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

<span class="kw">let </span><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">f1</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f1</font></span>)</span></span><br/>



<span class="kw">set </span><font color="Maroon" title="c5">c1</font> =  <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">f1</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">c2</font> = <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f1</font></span>)</span>;<br/>
<span class="lab"><font color="Green" title="E17">A1</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">f1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#D4" target="_self" onmouseover="rs('ltlaxio2/D4')" onmouseout="rh()">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#D4" target="_self" onmouseover="rs('ltlaxio2/D4')" onmouseout="rh()">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#D4" target="_self" onmouseover="rs('ltlaxio2/D4')" onmouseout="rh()">Def4</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>
;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="24_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">j</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:24_2"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><a NAME="E2:24_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A3</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E3:24_2"/>
<font color="Maroon" title="c7">j</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> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_2"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E4:24_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<font color="Maroon" title="c7">j</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">f1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#D4" target="_self" onmouseover="rs('ltlaxio2/D4')" onmouseout="rh()">Def4</a></span>;<br/><a NAME="E5:24_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A5</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">f1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a>, <a class="txt" href="#E2:24_2"><span class="lab"><font color="Green" title="E19">A3</font></span></a></span>;<br/><a NAME="E6:24_2"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_2"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="finseq_3.html#T29" onmouseover="rs('finseq_3/T29')" onmouseout="rh()">FINSEQ_3:29</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_2_1"/>
( <font color="Maroon" title="c7">j</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">f</font> or <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_2_1_1"/><span class="lab"><font color="Green" title="E23">A7</font></span>: 
<font color="Maroon" title="c7">j</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">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:24_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">j</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> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#D4" target="_self" onmouseover="rs('ltlaxio2/D4')" onmouseout="rh()">Def4</a></span>;<br/><a NAME="E3:24_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A8</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_2"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E4:24_2_1_1"/><span class="lab"><font color="Green" title="E25">A9</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_2_1_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E2:24_2"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a>, <a class="txt" href="#E1:24_2"><span class="lab"><font color="Green" title="E18">A2</font></span></a></span>
<br/>.= 
 <a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T8" target="_self" onmouseover="rs('ltlaxio2/T8')" onmouseout="rh()">Th8</a>, <a class="txt" href="#E5:24_2"><span class="lab"><font color="Green" title="E21">A5</font></span></a></span>
<br/>.= 
 <a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a>, <a class="txt" href="#E4:24_2_1_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T8" target="_self" onmouseover="rs('ltlaxio2/T8')" onmouseout="rh()">Th8</a>, <a class="txt" href="#E4:24_2_1_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a>, <a class="txt" href="#E3:24_2_1_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a>, <a class="txt" href="#E6:24_2"><span class="lab"><font color="Green" title="E22">A6</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_2_1_2"/><span class="lab"><font color="Green" title="E23">A10</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:24_2_1_2"/><span class="lab"><font color="Green" title="E24">A11</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" onmouseover="rs('nat_1/T10')" onmouseout="rh()">NAT_1:10</a></span>;<br/><div><span class="lab"><font color="Green" title="E25">A12</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="24_2_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f1</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:24_2_1_2_1"/>
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:24_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a>, <a class="txt" href="#E3:24_2_1_2"><span class="lab"><font color="Green" title="E24">A11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:24_2_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_2"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><a NAME="E5:24_2_1_2"/>
( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 ;<br/><a NAME="E6:24_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A13</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T25" onmouseover="rs('nat_1/T25')" onmouseout="rh()">NAT_1:25</a>, <a class="txt" href="#E3:24_2_1_2"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="txt" href="#E1:24_2_1_2"><span class="lab"><font color="Green" title="E23">A10</font></span></a></span>;<br/><a NAME="E7:24_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A14</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_2_1_2"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E8:24_2_1_2"/><span class="lab"><font color="Green" title="E28">A15</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_2_1_2"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="ref" href="ltlaxio2.html#D4" target="_self" onmouseover="rs('ltlaxio2/D4')" onmouseout="rh()">Def4</a></span>;<br/><a NAME="E9:24_2_1_2"/>
<font color="Maroon" title="c8">k</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> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#D4" target="_self" onmouseover="rs('ltlaxio2/D4')" onmouseout="rh()">Def4</a>, <a class="txt" href="#E4:24_2_1_2"><span class="lab"><font color="Green" title="E25">A12</font></span></a></span>;<br/><a NAME="E10:24_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A16</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24_2_1_2"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a>, <a class="txt" href="#E1:24_2"><span class="lab"><font color="Green" title="E18">A2</font></span></a></span>
<br/>.= 
 <a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T8" target="_self" onmouseover="rs('ltlaxio2/T8')" onmouseout="rh()">Th8</a>, <a class="txt" href="#E5:24_2"><span class="lab"><font color="Green" title="E21">A5</font></span></a></span>
<br/>.= 
 <a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T69" onmouseover="rs('finseq_4/T69')" onmouseout="rh()">FINSEQ_4:69</a>, <a class="txt" href="#E7:24_2_1_2"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="#E3:24_2_1_2"><span class="lab"><font color="Green" title="E24">A11</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T8" target="_self" onmouseover="rs('ltlaxio2/T8')" onmouseout="rh()">Th8</a>, <a class="txt" href="#E7:24_2_1_2"><span class="lab"><font color="Green" title="E27">A14</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T69" onmouseover="rs('finseq_4/T69')" onmouseout="rh()">FINSEQ_4:69</a>, <a class="txt" href="#E10:24_2_1_2"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E8:24_2_1_2"><span class="lab"><font color="Green" title="E28">A15</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a>, <a class="txt" href="#E6:24_2"><span class="lab"><font color="Green" title="E22">A6</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:24"/>
 <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <font color="Maroon" title="c4">f1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T9" onmouseover="rs('finseq_2/T9')" onmouseout="rh()">FINSEQ_2:9</a>, <a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E17">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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