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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">t</font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">t</font>];<br/>
<a NAME="E1:6_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E2:6_1"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:6_1_1"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">t</font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">t</font>]
 ;<br/>

<b>let </b><font color="Maroon">t</font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E2:6_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:6_1_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</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#T50">FINSEQ_1:50</a></i>;<br/>
<b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i>, <a class="ref" href="trees_1.html#T46">TREES_1:46</a></i>;<br/>
<a NAME="E7:6_1_1"/>
(  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">t</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">t</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T43">FINSEQ_1:43</a></i>;<br/>
<a NAME="E8:6_1_1"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E10:6_1_1"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">v</font>] &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E11:6_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">t</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:6_1"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>let </b><font color="Maroon">t</font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<a NAME="E4:6_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font>
 
;<br/>
<b>hence </b><a NAME="E5:6_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">t</font>]
 <b>by </b><i/>;<br/>


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