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">H<sub>2</sub></font>(<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:158"><span class="lab"><font color="Green" title="E127">A1</font></span></a>, <a class="ref" href="modelc_3.html#T74" target="_self" title="MODELC_3:th.74">Th74</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="E12:158_1"/><span class="lab"><font color="Green" title="E136">A25</font></span>: 
for <font color="Olive" title="b1">FSet</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">FSet</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K33" title="MODELC_3:func.33">FinalS_LTL</a> <font color="Maroon" title="c2">v</font> holds <br/><span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">FSet</font> </span>}</span>  is   <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="158_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">FSet</font> be    <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"> ( <font color="Maroon" title="c10">FSet</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K33" title="MODELC_3:func.33">FinalS_LTL</a> <font color="Maroon" title="c2">v</font> implies <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font> </span>}</span>  is   <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c11">FK</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font> </span>}</span> ;<br/>
<span class="kw">assume </span><a NAME="E1:158_1_7"/>
<font color="Maroon" title="c10">FSet</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K33" title="MODELC_3:func.33">FinalS_LTL</a> <font color="Maroon" title="c2">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font> </span>}</span>  is   <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:158_1_7"/><span class="lab"><font color="Green" title="E137">A26</font></span>: 
<font color="Maroon" title="c10">FSet</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font>
 <span class="kw">and </span><br/><a NAME="E4:158_1_7"/><span class="lab"><font color="Green" title="E138">A27</font></span>: 
 ex <font color="Olive" title="b1">F</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <br/>( <font color="Olive" title="b1">F</font> <a href="modelc_2.html#R2" title="MODELC_2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Olive" title="b1">F</font> is  <a href="modelc_2.html#V7" title="MODELC_2:attr.7">Until</a>  &amp; <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c2">v</font> )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">F</font> being   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a><span class="kw"> such that </span><br/><a NAME="E6:158_1_7"/>
<font color="Maroon" title="c13">F</font> <a href="modelc_2.html#R2" title="MODELC_2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">and </span><br/><a NAME="E7:158_1_7"/><span class="lab"><font color="Green" title="E139">A28</font></span>: 
<font color="Maroon" title="c13">F</font> is  <a href="modelc_2.html#V7" title="MODELC_2:attr.7">Until</a> 
 <span class="kw">and </span><br/><a NAME="E8:158_1_7"/><span class="lab"><font color="Green" title="E140">A29</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_3.html#K32" title="MODELC_3:func.32">FinalS_LTL</a> <font color="Maroon" title="c13">F</font>,<font color="Maroon" title="c2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:158_1_7"><span class="lab"><font color="Green" title="E138">A27</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">F2</font> =  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c13">F</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">F1</font> =  <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c13">F</font>;<br/>
<a NAME="E9:158_1_7"/><span class="lab"><font color="Green" title="E141">A30</font></span>: 
<font color="Maroon" title="c13">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c13">F</font></span>)</span> <a href="modelc_2.html#K7" title="MODELC_2:func.7">'U'</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c13">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:158_1_7"><span class="lab"><font color="Green" title="E139">A28</font></span></a>, <a class="ref" href="modelc_2.html#T8" title="MODELC_2:th.8">MODELC_2:8</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="158_1_7_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:158_1_7_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font> </span>}</span>  is  not   <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</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"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">L</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E3:158_1_7_1"/><span class="lab"><font color="Green" title="E142">A31</font></span>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T73" title="FINSEQ_1:th.73">FINSEQ_1:73</a></span>;<br/><a NAME="E4:158_1_7_1"/>
 ex <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> holds <br/> not <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">k</font> where <font color="Olive" title="b3">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="158_1_7_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:158_1_7_1_1"/><span class="lab"><font color="Green" title="E143">A32</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">L</font></span>)</span> )
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="158_1_7_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="158_1_7_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:158_1_7_1_1_1_1"/>
( 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="c16">L</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:158_1_7_1_1"><span class="lab"><font color="Green" title="E143">A32</font></span></a>, <a class="ref" href="nat_1.html#T19" title="NAT_1:th.19">NAT_1:19</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="158_1_7_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:158_1_7_1_1_1_1_1"/><span class="lab"><font color="Green" title="E144">A33</font></span>: 
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="c16">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> holds <br/> not <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">k</font> where <font color="Olive" title="b3">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font> </span>}</span> </span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c17">LEN</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">L</font>;<br/><a NAME="E2:158_1_7_1_1_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="158_1_7_1_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font> </span>}</span>  or <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:158_1_7_1_1_1_1_1_1"/>
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</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="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><br/>

<a NAME="E2:158_1_7_1_1_1_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:158_1_7_1_1_1_1_1_1"/>
