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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> ex <font color="Olive">X</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">T</font> &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">q</font> holds <br/> not <font color="Olive">p</font>,<font color="Olive">q</font> <a href="xboole_0.html#R3">are_c=-comparable</a>  ) );<br/>
<a NAME="E1:67_2_1"/>
( 0 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">T</font> &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">q</font> holds <br/> not <font color="Olive">p</font>,<font color="Olive">q</font> <a href="xboole_0.html#R3">are_c=-comparable</a>  ) )
 
<b>by </b><i><a class="ref" href="card_1.html#T78">CARD_1:78</a>, <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E2:67_2_1"/><b>then </b><i><font color="Green">E87</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
;<br/>
<a NAME="E3:67_2_1"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>given </b><font color="Maroon">X</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:67_2_1_1"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">T</font> )
 <b>and </b><br/><a NAME="E2:67_2_1_1"/>
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">q</font> holds <br/> not <font color="Olive">p</font>,<font color="Olive">q</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 ;<br/>

<a NAME="E3:67_2_1_1"/><i><font color="Green">E90</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="trees_1.html#T8" target="_self">Th8</a>, <a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E4:67_2_1_1"/>
(  <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">n</font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">T</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="trees_1.html#T8" target="_self">Th8</a>, <a class="ref" href="card_1.html#D11">CARD_1:def 11</a></i>;<br/>
<b>hence </b><a NAME="E5:67_2_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="trees_1.html#T15" target="_self">Th15</a>, <a class="ref" href="card_1.html#T72">CARD_1:72</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:67_2_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 <b>and </b><br/><a NAME="E6:67_2_1"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(<a class="ref" href="trees_1.html#D1" target="_self">Def1</a>, <a class="ref" href="trees_1.html#T6" target="_self">Th6</a>);<br/></i>
<b>consider </b><font color="Maroon">X</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:67_2_1"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">T</font> &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">q</font> holds <br/> not <font color="Olive">p</font>,<font color="Olive">q</font> <a href="xboole_0.html#R3">are_c=-comparable</a>  ) )
 <b>by </b><i><a class="ref" href="trees_1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E9:67_2_1"/>
<font color="Maroon">X</font> is <a href="trees_1.html#V2" target="_self">AntiChain_of_Prefixes-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:67_2_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">x</font> is   <a href="finseq_1.html#NM1">FinSequence</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="trees_1.html#D13" target="_self">TREES_1:def 13</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:67_2_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:67_2_1_2_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp; <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a>  )
 
<b>by </b><i><a class="txt" href="trees_1.html#E18"><i><font color="Green">Lemma43</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E3:67_2_1_2_1"/>
<font color="Maroon">x</font> is   <a href="finseq_1.html#NM1">FinSequence</a>
 <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>


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

<b>let </b><font color="Maroon">p1</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">p2</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>



<b>assume </b><a NAME="E2:67_2_1_2"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">p1</font>, <font color="Maroon">q2</font> = <font color="Maroon">p2</font> as    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">T</font> <b>by </b><i><a class="txt" href="trees_1.html#E18"><i><font color="Green">Lemma43</font></i></a></i>;<br/>
<a NAME="E4:67_2_1_2"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> )
 
;<br/>
<b>hence </b><a NAME="E5:67_2_1_2"/>
(  not <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font> or  not <font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="xboole_0.html#R3">are_c=-comparable</a>  )
 <b>by </b><i><a class="txt" href="trees_1.html#E18"><i><font color="Green">Lemma43</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as   <a href="trees_1.html#NM5">AntiChain_of_Prefixes</a> ;<br/>
<b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as    <a href="trees_1.html#M4" target="_self">AntiChain_of_Prefixes</a> of <font color="Maroon">T</font> <b>by </b><i><a class="txt" href="trees_1.html#E18"><i><font color="Green">Lemma43</font></i></a>, </i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>take </b>
<font color="Maroon">X</font>
;<br/>

<b>thus </b><a NAME="E12:67_2_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="trees_1.html#E18"><i><font color="Green">Lemma43</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">Y</font> be    <a href="trees_1.html#M4" target="_self">AntiChain_of_Prefixes</a> of <font color="Maroon">T</font>;<br/>

<a NAME="E13:67_2_1"/>
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">T</font> &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">q</font> holds <br/> not <font color="Olive">p</font>,<font color="Olive">q</font> <a href="xboole_0.html#R3">are_c=-comparable</a>  ) )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E14:67_2_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">Y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="trees_1.html#T15" target="_self">Th15</a>, <a class="txt" href="trees_1.html#E18"><i><font color="Green">Lemma43</font></i></a></i>;<br/>


</div>
