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<span class="kw">let </span><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">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">L</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><br/> for <font color="Olive" title="b2">v</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a>  st <font color="Olive" title="b1">L</font> <a href="modelc_3.html#R5" title="MODELC_3:pred.5">is_Finseq_for</a> <font color="Olive" title="b2">v</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c2">n</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="Olive" title="b1">L</font> holds <br/>( 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> <span class="p2">(<span class="default"><font color="Olive" title="b1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Olive" title="b2">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Olive" title="b1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Olive" title="b2">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Olive" title="b2">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Olive" title="b2">v</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">L</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</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_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a>  st <font color="Maroon" title="c3">L</font> <a href="modelc_3.html#R5" title="MODELC_3:pred.5">is_Finseq_for</a> <font color="Olive" title="b1">v</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c2">n</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="c3">L</font> holds <br/>( 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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Olive" title="b1">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Olive" title="b1">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Olive" title="b1">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Olive" title="b1">v</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">v</font> be   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">L</font> <a href="modelc_3.html#R5" title="MODELC_3:pred.5">is_Finseq_for</a> <font color="Maroon" title="c4">v</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c2">n</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="c3">L</font> implies ( 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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:86"/><span class="lab"><font color="Green" title="E75">A1</font></span>: 
( <font color="Maroon" title="c3">L</font> <a href="modelc_3.html#R5" title="MODELC_3:pred.5">is_Finseq_for</a> <font color="Maroon" title="c4">v</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c2">n</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="c3">L</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>for <font color="Olive" title="b1">n1</font>, <font color="Olive" title="b2">m1</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b3">L1</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">L1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; <font color="Olive" title="b3">L1</font> <a href="modelc_3.html#R5" title="MODELC_3:pred.5">is_Finseq_for</a> <font color="Maroon" title="c4">v</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m1</font> &amp; <font color="Olive" title="b2">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n1</font> &amp; <font color="Olive" title="b1">n1</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="Olive" title="b3">L1</font> holds <br/>( 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> <span class="p2">(<span class="default"><font color="Olive" title="b3">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Olive" title="b3">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> );<br/>
<a NAME="E2:86"/><span class="lab"><font color="Green" title="E76">A2</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">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">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">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:86_1"/><span class="lab"><font color="Green" title="E77">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">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">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:86_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">n1</font>, <font color="Maroon" title="c7">m1</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">L1</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">L1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Olive" title="b1">L1</font> <a href="modelc_3.html#R5" title="MODELC_3:pred.5">is_Finseq_for</a> <font color="Maroon" title="c4">v</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m1</font> &amp; <font color="Maroon" title="c7">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> &amp; <font color="Maroon" title="c6">n1</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="Olive" title="b1">L1</font> holds <br/>( 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> <span class="p2">(<span class="default"><font color="Olive" title="b1">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Olive" title="b1">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">L1</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">L1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c8">L1</font> <a href="modelc_3.html#R5" title="MODELC_3:pred.5">is_Finseq_for</a> <font color="Maroon" title="c4">v</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m1</font> &amp; <font color="Maroon" title="c7">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> &amp; <font color="Maroon" title="c6">n1</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="c8">L1</font> implies ( 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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:86_1_1"/><span class="lab"><font color="Green" title="E78">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">L1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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="c8">L1</font> <a href="modelc_3.html#R5" title="MODELC_3:pred.5">is_Finseq_for</a> <font color="Maroon" title="c4">v</font> or  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m1</font> or  not <font color="Maroon" title="c7">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> or  not <font color="Maroon" title="c6">n1</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="c8">L1</font> or ( 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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> ) )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_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="E1:86_1_1_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">L1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1_1"><span class="lab"><font color="Green" title="E78">A4</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:86_1_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">L1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">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="c8">L1</font> <a href="modelc_3.html#R5" title="MODELC_3:pred.5">is_Finseq_for</a> <font color="Maroon" title="c4">v</font> or  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m1</font> or  not <font color="Maroon" title="c7">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> or  not <font color="Maroon" title="c6">n1</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="c8">L1</font> or ( 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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> ) )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:86_1_1_1_1_1"/>
