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

<b>let </b><font color="Maroon" title="c1">H</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"> for <font color="Olive" title="b1">W</font>, <font color="Olive" title="b2">W1</font>, <font color="Olive" title="b3">W2</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_2.html#K13" title="MODELC_2:func.13">Subformulae</a> <font color="Maroon" title="c1">H</font></span>)</span>  st <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b3">W2</font> holds <br/> <a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Olive" title="b1">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Olive" title="b2">W1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Olive" title="b3">W2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">W</font>, <font color="Maroon" title="c3">W1</font>, <font color="Maroon" title="c4">W2</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_2.html#K13" title="MODELC_2:func.13">Subformulae</a> <font color="Maroon" title="c1">H</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="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">W2</font> implies  <a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c2">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c3">W1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c4">W2</font></span>)</span> )</span><br/>







<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">W</font>, <font color="Olive" title="b2">W1</font>, <font color="Olive" title="b3">W2</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_2.html#K13" title="MODELC_2:func.13">Subformulae</a> <font color="Maroon" title="c1">H</font></span>)</span>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">W1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b3">W2</font> holds <br/> <a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Olive" title="b1">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Olive" title="b2">W1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Olive" title="b3">W2</font></span>)</span>;<br/>
<a NAME="E1:69"/><i><font color="Green" title="E59">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">W</font>, <font color="Maroon" title="c6">W1</font>, <font color="Maroon" title="c7">W2</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_2.html#K13" title="MODELC_2:func.13">Subformulae</a> <font color="Maroon" title="c1">H</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">W1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">W2</font> implies  <a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c5">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c6">W1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c7">W2</font></span>)</span> )</span><br/>





<b>assume </b><a NAME="E1:69_1"/><i><font color="Green" title="E60">B1</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">W1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">W2</font> or  <a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c5">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c6">W1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c7">W2</font></span>)</span> )</span><br/>

<a NAME="E2:69_1"/><i><font color="Green" title="E61">B2</font></i>: 
<font color="Maroon" title="c6">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_3.html#K7" title="MODELC_3:func.7">{}</a> <font color="Maroon" title="c1">H</font>
 
<b>by </b><i><a class="txt" href="#E1:69_1"><i><font color="Green" title="E60">B1</font></i></a></i>;<br/>
<a NAME="E3:69_1"/><b>then </b>
 <a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c6">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="modelc_3.html#T13" target="_self" title="MODELC_3:th.13">Thlen10</a></i>;<br/>
<b>hence </b><a NAME="E4:69_1"/>
(  not <font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">W2</font> or  <a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c5">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c6">W1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c7">W2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E2:69_1"><i><font color="Green" title="E61">B2</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>
<a NAME="E2:69"/><i><font color="Green" title="E60">A2</font></i>: 
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="69_2"><b>proof </b></a><div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><a NAME="E1:69_2"/><i><font color="Green" title="E61">B1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="E2:69_2"/>
<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="69_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">W</font>, <font color="Maroon" title="c7">W1</font>, <font color="Maroon" title="c8">W2</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_2.html#K13" title="MODELC_2:func.13">Subformulae</a> <font color="Maroon" title="c1">H</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">W1</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="c6">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">W1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">W2</font> implies  <a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c6">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c7">W1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c8">W2</font></span>)</span> )</span><br/>





<b>assume </b><a NAME="E1:69_2_1"/><i><font color="Green" title="E62">B2</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">W1</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">W1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">W2</font> or  <a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c6">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c7">W1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c8">W2</font></span>)</span> )</span><br/>

