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

<b>let </b><font color="Maroon">T0</font> be   <a href="trees_1.html#NM1">Tree</a>, <font color="Maroon">T1</font> be   <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">t</font> be    <a href="trees_1.html#M1">Element</a> of  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font>;<br/>



<b>set </b><font color="Maroon">RT</font> =  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:12_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:12_1"/><i><font color="Green">E30</font></i>: 
 <a href="trees_2.html#K1">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="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">n</font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="trees_2.html#D5">TREES_2:def 5</a></i>;<br/><a NAME="E3:12_1"/>
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">T0</font> &amp; <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a>, <a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/><a NAME="E4:12_1"/><b>then </b>
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font>
 <b>by </b><i><a class="ref" href="trees_3.html#T71">TREES_3:71</a></i>;<br/><a NAME="E5:12_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><a NAME="E6:12_1"/>
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> &amp; <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a>, <a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/><a NAME="E7:12_1"/><b>then </b>
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font>
 <b>by </b><i><a class="ref" href="trees_3.html#T71">TREES_3:71</a></i>;<br/><a NAME="E8:12_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:12_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font>
 ;<br/><b>then consider </b><font color="Maroon">n</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="E3:12_1_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> &amp; <font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font> )
 <b>by </b><i/>;<br/><a NAME="E4:12_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x1</font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">p</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E7:12_1_1_1"/><i><font color="Green">E39</font></i>: 
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T0</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p</font> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p</font> ) )
 <b>by </b><i>, <a class="ref" href="trees_3.html#T71">TREES_3:71</a></i>;<br/><a NAME="E8:12_1_1_1"/>
( <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/><a NAME="E9:12_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E10:12_1_1_1"/><b>then </b>
( <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="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> or <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="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><b>hence </b><a NAME="E11:12_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><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="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:12_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><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="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 ;<br/><a NAME="E3:12_1_1"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> or <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E4:12_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:12_1"/>
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><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="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T0</font>;<br/><b>assume </b><a NAME="E1:12_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font>
 ;<br/><b>let </b><font color="Maroon">sp</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:12_2_1"/>
<font color="Maroon">sp</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:12_2_1"/><b>then </b>
<font color="Maroon">sp</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">n</font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</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="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T0</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="trees_2.html#D5">TREES_2:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">n</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="E4:12_2_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">sp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> &amp; <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T0</font> )
 ;<br/><a NAME="E5:12_2_1"/>
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font>
 <b>by </b><i>, <a class="ref" href="trees_3.html#T72">TREES_3:72</a></i>;<br/><a NAME="E6:12_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><a NAME="E7:12_2_1"/><b>then </b>
<font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <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="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="lang1.html#K3">*&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="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font> </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E8:12_2_1"/><b>then </b>
<font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="trees_2.html#D5">TREES_2:def 5</a></i>;<br/><b>hence </b><a NAME="E9:12_2_1"/>
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sp</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/></div>
<b>end;</b></div><b>set </b><font color="Maroon">zsp</font> = <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sp</font>;<br/><b>assume </b><a NAME="E3:12_2"/>
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sp</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font>
 ;<br/><a NAME="E4:12_2"/><b>then </b>
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sp</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="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">n</font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="trees_2.html#D5">TREES_2:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">n</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="E6:12_2"/><i><font color="Green">E42</font></i>: 
( <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> &amp; <font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font> )
 ;<br/><a NAME="E7:12_2"/>
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><a NAME="E8:12_2"/><b>then </b>
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T0</font>
 <b>by </b><i><a class="ref" href="trees_3.html#T72">TREES_3:72</a></i>;<br/><a NAME="E9:12_2"/><b>then </b>
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="lang1.html#K3">*&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="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T0</font> </span>}</span> 
 ;<br/><a NAME="E10:12_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="trees_2.html#D5">TREES_2:def 5</a></i>;<br/><a NAME="E11:12_2"/>
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sp</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><b>hence </b><a NAME="E12:12_2"/>
<font color="Maroon">sp</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">p</font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T1</font>;<br/>

<b>assume </b><a NAME="E3:12"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font>
 ;<br/>

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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:12_3"/>
<font color="Maroon">sp</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:12_3"/><b>then </b>
<font color="Maroon">sp</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">n</font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</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="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="trees_2.html#D5">TREES_2:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">n</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="E4:12_3"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">sp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> &amp; <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> )
 ;<br/><a NAME="E5:12_3"/>
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font>
 <b>by </b><i>, <a class="ref" href="trees_3.html#T73">TREES_3:73</a></i>;<br/><a NAME="E6:12_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><a NAME="E7:12_3"/><b>then </b>
<font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <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="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="lang1.html#K3">*&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="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font> </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E8:12_3"/><b>then </b>
<font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="trees_2.html#D5">TREES_2:def 5</a></i>;<br/><b>hence </b><a NAME="E9:12_3"/>
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sp</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/></div>
<b>end;</b></div>

<b>set </b><font color="Maroon">zsp</font> = <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sp</font>;<br/>
<b>assume </b><a NAME="E5:12"/>
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sp</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">t</font>
 ;<br/>

<a NAME="E6:12"/><b>then </b>
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sp</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="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">n</font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="trees_2.html#D5">TREES_2:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</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="E8:12"/><i><font color="Green">E57</font></i>: 
( <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> &amp; <font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font> )
 ;<br/>
<a NAME="E9:12"/>
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K15">tree</a> <font color="Maroon">T0</font>,<font color="Maroon">T1</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/>
<a NAME="E10:12"/><b>then </b>
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font>
 
<b>by </b><i><a class="ref" href="trees_3.html#T73">TREES_3:73</a></i>;<br/>
<a NAME="E11:12"/><b>then </b>
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="lang1.html#K3">*&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="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">k</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> </span>}</span> 
 
;<br/>
<a NAME="E12:12"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="trees_2.html#D5">TREES_2:def 5</a></i>;<br/>
<a NAME="E13:12"/>
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">sp</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/>
<b>hence </b><a NAME="E14:12"/>
<font color="Maroon">sp</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/>


</div>