(  not <font color="Maroon" title="c8">L1</font> <a href="modelc_3.html#R5" title="MODELC_3:pred.5">is_Finseq_for</a> <font color="Maroon" title="c4">v</font> or  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m1</font> or  not <font color="Maroon" title="c7">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> or  not <font color="Maroon" title="c6">n1</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="c8">L1</font> or ( 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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1"><span class="lab"><font color="Green" title="E77">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:86_1_1_1_1_2"/><span class="lab"><font color="Green" title="E79">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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="c8">L1</font> <a href="modelc_3.html#R5" title="MODELC_3:pred.5">is_Finseq_for</a> <font color="Maroon" title="c4">v</font> or  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m1</font> or  not <font color="Maroon" title="c7">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> or  not <font color="Maroon" title="c6">n1</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="c8">L1</font> or ( 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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> ) )</span><br/><div class="add"><a NAME="E2:86_1_1_1_1_2"/>
( <font color="Maroon" title="c8">L1</font> <a href="modelc_3.html#R5" title="MODELC_3:pred.5">is_Finseq_for</a> <font color="Maroon" title="c4">v</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m1</font> &amp; <font color="Maroon" title="c7">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> &amp; <font color="Maroon" title="c6">n1</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="c8">L1</font> implies ( 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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:86_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E80">A6</font></span>: 
<font color="Maroon" title="c8">L1</font> <a href="modelc_3.html#R5" title="MODELC_3:pred.5">is_Finseq_for</a> <font color="Maroon" title="c4">v</font>
 <span class="kw">and </span><br/><a NAME="E2:86_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E81">A7</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m1</font>
 <span class="kw">and </span><br/><a NAME="E3:86_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E82">A8</font></span>: 
<font color="Maroon" title="c7">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font>
 <span class="kw">and </span><br/><a NAME="E4:86_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E83">A9</font></span>: 
<font color="Maroon" title="c6">n1</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="c8">L1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> )</span><br/>

<a NAME="E5:86_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E84">A10</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E81">A7</font></span></a>, <a class="txt" href="#E3:86_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E82">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_1_1_1_2_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_1_1_2_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="E1:86_1_1_1_1_2_1_1_1"/>
( <font color="Maroon" title="c6">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> or <font color="Maroon" title="c6">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1_1_1_1_2"><span class="lab"><font color="Green" title="E79">A5</font></span></a>, <a class="txt" href="#E4:86_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E83">A9</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_1_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:86_1_1_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E85">A11</font></span>: 
<font color="Maroon" title="c6">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c9">L2</font> = <font color="Maroon" title="c8">L1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k</font></span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">L2</font> = <font color="Maroon" title="c8">L1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k</font></span>)</span> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T19" title="FINSEQ_1:th.19">FINSEQ_1:19</a></span>;<br/><a NAME="E3:86_1_1_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E86">A12</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <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="Maroon" title="c5">k</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="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E4:86_1_1_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A13</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1_1_1_1_2"><span class="lab"><font color="Green" title="E79">A5</font></span></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></span>;<br/><a NAME="E5:86_1_1_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">n1</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="c10">L2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:86_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E84">A10</font></span></a>, <a class="txt" href="#E1:86_1_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E85">A11</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E6:86_1_1_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A14</font></span>: 
