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<b>let </b><font color="Maroon">t</font> be  <a href="trees_9.html#V2" target="_self">finite-branching</a> <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">t</font>;<br/>



<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  = <font color="Maroon">p</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">a<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/>
<a NAME="E1:28"/><i><font color="Green">E22</font></i>: 
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>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">p</font> holds <br/> 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="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_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="E1:28_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:28_1"/><b>then </b>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">t</font> )
 
;<br/>
<b>hence </b><a NAME="E3:28_1"/>
 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">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:28"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">j</font>) <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>) <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:28_2"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">p</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">j</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">p</font> )
 ;<br/>

<a NAME="E2:28_2"/><b>then </b>
<font color="Maroon">p</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">i</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="trees_1.html#D5">TREES_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:28_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">i</font>) <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">p</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:28"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">j</font>) holds <br/><font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:28_3"/>
<font color="Maroon">p</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">i</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</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">j</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/>

<b>hence </b><font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</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">j</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
<br/>.= 
<font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:28"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>) <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">p</font> iff <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="trees_2.html#K1">succ</a> <font color="Maroon">p</font></span>)</span> )
 <b>from </b><i><a class="ref" href="trees_9.html#S1" target="_self">TREES_9:sch 1</a>(, , );</i><br/>


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