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ; <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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">L</font> = <font color="Maroon" title="c16">L</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:158_1_7_1"><span class="lab"><font color="Green" title="E142">A31</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c19">realMAX</font> =  <a href="rfinseq2.html#K3" title="RFINSEQ2:func.3">max</a> <font color="Maroon" title="c18">L</font>;<br/><span class="kw">set </span><font color="Maroon" title="c20">iMAX</font> = <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="rfinseq2.html#K3" title="RFINSEQ2:func.3">max</a> <font color="Maroon" title="c18">L</font></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c21">natMAX</font> = <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="rfinseq2.html#K3" title="RFINSEQ2:func.3">max</a> <font color="Maroon" title="c18">L</font></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1;<br/><a NAME="E4:158_1_7_1_1_1_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rfinseq2.html#K3" title="RFINSEQ2:func.3">max</a> <font color="Maroon" title="c18">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="158_1_7_1_1_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c22">b</font> = <font color="Maroon" title="c18">L</font> <a href="seq_1.html#K1" title="SEQ_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="c16">L</font></span>)</span>;<br/>
<a NAME="E1:158_1_7_1_1_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:158_1_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E144">A33</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E2:158_1_7_1_1_1_1_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E3:158_1_7_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">L</font> <a href="seq_1.html#K1" title="SEQ_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="c16">L</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c18">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E4:158_1_7_1_1_1_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">L</font> <a href="seq_1.html#K1" title="SEQ_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="c16">L</font></span>)</span> &amp; <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:158_1_7_1"><span class="lab"><font color="Green" title="E142">A31</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:158_1_7_1_1_1_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rfinseq2.html#K3" title="RFINSEQ2:func.3">max</a> <font color="Maroon" title="c18">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:158_1_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E144">A33</font></span></a>, <a class="ref" href="rfinseq2.html#T1" title="RFINSEQ2:th.1">RFINSEQ2:1</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">iMAX</font> = <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="rfinseq2.html#K3" title="RFINSEQ2:func.3">max</a> <font color="Maroon" title="c18">L</font></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T80" title="INT_1:th.80">INT_1:80</a></span>;<br/><a NAME="E6:158_1_7_1_1_1_1_1"/>
<font color="Maroon" title="c22">iMAX</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 is   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c23">natMAX</font> = <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="rfinseq2.html#K3" title="RFINSEQ2:func.3">max</a> <font color="Maroon" title="c18">L</font></span>)</span></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><a NAME="E8:158_1_7_1_1_1_1_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c23">natMAX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/> not <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="158_1_7_1_1_1_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">k</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"> ( <font color="Maroon" title="c23">natMAX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">k</font> implies  not <font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font> </span>}</span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:158_1_7_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E145">A34</font></span>: 
<font color="Maroon" title="c23">natMAX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font> </span>}</span> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="158_1_7_1_1_1_1_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:158_1_7_1_1_1_1_1_3_1"/>
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">i1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:158_1_7_1_1_1_1_1_3_1"/><span class="lab"><font color="Green" title="E146">A35</font></span>: 
<font color="Maroon" title="c25">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">L</font>
 <span class="kw">and </span><br/><a NAME="E4:158_1_7_1_1_1_1_1_3_1"/><span class="lab"><font color="Green" title="E147">A36</font></span>: 
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c25">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:158_1_7_1"><span class="lab"><font color="Green" title="E142">A31</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">i1</font> = <font color="Maroon" title="c25">i1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:158_1_7_1_1_1_1_1_3_1"><span class="lab"><font color="Green" title="E146">A35</font></span></a></span>;<br/><a NAME="E6:158_1_7_1_1_1_1_1_3_1"/>
<font color="Maroon" title="c26">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">L</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:158_1_7_1_1_1_1_1_3_1"><span class="lab"><font color="Green" title="E146">A35</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E7:158_1_7_1_1_1_1_1_3_1"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">i1</font> &amp; <font color="Maroon" title="c26">i1</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="c16">L</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E8:158_1_7_1_1_1_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rfinseq2.html#K3" title="RFINSEQ2:func.3">max</a> <font color="Maroon" title="c18">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:158_1_7_1_1_1_1_1_3_1"><span class="lab"><font color="Green" title="E147">A36</font></span></a>, <a class="ref" href="rfinseq2.html#T1" title="RFINSEQ2:th.1">RFINSEQ2:1</a></span>;<br/><span class="kw">hence </span><a NAME="E9:158_1_7_1_1_1_1_1_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:158_1_7_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E145">A34</font></span></a>, <a class="ref" href="int_1.html#T57" title="INT_1:th.57">INT_1:57</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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="E3:158_1_7_1_1_1_1_1_3"/>
