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

<b>let </b><font color="Maroon" title="c1">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">L</font>, <font color="Olive" title="b2">L1</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><br/> for <font color="Olive" title="b3">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="b3">v</font> &amp; <font color="Maroon" title="c1">m</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> &amp; <font color="Olive" title="b2">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">L</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="c1">m</font></span>)</span> holds <br/><font color="Olive" title="b2">L1</font> <a href="modelc_3.html#R5" title="MODELC_3:pred.5">is_Finseq_for</a> <font color="Olive" title="b3">v</font></span><br/><b>let </b><font color="Maroon" title="c2">L</font>, <font color="Maroon" title="c3">L1</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c2">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; <font color="Maroon" title="c1">m</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="c2">L</font> &amp; <font color="Maroon" title="c3">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">L</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="c1">m</font></span>)</span> holds <br/><font color="Maroon" title="c3">L1</font> <a href="modelc_3.html#R5" title="MODELC_3:pred.5">is_Finseq_for</a> <font color="Olive" title="b1">v</font></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">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; <font color="Maroon" title="c1">m</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="c2">L</font> &amp; <font color="Maroon" title="c3">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">L</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="c1">m</font></span>)</span> implies <font color="Maroon" title="c3">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><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:81"/><i><font color="Green" title="E70">A1</font></i>: 
<font color="Maroon" title="c2">L</font> <a href="modelc_3.html#R5" title="MODELC_3:pred.5">is_Finseq_for</a> <font color="Maroon" title="c4">v</font>
 <b>and </b><br/><a NAME="E2:81"/><i><font color="Green" title="E71">A2</font></i>: 
<font color="Maroon" title="c1">m</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="c2">L</font>
 <b>and </b><br/><a NAME="E3:81"/><i><font color="Green" title="E72">A3</font></i>: 
<font color="Maroon" title="c3">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">L</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="c1">m</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">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><br/>

<b>reconsider </b><font color="Maroon" title="c5">L1</font> = <font color="Maroon" title="c3">L1</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<a NAME="E5:81"/><i><font color="Green" title="E73">A4</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">L1</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="c1">m</font> )
 
<b>by </b><i><a class="txt" href="#E2:81"><i><font color="Green" title="E71">A2</font></i></a>, <a class="txt" href="#E3:81"><i><font color="Green" title="E72">A3</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></i>;<br/>
<a NAME="E6:81"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">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="c5">L1</font> holds <br/> ex <font color="Olive" title="b2">N1</font>, <font color="Olive" title="b3">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> st <br/>( <font color="Olive" title="b2">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b3">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b3">N2</font> <a href="modelc_3.html#R4" title="MODELC_3:pred.4">is_succ_of</a> <font color="Olive" title="b2">N1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c6">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="c5">L1</font> implies  ex <font color="Olive" title="b1">N1</font>, <font color="Olive" title="b2">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> st <br/>( <font color="Olive" title="b1">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Olive" title="b2">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">N2</font> <a href="modelc_3.html#R4" title="MODELC_3:pred.4">is_succ_of</a> <font color="Olive" title="b1">N1</font> ) )</span><br/>

<b>assume </b><a NAME="E1:81_1"/><i><font color="Green" title="E74">B1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c6">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="c5">L1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">N1</font>, <font color="Olive" title="b2">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> st <br/>( <font color="Olive" title="b1">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Olive" title="b2">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">N2</font> <a href="modelc_3.html#R4" title="MODELC_3:pred.4">is_succ_of</a> <font color="Olive" title="b1">N1</font> )</span><br/>

<a NAME="E2:81_1"/>
<font color="Maroon" title="c6">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="c2">L</font>
 
<b>by </b><i><a class="txt" href="#E1:81_1"><i><font color="Green" title="E74">B1</font></i></a>, <a class="txt" href="#E5:81"><i><font color="Green" title="E73">A4</font></i></a>, <a class="txt" href="#E2:81"><i><font color="Green" title="E71">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">N1</font>, <font color="Maroon" title="c8">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><b> such that </b><br/><a NAME="E4:81_1"/><i><font color="Green" title="E75">B3</font></i>: 
( <font color="Maroon" title="c7">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c8">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c8">N2</font> <a href="modelc_3.html#R4" title="MODELC_3:pred.4">is_succ_of</a> <font color="Maroon" title="c7">N1</font> )
 <b>by </b><i><a class="txt" href="#E1:81_1"><i><font color="Green" title="E74">B1</font></i></a>, <a class="txt" href="#E1:81"><i><font color="Green" title="E70">A1</font></i></a>, <a class="ref" href="modelc_3.html#D19" target="_self" title="MODELC_3:def.19">DefFinseq</a></i>;<br/>
<a NAME="E5:81_1"/>
<font color="Maroon" title="c6">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="c5">L1</font>
 
<b>by </b><i><a class="txt" href="#E1:81_1"><i><font color="Green" title="E74">B1</font></i></a>, <a class="txt" href="#E5:81"><i><font color="Green" title="E73">A4</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:81_1"/><b>then </b><i><font color="Green" title="E76">B4</font></i>: 
<font color="Maroon" title="c5">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
 
<b>by </b><i><a class="txt" href="#E3:81"><i><font color="Green" title="E72">A3</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E7:81_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> )
 
<b>by </b><i><a class="txt" href="#E1:81_1"><i><font color="Green" title="E74">B1</font></i></a>, <a class="txt" href="#E5:81"><i><font color="Green" title="E73">A4</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E8:81_1"/><b>then </b>
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c5">L1</font>
 
<b>by </b><i><a class="txt" href="#E5:81"><i><font color="Green" title="E73">A4</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:81_1"/><b>then </b>
<font color="Maroon" title="c5">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:81"><i><font color="Green" title="E72">A3</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E10:81_1"/>
 ex <font color="Olive" title="b1">N1</font>, <font color="Olive" title="b2">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> st <br/>( <font color="Olive" title="b1">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Olive" title="b2">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">N2</font> <a href="modelc_3.html#R4" title="MODELC_3:pred.4">is_succ_of</a> <font color="Olive" title="b1">N1</font> )
 <b>by </b><i><a class="txt" href="#E6:81_1"><i><font color="Green" title="E76">B4</font></i></a>, <a class="txt" href="#E4:81_1"><i><font color="Green" title="E75">B3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:81"/>
<font color="Maroon" title="c3">L1</font> <a href="modelc_3.html#R5" title="MODELC_3:pred.5">is_Finseq_for</a> <font color="Maroon" title="c4">v</font>
 <b>by </b><i><a class="ref" href="modelc_3.html#D19" target="_self" title="MODELC_3:def.19">DefFinseq</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
