<?xml version="1.0"?>
<div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a<sub>1</sub></font> </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a<sub>1</sub></font>));<br/>
<a NAME="E1:71_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">k1</font>, <font color="Olive">k2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k2</font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">d</font>) holds <br/><font color="Olive">k1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">d</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon">k1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">k2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:71_1_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k2</font> &amp; <font color="Maroon">k2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">d</font>) </span>}</span>  )
 ;<br/>

<a NAME="E2:71_1_1"/><b>then </b>
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">d</font>) )
 
;<br/>
<a NAME="E3:71_1_1"/><b>then </b>
<font color="Maroon">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">d</font>)
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E4:71_1_1"/>
<font color="Maroon">k1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">d</font>)
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">T</font> being   <a href="trees_2.html#NM6">DecoratedTree</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:71_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive">t</font> being   <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font> holds <br/> (  <a href="trees_2.html#K1" target="_self">succ</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">T</font> <a href="trees_2.html#K3" target="_self">.</a> <font color="Olive">t</font>) </span>}</span>  &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">T</font> <a href="trees_2.html#K3" target="_self">.</a> <font color="Olive">t</font>) holds <br/><font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3" target="_self">.</a> <font color="Olive">t</font></span>)</span>,<font color="Olive">n</font> ) ) ) )
 <b>from </b><i><a class="ref" href="trees_2.html#S8" target="_self">TREES_2:sch 8</a>();<br/></i>
<b>take </b>
<font color="Maroon">T</font>
;<br/>

<b>thus </b><a NAME="E4:71_1"/>
<font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">t</font> be    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font>;<br/>

<a NAME="E5:71_1"/><i><font color="Green">E60</font></i>: 
 <a href="trees_2.html#K1" target="_self">succ</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3" target="_self">.</a> <font color="Maroon">t</font></span>)</span>)) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E6:71_1"/>
 <a href="trees_2.html#K1" target="_self">succ</a> <font color="Maroon">t</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">i</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3" target="_self">.</a> <font color="Maroon">t</font></span>)</span>) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_2"><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="E1:71_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1" target="_self">succ</a> <font color="Maroon">t</font>
 ;<br/>

<b>then consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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">l</font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3" target="_self">.</a> <font color="Maroon">t</font></span>)</span>)) )
 <b>by </b><i/>;<br/>
<a NAME="E4:71_1_2"/>
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3" target="_self">.</a> <font color="Maroon">t</font></span>)</span>) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:71_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">i</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3" target="_self">.</a> <font color="Maroon">t</font></span>)</span>) </span>}</span> 
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E7:71_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">i</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3" target="_self">.</a> <font color="Maroon">t</font></span>)</span>) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="trees_2.html#K1" target="_self">succ</a> <font color="Maroon">t</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_3"><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="E1:71_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">i</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3" target="_self">.</a> <font color="Maroon">t</font></span>)</span>) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71_1_3"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</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">l</font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3" target="_self">.</a> <font color="Maroon">t</font></span>)</span>) )
 ;<br/>
<a NAME="E4:71_1_3"/>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3" target="_self">.</a> <font color="Maroon">t</font></span>)</span>))
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:71_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1" target="_self">succ</a> <font color="Maroon">t</font>
 <b>by </b><i>, </i>;<br/>


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

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

<b>assume </b><a NAME="E8:71_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3" target="_self">.</a> <font color="Maroon">t</font></span>)</span>)
 ;<br/>

<a NAME="E9:71_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3" target="_self">.</a> <font color="Maroon">t</font></span>)</span>))
 
;<br/>
<b>hence </b><a NAME="E10:71_1"/>
<font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">t</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">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_2.html#K3" target="_self">.</a> <font color="Maroon">t</font></span>)</span>,<font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


</div>
