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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>for <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="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> holds <br/><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>;<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:45_2_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:45_2_1"/>
(  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a>  &amp; ( for <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a> <font color="Olive">q</font> holds <br/><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> ) )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/>
<a NAME="E5:45_2_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>thus </b><a NAME="E1:45_2_1_1"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a>  )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E85</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_1_2"><b>now </b></a><div class="add"><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="E1:45_2_1_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>thus </b><a NAME="E2:45_2_1_2"/>
 <a href="trees_1.html#K1" target="_self">ProperPrefixes</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:45_2_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K1" target="_self">ProperPrefixes</a> <font color="Maroon">q</font>
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:45_2_1_2_1"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">q</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:45_2_1_2_1"/>
<font color="Maroon">r</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E5:45_2_1_2_1"/><b>then </b><i><font color="Green">E88</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">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Olive">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/>
<b>consider </b><font color="Maroon">s</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E8:45_2_1_2_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E9:45_2_1_2_1"/>
( <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp; <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font> )
 
<b>by </b><i>, , <a class="txt" href="trees_1.html#E1:45"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/>
<a NAME="E10:45_2_1_2_1"/><b>then </b>
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a>  &amp; ( for <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="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> holds <br/><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> ) )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>hence </b><a NAME="E11:45_2_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_1_3"><b>now </b></a><div class="add"><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>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><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>assume </b><a NAME="E1:45_2_1_3"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 ;<br/><a NAME="E2:45_2_1_3"/><b>then </b>
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 <b>by </b><i/>;<br/><a NAME="E3:45_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><a NAME="E4:45_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:45_2_1_3"/><b>then </b>
( <font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a>  &amp; ( for <font color="Olive">r</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> holds <br/><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> ) )
 <b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><b>hence </b><a NAME="E6:45_2_1_3"/>
<font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><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#NM1">Tree</a> <b>by </b><i>, , </i>;<br/>
<b>take </b>
<font color="Maroon">X</font>
;<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>thus </b><a NAME="E9:45_2_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> implies <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> )
 <b>by </b><i/>;<br/>

<b>assume </b><a NAME="E10:45_2_1"/>
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 ;<br/>

<a NAME="E11:45_2_1"/><b>then </b>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a>  &amp; ( for <font color="Olive">r</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> holds <br/><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> ) )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>hence </b><a NAME="E12:45_2_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div>