<a NAME="E2:69_2_1"/>
( <font color="Maroon" title="c6">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">W1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">W2</font> implies  <a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c6">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c7">W1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c8">W2</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_2_1_1"/><i><font color="Green" title="E63">C1</font></i>: 
<font color="Maroon" title="c6">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">W1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">W2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c6">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c7">W1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c8">W2</font></span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_2_1_1_1_1"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">W1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> or  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">W1</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><b>by </b><i><a class="txt" href="#E1:69_2_1"><i><font color="Green" title="E62">B2</font></i></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:69_2_1_1_1_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">W1</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c6">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c7">W1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c8">W2</font></span>)</span></span><br/><div class="add"><b>hence </b><a NAME="E2:69_2_1_1_1_1_1"/>
 <a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c6">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c7">W1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c8">W2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:69_2"><i><font color="Green" title="E61">B1</font></i></a>, <a class="txt" href="#E1:69_2_1_1"><i><font color="Green" title="E63">C1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:69_2_1_1_1_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">W1</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c6">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c7">W1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c8">W2</font></span>)</span></span><br/><div class="add"><a NAME="E2:69_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">W1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:69_2_1_1_1_1_2"/><i><font color="Green" title="E64">D1</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">W1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E5:69_2_1_1_1_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_3.html#K1" title="MODELC_3:func.1">Subformulae</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="txt" href="#E4:69_2_1_1_1_1_2"><i><font color="Green" title="E64">D1</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c9">x</font> as   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> <b>by </b><i><a class="ref" href="modelc_2.html#T1" title="MODELC_2:th.1">MODELC_2:1</a></i>;<br/><b>set </b><font color="Maroon" title="c11">X</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><b>set </b><font color="Maroon" title="c12">Y</font> = <font color="Maroon" title="c7">W1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><b>set </b><font color="Maroon" title="c13">Z</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">W1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">W2</font>;<br/><a NAME="E7:69_2_1_1_1_1_2"/><i><font color="Green" title="E65">D4</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">W1</font>
 <b>by </b><i><a class="txt" href="#E4:69_2_1_1_1_1_2"><i><font color="Green" title="E64">D1</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>then </b> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">W1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">W1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">W1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
<b>by </b><i><a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a></i>
;<br/><a NAME="E9:69_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">W1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">W1</font>
 ;<br/><a NAME="E10:69_2_1_1_1_1_2"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">W1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="#E1:69_2_1"><i><font color="Green" title="E62">B2</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E11:69_2_1_1_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">W1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">W1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">W2</font> implies  <a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">W1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">W2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">W1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c8">W2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:69_2"><i><font color="Green" title="E61">B1</font></i></a></i>;<br/><a NAME="E12:69_2_1_1_1_1_2"/><b>then </b>
 <a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">W1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">W2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c7">W1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c8">W2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:69_2_1_1_1_1_2"><i><font color="Green" title="E64">D1</font></i></a>, <a class="ref" href="modelc_3.html#T10" target="_self" title="MODELC_3:th.10">Thlen07</a></i>;<br/><a NAME="E13:69_2_1_1_1_1_2"/><b>then </b>
 <a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">W1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">W2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c7">W1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c8">W2</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">x</font></span>)</span>
 ;<br/><a NAME="E14:69_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E66">D5</font></i>: 
<span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">W1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">W2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c7">W1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c8">W2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">W1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">W2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">W1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">W2</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>
<br/>.= 
<font color="Maroon" title="c7">W1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">W2</font>
<b>by </b><i><a class="txt" href="#E7:69_2_1_1_1_1_2"><i><font color="Green" title="E65">D4</font></i></a>, <a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></i>
;<br/><a NAME="E16:69_2_1_1_1_1_2"/><b>then </b>
 <a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c6">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">W1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">W2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:69_2_1_1"><i><font color="Green" title="E63">C1</font></i></a>, <a class="ref" href="modelc_3.html#T12" target="_self" title="MODELC_3:th.12">Thlen09</a></i>;<br/><b>hence </b><a NAME="E17:69_2_1_1_1_1_2"/>
 <a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c6">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c7">W1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c8">W2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E14:69_2_1_1_1_1_2"><i><font color="Green" title="E66">D5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:69_2_1_1"/>
 <a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c6">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c7">W1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c8">W2</font></span>)</span>
 ; <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="E3:69_2_1"/>
(  not <font color="Maroon" title="c6">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">W1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">W2</font> or  <a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c6">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c7">W1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c8">W2</font></span>)</span> )
 ; <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="E3:69_2"/>
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:69"/><i><font color="Green" title="E61">A3</font></i>: 
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>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E1:69"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="#E2:69"><i><font color="Green" title="E60">A2</font></i></a>);<br/></i>
<b>set </b><font color="Maroon" title="c5">k</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">W1</font>;<br/>
<a NAME="E4:69"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">W1</font>]
 
<b>by </b><i><a class="txt" href="#E3:69"><i><font color="Green" title="E61">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:69"/>
( <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">W2</font> implies  <a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c2">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c3">W1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="modelc_3.html#K18" title="MODELC_3:func.18">len</a> <font color="Maroon" title="c4">W2</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
