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<div class="add">

<b>let </b><font color="Maroon" title="c1">E</font>, <font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span>;<br/>





<b>thus </b><a NAME="E1:80"/>
( <span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a>  implies <span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:80_1"/><i><font color="Green" title="E54">A1</font></i>: 
( <span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a>  &amp; not <span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )
 ;<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><span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c4">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:80_1"/><i><font color="Green" title="E55">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>]
 <b>by </b><i><a class="txt" href="flang_1.html#E1:80_1"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="flang_1.html#T42" target="_self" title="FLANG_1:th.42">Th42</a></i>;<br/>
<a NAME="E4:80_1"/><i><font color="Green" title="E56">A3</font></i>: 
 ex <font color="Olive" title="b1">i</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">i</font>]
 
<b>by </b><i><a class="txt" href="flang_1.html#E3:80_1"><i><font color="Green" title="E55">A2</font></i></a></i>;<br/>
<a NAME="E5:80_1"/>
 ex <font color="Olive" title="b1">n1</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n1</font>] &amp; ( for <font color="Olive" title="b2">n2</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="b2">n2</font>] holds <br/><font color="Olive" title="b1">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n2</font> ) )
 
<b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="flang_1.html#E4:80_1"><i><font color="Green" title="E56">A3</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c5">n1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:80_1"/><i><font color="Green" title="E57">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n1</font>]
 <b>and </b><br/><a NAME="E8:80_1"/><i><font color="Green" title="E58">A5</font></i>: 
for <font color="Olive" title="b1">n2</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">n2</font>] holds <br/><font color="Maroon" title="c5">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n2</font>
 ;<br/>
<div><i><font color="Green" title="E59">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:80_1_1"/>
<font color="Maroon" title="c5">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><b>then consider </b><font color="Maroon" title="c6">n2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:80_1_1"/><i><font color="Green" title="E60">A7</font></i>: 
<font color="Maroon" title="c6">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><a NAME="E4:80_1_1"/>
<span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c6">n2</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="flang_1.html#E7:80_1"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="flang_1.html#E3:80_1_1"><i><font color="Green" title="E60">A7</font></i></a>, <a class="ref" href="flang_1.html#T24" target="_self" title="FLANG_1:th.24">Th24</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> being    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> <b> such that </b><br/><a NAME="E6:80_1_1"/><i><font color="Green" title="E61">A8</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c6">n2</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c8">b</font> )
 <b>by </b><i><a class="ref" href="flang_1.html#D1" target="_self" title="FLANG_1:def.1">Def1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:80_1_1_1"/><i><font color="Green" title="E62">A9</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c9">n2</font> = <font color="Maroon" title="c6">n2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:80_1_1_1"/>
 ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] &amp; <font color="Maroon" title="c5">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">i</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c9">n2</font>
;<br/>

<b>thus </b><a NAME="E1:80_1_1_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">n2</font>] &amp; <font color="Maroon" title="c5">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">n2</font> )
 <b>by </b><i><a class="txt" href="flang_1.html#E3:80_1_1"><i><font color="Green" title="E60">A7</font></i></a>, <a class="txt" href="flang_1.html#E6:80_1_1"><i><font color="Green" title="E61">A8</font></i></a>, <a class="txt" href="flang_1.html#E1:80_1_1_1"><i><font color="Green" title="E62">A9</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:80_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="flang_1.html#E8:80_1"><i><font color="Green" title="E58">A5</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:80_1_1"/>
<span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="flang_1.html#E6:80_1_1"><i><font color="Green" title="E61">A8</font></i></a>, <a class="ref" href="flang_1.html#T4" target="_self" title="FLANG_1:th.4">Th4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:80_1"/><i><font color="Green" title="E60">A10</font></i>: 
( <font color="Maroon" title="c5">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )
 
<b>by </b><i><a class="txt" href="flang_1.html#E7:80_1"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="flang_1.html#T26" target="_self" title="FLANG_1:th.26">Th26</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:80_1_2"/>
<font color="Maroon" title="c5">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:80_1_2"/><b>then </b>
<span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="flang_1.html#K4" title="FLANG_1:func.4">{</a><span class="default"><span class="p2">(<span class="default"><a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font></span>)</span></span><a href="flang_1.html#K4" title="FLANG_1:func.4">}</a></span>
 <b>by </b><i><a class="txt" href="flang_1.html#E7:80_1"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="flang_1.html#T25" target="_self" title="FLANG_1:th.25">Th25</a></i>;<br/><b>hence </b><a NAME="E3:80_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:80_1"/>
contradiction
 <b>by </b><i><a class="txt" href="flang_1.html#E1:80_1"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="flang_1.html#E9:80_1"><i><font color="Green" title="E59">A6</font></i></a>, <a class="txt" href="flang_1.html#E10:80_1"><i><font color="Green" title="E60">A10</font></i></a>, <a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:80"/>
<span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ;<br/>

<a NAME="E3:80"/><b>then </b>
<span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> 1
 
<b>by </b><i><a class="ref" href="flang_1.html#T26" target="_self" title="FLANG_1:th.26">Th26</a></i>;<br/>
<b>hence </b><a NAME="E4:80"/>
<span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K8" title="FLANG_1:func.8">*</a> 
 <b>by </b><i><a class="ref" href="flang_1.html#T42" target="_self" title="FLANG_1:th.42">Th42</a></i>;<br/>


</div>