<font color="Maroon" title="c10">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/><a NAME="E7:86_1_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c7">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:86_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E82">A8</font></span></a>, <a class="txt" href="#E1:86_1_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E85">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:86_1_1_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">m1</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="c10">L2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E81">A7</font></span></a>, <a class="txt" href="#E4:86_1_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E87">A13</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E9:86_1_1_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A15</font></span>: 
<font color="Maroon" title="c10">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/><a NAME="E10:86_1_1_1_1_2_1_1_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon" title="c10">L2</font> <a href="modelc_3.html#R5" title="MODELC_3:pred.5">is_Finseq_for</a> <font color="Maroon" title="c4">v</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1_1_1_1_2"><span class="lab"><font color="Green" title="E79">A5</font></span></a>, <a class="txt" href="#E1:86_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E80">A6</font></span></a>, <a class="txt" href="#E3:86_1_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E86">A12</font></span></a>, <a class="ref" href="modelc_3.html#T26" target="_self" title="MODELC_3:th.26">Th26</a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></span>;<br/><span class="kw">hence </span><a NAME="E11:86_1_1_1_1_2_1_1_1_1"/>
( 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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1"><span class="lab"><font color="Green" title="E77">A3</font></span></a>, <a class="txt" href="#E2:86_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E81">A7</font></span></a>, <a class="txt" href="#E3:86_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E82">A8</font></span></a>, <a class="txt" href="#E1:86_1_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E85">A11</font></span></a>, <a class="txt" href="#E9:86_1_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E89">A15</font></span></a>, <a class="txt" href="#E6:86_1_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E88">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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_1_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:86_1_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E85">A16</font></span>: 
<font color="Maroon" title="c6">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_1_1_1_2_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_1_1_2_1_1_1_2_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="E1:86_1_1_1_1_2_1_1_1_2_1_1"/>
( <font color="Maroon" title="c7">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n1</font> or <font color="Maroon" title="c7">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n1</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:86_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E82">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_1_1_2_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:86_1_1_1_1_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E86">A17</font></span>: 
<font color="Maroon" title="c7">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c9">L2</font> = <font color="Maroon" title="c8">L1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k</font></span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">L2</font> = <font color="Maroon" title="c8">L1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k</font></span>)</span> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T19" title="FINSEQ_1:th.19">FINSEQ_1:19</a></span>;<br/><a NAME="E3:86_1_1_1_1_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E87">A18</font></span>: 
<font color="Maroon" title="c7">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E85">A16</font></span></a>, <a class="txt" href="#E1:86_1_1_1_1_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E86">A17</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:86_1_1_1_1_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E88">A19</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <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="Maroon" title="c5">k</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="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E5:86_1_1_1_1_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A20</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1_1_1_1_2"><span class="lab"><font color="Green" title="E79">A5</font></span></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></span>;<br/><a NAME="E6:86_1_1_1_1_2_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">m1</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="c10">L2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E81">A7</font></span></a>, <a class="txt" href="#E3:86_1_1_1_1_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E87">A18</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E7:86_1_1_1_1_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A21</font></span>: 
<font color="Maroon" title="c10">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/><a NAME="E8:86_1_1_1_1_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E91">A22</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E81">A7</font></span></a>, <a class="txt" href="#E3:86_1_1_1_1_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E87">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:86_1_1_1_1_2_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">k</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="c10">L2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:86_1_1_1_1_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E89">A20</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E10:86_1_1_1_1_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A23</font></span>: 
<font color="Maroon" title="c10">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/><a NAME="E11:86_1_1_1_1_2_1_1_1_2_1_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon" title="c10">L2</font> <a href="modelc_3.html#R5" title="MODELC_3:pred.5">is_Finseq_for</a> <font color="Maroon" title="c4">v</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1_1_1_1_2"><span class="lab"><font color="Green" title="E79">A5</font></span></a>, <a class="txt" href="#E1:86_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E80">A6</font></span></a>, <a class="txt" href="#E4:86_1_1_1_1_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E88">A19</font></span></a>, <a class="ref" href="modelc_3.html#T26" target="_self" title="MODELC_3:th.26">Th26</a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></span>;<br/><a NAME="E12:86_1_1_1_1_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A24</font></span>: 
