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

<b>set </b><font color="Maroon" title="c1">a</font> =  <a href="modelc_2.html#K2" title="MODELC_2:func.2">atom.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<a NAME="E1:132_1_1"/><i><font color="Green" title="E111">A1</font></i>: 
 <a href="modelc_2.html#K2" title="MODELC_2:func.2">atom.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is <a href="modelc_2.html#V2" title="MODELC_2:attr.2">atomic</a>
 
<b>by </b><i><a class="ref" href="modelc_2.html#D11" title="MODELC_2:def.11">MODELC_2:def 11</a></i>;<br/>
<b>take </b>
 <a href="modelc_2.html#K2" title="MODELC_2:func.2">atom.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="modelc_2.html#K2" title="MODELC_2:func.2">atom.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is <a href="modelc_3.html#V5" title="MODELC_3:attr.5">neg-inner-most</a></span><br/>

<a NAME="E2:132_1_1"/>
for <font color="Olive" title="b1">G</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a>  st <font color="Olive" title="b1">G</font> <a href="modelc_2.html#R2" title="MODELC_2:pred.2">is_subformula_of</a>  <a href="modelc_2.html#K2" title="MODELC_2:func.2">atom.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">G</font> is <a href="modelc_2.html#V3" title="MODELC_2:attr.3">negative</a> holds <br/> <a href="modelc_2.html#K10" title="MODELC_2:func.10">the_argument_of</a> <font color="Olive" title="b1">G</font> is <a href="modelc_2.html#V2" title="MODELC_2:attr.2">atomic</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="132_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">G</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">G</font> <a href="modelc_2.html#R2" title="MODELC_2:pred.2">is_subformula_of</a>  <a href="modelc_2.html#K2" title="MODELC_2:func.2">atom.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">G</font> is <a href="modelc_2.html#V3" title="MODELC_2:attr.3">negative</a> implies  <a href="modelc_2.html#K10" title="MODELC_2:func.10">the_argument_of</a> <font color="Maroon" title="c2">G</font> is <a href="modelc_2.html#V2" title="MODELC_2:attr.2">atomic</a> )</span><br/>

<b>assume </b><a NAME="E1:132_1_1_1"/><i><font color="Green" title="E112">B1</font></i>: 
<font color="Maroon" title="c2">G</font> <a href="modelc_2.html#R2" title="MODELC_2:pred.2">is_subformula_of</a>  <a href="modelc_2.html#K2" title="MODELC_2:func.2">atom.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">G</font> is <a href="modelc_2.html#V3" title="MODELC_2:attr.3">negative</a> or  <a href="modelc_2.html#K10" title="MODELC_2:func.10">the_argument_of</a> <font color="Maroon" title="c2">G</font> is <a href="modelc_2.html#V2" title="MODELC_2:attr.2">atomic</a> )</span><br/>

<b>consider </b><font color="Maroon" title="c3">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>, <font color="Maroon" title="c4">L</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E3:132_1_1_1"/><i><font color="Green" title="E113">B2</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K2" title="MODELC_2:func.2">atom.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( 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> <font color="Maroon" title="c3">n</font> holds <br/> ex <font color="Olive" title="b2">H1</font>, <font color="Olive" title="b3">G1</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> st <br/>( <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H1</font> &amp; <font color="Maroon" title="c4">L</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">G1</font> &amp; <font color="Olive" title="b2">H1</font> <a href="modelc_2.html#R1" title="MODELC_2:pred.1">is_immediate_constituent_of</a> <font color="Olive" title="b3">G1</font> ) ) )
 <b>by </b><i><a class="txt" href="#E1:132_1_1_1"><i><font color="Green" title="E112">B1</font></i></a>, <a class="ref" href="modelc_2.html#D22" title="MODELC_2:def.22">MODELC_2:def 22</a></i>;<br/>
<a NAME="E4:132_1_1_1"/><i><font color="Green" title="E114">B3</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="132_1_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c5">k</font> = <font color="Maroon" title="c3">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1;<br/>
<b>reconsider </b><font color="Maroon" title="c6">k</font> = <font color="Maroon" title="c3">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="#E3:132_1_1_1"><i><font color="Green" title="E113">B2</font></i></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="132_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:132_1_1_1_1_1"/>
not <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:132_1_1_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="#E3:132_1_1_1"><i><font color="Green" title="E113">B2</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E3:132_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E115">C0</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T19" title="NAT_1:th.19">NAT_1:19</a></i>;<br/><a NAME="E4:132_1_1_1_1_1"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">H1</font>, <font color="Maroon" title="c8">G1</font> being   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a><b> such that </b><br/><a NAME="E6:132_1_1_1_1_1"/><i><font color="Green" title="E116">C1</font></i>: 
( <font color="Maroon" title="c4">L</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="c7">H1</font> &amp; <font color="Maroon" title="c4">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">G1</font> &amp; <font color="Maroon" title="c7">H1</font> <a href="modelc_2.html#R1" title="MODELC_2:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c8">G1</font> )
 <b>by </b><i><a class="txt" href="#E3:132_1_1_1_1_1"><i><font color="Green" title="E115">C0</font></i></a>, <a class="txt" href="#E3:132_1_1_1"><i><font color="Green" title="E113">B2</font></i></a></i>;<br/><b>thus </b><a NAME="E7:132_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:132_1_1"><i><font color="Green" title="E111">A1</font></i></a>, <a class="txt" href="#E3:132_1_1_1"><i><font color="Green" title="E113">B2</font></i></a>, <a class="txt" href="#E6:132_1_1_1_1_1"><i><font color="Green" title="E116">C1</font></i></a>, <a class="ref" href="modelc_2.html#T19" title="MODELC_2:th.19">MODELC_2:19</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="E3:132_1_1_1_1"/>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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>thus </b><a NAME="E5:132_1_1_1"/>
(  not <font color="Maroon" title="c2">G</font> is <a href="modelc_2.html#V3" title="MODELC_2:attr.3">negative</a> or  <a href="modelc_2.html#K10" title="MODELC_2:func.10">the_argument_of</a> <font color="Maroon" title="c2">G</font> is <a href="modelc_2.html#V2" title="MODELC_2:attr.2">atomic</a> )
 <b>by </b><i><a class="txt" href="#E1:132_1_1"><i><font color="Green" title="E111">A1</font></i></a>, <a class="txt" href="#E3:132_1_1_1"><i><font color="Green" title="E113">B2</font></i></a>, <a class="txt" href="#E4:132_1_1_1"><i><font color="Green" title="E114">B3</font></i></a>, <a class="ref" href="modelc_2.html#T78" title="MODELC_2:th.78">MODELC_2:78</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="E3:132_1_1"/>
 <a href="modelc_2.html#K2" title="MODELC_2:func.2">atom.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is <a href="modelc_3.html#V5" title="MODELC_3:attr.5">neg-inner-most</a>
 <b>by </b><i><a class="ref" href="modelc_3.html#D37" target="_self" title="MODELC_3:def.37">DefNegIM</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>