not <font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font> </span>}</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="E9:158_1_7_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> holds <br/> not <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">k</font> where <font color="Olive" title="b3">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font> </span>}</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="158_1_7_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:158_1_7_1_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> holds <br/> not <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">k</font> where <font color="Olive" title="b3">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font> </span>}</span> </span><br/><div class="add"><a NAME="E2:158_1_7_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:158_1_7_1_1_1_1_2"/><span class="kw">then </span>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/> not <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:158_1_7_1"><span class="lab"><font color="Green" title="E142">A31</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:158_1_7_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> holds <br/> not <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">k</font> where <font color="Olive" title="b3">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font> </span>}</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:158_1_7_1_1"/>
 ex <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> holds <br/> not <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">k</font> where <font color="Olive" title="b3">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font> </span>}</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">then </span><span class="kw">consider </span><font color="Maroon" title="c17">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:158_1_7_1"/><span class="lab"><font color="Green" title="E143">A37</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c17">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/> not <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font> </span>}</span> 
 ;<br/><a NAME="E7:158_1_7_1"/><span class="lab"><font color="Green" title="E144">A38</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c17">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/>( <font color="Maroon" title="c13">F</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">H<sub>2</sub></font>(<font color="Olive" title="b1">k</font>) &amp; not  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c13">F</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">H<sub>2</sub></font>(<font color="Olive" title="b1">k</font>) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="158_1_7_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">k</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"> ( <font color="Maroon" title="c17">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">k</font> implies ( <font color="Maroon" title="c13">F</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">H<sub>2</sub></font>(<font color="Maroon" title="c18">k</font>) &amp; not  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c13">F</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">H<sub>2</sub></font>(<font color="Maroon" title="c18">k</font>) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:158_1_7_1_2"/>
<font color="Maroon" title="c17">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">k</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="c13">F</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">H<sub>2</sub></font>(<font color="Maroon" title="c18">k</font>) &amp; not  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c13">F</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">H<sub>2</sub></font>(<font color="Maroon" title="c18">k</font>) )</span><br/>

<a NAME="E2:158_1_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E145">A39</font></span>: 
not <font color="Maroon" title="c18">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:158_1_7_1"><span class="lab"><font color="Green" title="E143">A37</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">k</font> = <font color="Maroon" title="c18">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">r</font> = <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">k</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">r</font> = <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">k</font> as   <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="c2">v</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:158_1"><span class="lab"><font color="Green" title="E130">A4</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="158_1_7_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:158_1_7_1_2_1"/>
(  not <font color="Maroon" title="c13">F</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 <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c2">v</font></span>)</span> or  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c13">F</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 <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c2">v</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:158_1_7_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:158_1_7"><span class="lab"><font color="Green" title="E137">A26</font></span></a>, <a class="txt" href="#E8:158_1_7"><span class="lab"><font color="Green" title="E140">A29</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:158_1_7_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:158_1_7_1_2"><span class="lab"><font color="Green" title="E145">A39</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="E6:158_1_7_1_2"/>
( <font color="Maroon" title="c13">F</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">H<sub>2</sub></font>(<font color="Maroon" title="c18">k</font>) &amp; not  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c13">F</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">H<sub>2</sub></font>(<font color="Maroon" title="c18">k</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><span class="kw">set </span><font color="Maroon" title="c18">w1</font> =  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c17">m</font>;<br/><span class="kw">set </span><font color="Maroon" title="c19">m1</font> = <font color="Maroon" title="c17">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><a NAME="E8:158_1_7_1"/><span class="lab"><font color="Green" title="E145">A40</font></span>: 
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c17">m</font> <a href="modelc_2.html#R8" title="MODELC_2:pred.8">|=</a>  <a href="modelc_3.html#K14" title="MODELC_3:func.14">*</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c17">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:158_1"><span class="lab"><font color="Green" title="E132">A8</font></span></a></span>;<br/><a NAME="E9:158_1_7_1"/>
