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

<span class="kw">set </span><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/>
<span class="kw">take </span>
 <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E1:132_1_1"/><span class="lab"><font color="Green" title="E111">A1</font></span>: 
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#D11" title="MODELC_2:def.11">MODELC_2:def 11</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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:132_1_1_1"/>
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:132_1_1_1"/><span class="lab"><font color="Green" title="E112">A2</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">and </span><br/><a NAME="E4:132_1_1_1"/>
 <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>
 <span class="kw">and </span><br/><a NAME="E5:132_1_1_1"/><span class="lab"><font color="Green" title="E113">A3</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E6:132_1_1_1"/><span class="lab"><font color="Green" title="E114">A4</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:132_1_1_1"/><span class="lab"><font color="Green" title="E115">A5</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 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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#D22" title="MODELC_2:def.22">MODELC_2:def 22</a></span>;<br/>
<a NAME="E8:132_1_1_1"/>
<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"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><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/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:132_1_1_1"><span class="lab"><font color="Green" title="E112">A2</font></span></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="132_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:132_1_1_1_1_1"/><span class="kw">then </span>
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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:132_1_1_1"><span class="lab"><font color="Green" title="E112">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:132_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A6</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T19" title="NAT_1:th.19">NAT_1:19</a></span>;<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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/><a NAME="E5:132_1_1_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">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="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="Olive" title="b2">G1</font> &amp; <font color="Olive" title="b1">H1</font> <a href="modelc_2.html#R1" title="MODELC_2:pred.1">is_immediate_constituent_of</a> <font color="Olive" title="b2">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:132_1_1_1"><span class="lab"><font color="Green" title="E115">A5</font></span></a>, <a class="txt" href="#E3:132_1_1_1_1_1"><span class="lab"><font color="Green" title="E116">A6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:132_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:132_1_1"><span class="lab"><font color="Green" title="E111">A1</font></span></a>, <a class="txt" href="#E6:132_1_1_1"><span class="lab"><font color="Green" title="E114">A4</font></span></a>, <a class="ref" href="modelc_2.html#T19" title="MODELC_2:th.19">MODELC_2:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:132_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:132_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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:132_1_1"><span class="lab"><font color="Green" title="E111">A1</font></span></a>, <a class="txt" href="#E5:132_1_1_1"><span class="lab"><font color="Green" title="E113">A3</font></span></a>, <a class="txt" href="#E6:132_1_1_1"><span class="lab"><font color="Green" title="E114">A4</font></span></a>, <a class="ref" href="modelc_2.html#T78" title="MODELC_2:th.78">MODELC_2:78</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:132_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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#D37" target="_self" title="MODELC_3:def.37">Def37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