( 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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1"><span class="lab"><font color="Green" title="E77">A3</font></span></a>, <a class="txt" href="#E2:86_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E81">A7</font></span></a>, <a class="txt" href="#E3:86_1_1_1_1_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E87">A18</font></span></a>, <a class="txt" href="#E7:86_1_1_1_1_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E90">A21</font></span></a>, <a class="txt" href="#E10:86_1_1_1_1_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E92">A23</font></span></a></span>;<br/><a NAME="E13:86_1_1_1_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">L1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1_1_1_1_2"><span class="lab"><font color="Green" title="E79">A5</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">N1</font>, <font color="Maroon" title="c12">N2</font> being   <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="c4">v</font><span class="kw"> such that </span><br/><a NAME="E15:86_1_1_1_1_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E94">A25</font></span>: 
( <font color="Maroon" title="c11">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon" title="c12">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <span class="kw">and </span><br/><a NAME="E16:86_1_1_1_1_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E95">A26</font></span>: 
<font color="Maroon" title="c12">N2</font> <a href="modelc_3.html#R4" title="MODELC_3:pred.4">is_succ_of</a> <font color="Maroon" title="c11">N1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E80">A6</font></span></a>, <a class="txt" href="#E8:86_1_1_1_1_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E91">A22</font></span></a>, <a class="ref" href="modelc_3.html#D19" target="_self" title="MODELC_3:def.19">Def19</a></span>;<br/><a NAME="E17:86_1_1_1_1_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E96">A27</font></span>: 
( <font color="Maroon" title="c11">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <font color="Maroon" title="c11">N1</font>,<font color="Maroon" title="c4">v</font> &amp; <font color="Maroon" title="c12">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <font color="Maroon" title="c12">N2</font>,<font color="Maroon" title="c4">v</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/><a NAME="E18:86_1_1_1_1_2_1_1_1_2_1_1_1"/>
( the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c11">N1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U1" title="MODELC_3:sel.1">LTLold</a> of <font color="Maroon" title="c12">N2</font> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <font color="Maroon" title="c11">N1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <font color="Maroon" title="c12">N2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:86_1_1_1_1_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E95">A26</font></span></a>, <a class="ref" href="modelc_3.html#T25" target="_self" title="MODELC_3:th.25">Th25</a></span>;<br/><span class="kw">hence </span><a NAME="E19:86_1_1_1_1_2_1_1_1_2_1_1_1"/>
( 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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E85">A16</font></span></a>, <a class="txt" href="#E12:86_1_1_1_1_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E93">A24</font></span></a>, <a class="txt" href="#E15:86_1_1_1_1_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E94">A25</font></span></a>, <a class="txt" href="#E17:86_1_1_1_1_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E96">A27</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_1_1_2_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:86_1_1_1_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c7">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:86_1_1_1_1_2_1_1_1_2_1_1_2"/>
( 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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">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"> 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="E3:86_1_1_1_1_2_1_1_1_2"/>
( 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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">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"> 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="E7:86_1_1_1_1_2_1"/>
( 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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">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"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:86_1_1_1_1_2"/>
(  not <font color="Maroon" title="c8">L1</font> <a href="modelc_3.html#R5" title="MODELC_3:pred.5">is_Finseq_for</a> <font color="Maroon" title="c4">v</font> or  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m1</font> or  not <font color="Maroon" title="c7">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> or  not <font color="Maroon" title="c6">n1</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="c8">L1</font> or ( 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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">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"> 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="E3:86_1_1"/>
(  not <font color="Maroon" title="c8">L1</font> <a href="modelc_3.html#R5" title="MODELC_3:pred.5">is_Finseq_for</a> <font color="Maroon" title="c4">v</font> or  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m1</font> or  not <font color="Maroon" title="c7">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n1</font> or  not <font color="Maroon" title="c6">n1</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="c8">L1</font> or ( 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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n1</font></span>)</span>,<font color="Maroon" title="c4">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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:86_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:86"/><span class="lab"><font color="Green" title="E77">A28</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E4:86"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E3:86"><span class="lab"><font color="Green" title="E77">A28</font></span></a>, <a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E76">A2</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:86"/>
( 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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="modelc_3.html#U3" title="MODELC_3:sel.3">LTLnext</a> of <span class="p1">(<span class="default"><a href="modelc_3.html#K10" title="MODELC_3:func.10">CastNode</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">v</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E75">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
