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

<b>let </b><font color="Maroon">H</font> be   <a href="modelc_1.html#NM1">CTL-formula</a>;<br/>

<a NAME="E1:21"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">H</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a> 
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E2:21"/><i><font color="Green">E28</font></i>: 
( not <font color="Maroon">H</font> is <a href="modelc_1.html#V2" target="_self">atomic</a> &amp; not <font color="Maroon">H</font> is <a href="modelc_1.html#V3" target="_self">negative</a> &amp; not <font color="Maroon">H</font> is <a href="modelc_1.html#V4" target="_self">conjunctive</a> &amp; not <font color="Maroon">H</font> is <a href="modelc_1.html#V5" target="_self">ExistNext</a> &amp; not <font color="Maroon">H</font> is <a href="modelc_1.html#V6" target="_self">ExistGlobal</a> &amp; not <font color="Maroon">H</font> is <a href="modelc_1.html#V7" target="_self">ExistUntill</a> )
 ;<br/>

<a NAME="E3:21"/><i><font color="Green">E50</font></i>: 
not <a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span> is <a href="xboole_0.html#V1">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<a NAME="E1:21_1"/><i><font color="Green">E52</font></i>: 
 <a href="modelc_1.html#K7" target="_self">atom.</a> 0 <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E2:21_1"/>
 <a href="modelc_1.html#K7" target="_self">atom.</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:21_1"/><b>then </b>
not  <a href="modelc_1.html#K7" target="_self">atom.</a> 0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:21_1"/>
not <a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:21"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Olive">a</font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:21_2"/><i><font color="Green">E55</font></i>: 
 <a href="modelc_1.html#K7" target="_self">atom.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a> 
 <b>by </b><i/>;<br/><a NAME="E2:21_2"/>
 <a href="modelc_1.html#K7" target="_self">atom.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:21_2"/><b>then </b>
not  <a href="modelc_1.html#K7" target="_self">atom.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:21_2"/>
 <a href="modelc_1.html#K7" target="_self">atom.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1"><i><font color="Green">Lemma23</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:21_3"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:21_3"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="modelc_1.html#K8" target="_self">'not'</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a> 
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">H1</font> = <font color="Maroon">p</font> as   <a href="modelc_1.html#NM1">CTL-formula</a> <b>by </b><i>, </i>;<br/><a NAME="E4:21_3"/>
 <a href="modelc_1.html#K8" target="_self">'not'</a> <font color="Maroon">H1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:21_3"/><b>then </b>
not  <a href="modelc_1.html#K8" target="_self">'not'</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:21_3"/>
 <a href="modelc_1.html#K8" target="_self">'not'</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">q</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:21_4"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><a NAME="E2:21_4"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">p</font> <a href="modelc_1.html#K9" target="_self">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a> 
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">p</font>, <font color="Maroon">G</font> = <font color="Maroon">q</font> as   <a href="modelc_1.html#NM1">CTL-formula</a> <b>by </b><i>, </i>;<br/><a NAME="E4:21_4"/>
<font color="Maroon">F</font> <a href="modelc_1.html#K9" target="_self">'&amp;'</a> <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:21_4"/><b>then </b>
not <font color="Maroon">p</font> <a href="modelc_1.html#K9" target="_self">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:21_4"/>
<font color="Maroon">p</font> <a href="modelc_1.html#K9" target="_self">'&amp;'</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:21_5"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:21_5"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="modelc_1.html#K10" target="_self">EX</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a> 
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">H1</font> = <font color="Maroon">p</font> as   <a href="modelc_1.html#NM1">CTL-formula</a> <b>by </b><i>, </i>;<br/><a NAME="E4:21_5"/>
 <a href="modelc_1.html#K10" target="_self">EX</a> <font color="Maroon">H1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:21_5"/><b>then </b>
not  <a href="modelc_1.html#K10" target="_self">EX</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:21_5"/>
 <a href="modelc_1.html#K10" target="_self">EX</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:21"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E65</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:21_6"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:21_6"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="modelc_1.html#K11" target="_self">EG</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a> 
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">H1</font> = <font color="Maroon">p</font> as   <a href="modelc_1.html#NM1">CTL-formula</a> <b>by </b><i><a class="txt" href="modelc_1.html#E1:21"><i><font color="Green">E27</font></i></a>, </i>;<br/><a NAME="E4:21_6"/>
 <a href="modelc_1.html#K11" target="_self">EG</a> <font color="Maroon">H1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:21_6"/><b>then </b>
not  <a href="modelc_1.html#K11" target="_self">EG</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:21_6"/>
 <a href="modelc_1.html#K11" target="_self">EG</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E2:21"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">q</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:21_7"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><a NAME="E2:21_7"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">p</font> <a href="modelc_1.html#K12" target="_self">EU</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a> 
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">p</font>, <font color="Maroon">G</font> = <font color="Maroon">q</font> as   <a href="modelc_1.html#NM1">CTL-formula</a> <b>by </b><i><a class="txt" href="modelc_1.html#E1:21"><i><font color="Green">E27</font></i></a>, </i>;<br/><a NAME="E4:21_7"/>
<font color="Maroon">F</font> <a href="modelc_1.html#K12" target="_self">EU</a> <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:21_7"/><b>then </b>
not <font color="Maroon">p</font> <a href="modelc_1.html#K12" target="_self">EU</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:21_7"/>
<font color="Maroon">p</font> <a href="modelc_1.html#K12" target="_self">EU</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E2:21"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:21"/><b>then </b>
 <a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  <a href="tarski.html#R1">c=</a> <a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, , , <a class="txt" href="modelc_1.html#E1"><i><font color="Green">Lemma23</font></i></a>, , , , </i>;<br/>
<a NAME="E12:21"/><b>then </b>
<font color="Maroon">H</font> <a href="hidden.html#R2">in</a> <a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E13:21"/><b>then </b>
( not <font color="Maroon">H</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">H</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E14:21"/>
contradiction
 ;<br/>


</div>