<font color="Maroon" title="c17">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E10:158_1_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">F</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">H<sub>2</sub></font>(<font color="Maroon" title="c17">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:158_1_7_1"><span class="lab"><font color="Green" title="E144">A38</font></span></a></span>;<br/><a NAME="E11:158_1_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K14" title="MODELC_3:func.14">*</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c17">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="modelc_3.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/><a NAME="E12:158_1_7_1"/><span class="kw">then </span>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c17">m</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a> <font color="Maroon" title="c13">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:158_1_7_1"><span class="lab"><font color="Green" title="E145">A40</font></span></a>, <a class="ref" href="modelc_2.html#D66" title="MODELC_2:def.66">MODELC_2:def 66</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">h</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E14:158_1_7_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="c20">h</font> holds <br/> <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c17">m</font></span>)</span>,<font color="Olive" title="b1">j</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K11" title="MODELC_2:func.11">the_left_argument_of</a> <font color="Maroon" title="c13">F</font>
 <span class="kw">and </span><br/><a NAME="E15:158_1_7_1"/><span class="lab"><font color="Green" title="E146">A41</font></span>: 
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c17">m</font></span>)</span>,<font color="Maroon" title="c20">h</font> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c13">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:158_1_7"><span class="lab"><font color="Green" title="E141">A30</font></span></a>, <a class="ref" href="modelc_2.html#T68" title="MODELC_2:th.68">MODELC_2:68</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c21">m2</font> = <font color="Maroon" title="c17">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">h</font>;<br/><span class="kw">set </span><font color="Maroon" title="c22">m3</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">h</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><a NAME="E16:158_1_7_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">h</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><a NAME="E17:158_1_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">h</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E18:158_1_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E147">A42</font></span>: 
( <font color="Maroon" title="c13">F</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">H<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">h</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) &amp; not  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c13">F</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">H<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">h</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:158_1_7_1"><span class="lab"><font color="Green" title="E144">A38</font></span></a></span>;<br/><a NAME="E19:158_1_7_1"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c1">w</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">h</font></span>)</span> <a href="modelc_2.html#R7" title="MODELC_2:pred.7">|=</a>  <a href="modelc_2.html#K12" title="MODELC_2:func.12">the_right_argument_of</a> <font color="Maroon" title="c13">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:158_1_7_1"><span class="lab"><font color="Green" title="E146">A41</font></span></a>, <a class="ref" href="modelc_2.html#T80" title="MODELC_2:th.80">MODELC_2:80</a></span>;<br/><span class="kw">hence </span><a NAME="E20:158_1_7_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:158_1"><span class="lab"><font color="Green" title="E135">A24</font></span></a>, <a class="txt" href="#E7:158_1_7"><span class="lab"><font color="Green" title="E139">A28</font></span></a>, <a class="txt" href="#E18:158_1_7_1"><span class="lab"><font color="Green" title="E147">A42</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="E11:158_1_7"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">FSet</font> </span>}</span>  is   <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><a href="modelc_3.html#K37" title="MODELC_3:func.37">chosen_run</a> <font color="Maroon" title="c1">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c8">chf</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:158_1"><span class="lab"><font color="Green" title="E131">A7</font></span></a></span>
<br/>.= 
 <a href="modelc_3.html#K11" title="MODELC_3:func.11">init</a> <font color="Maroon" title="c2">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#D50" target="_self" title="MODELC_3:def.50">Def50</a></span>
;<br/>
<a NAME="E14:158_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K31" title="MODELC_3:func.31">InitS_LTL</a> <font color="Maroon" title="c2">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:158_1"/>
 ex <font color="Olive" title="b1">run</font> being  <a href="nat_1.html#NM2" title="NAT_1:NM.2">sequence</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K23" title="MODELC_3:func.23">LTLStates</a> <font color="Maroon" title="c2">v</font></span>)</span> st <br/>( <font color="Olive" title="b1">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K31" title="MODELC_3:func.31">InitS_LTL</a> <font color="Maroon" title="c2">v</font> &amp; ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">run</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c1">w</font>,<a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K30" title="MODELC_3:func.30">Tran_LTL</a> <font color="Maroon" title="c2">v</font> &amp; ( for <font color="Olive" title="b3">FSet</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">FSet</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K33" title="MODELC_3:func.33">FinalS_LTL</a> <font color="Maroon" title="c2">v</font> holds <br/><span class="p1">{<span class="default"> <font color="Olive" title="b4">k</font> where <font color="Olive" title="b4">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">run</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">FSet</font> </span>}</span>  is   <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:158_1"><span class="lab"><font color="Green" title="E134">A21</font></span></a>, <a class="txt" href="#E12:158_1"><span class="lab"><font color="Green" title="E136">A25</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="E3:158"/>
<font color="Maroon" title="c1">w</font> <a href="modelc_3.html#R9" title="MODELC_3:pred.9">is-accepted-by</a>  <a href="modelc_3.html#K34" title="MODELC_3:func.34">BAutomaton</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#D39" target="_self" title="MODELC_3:def.39">Def39</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>
