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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="trees_1.html#NM1">Tree</a>;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="trees_3.html#K15">tree</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:13_1"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="bintree1.html#V1" target="_self">binary</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="bintree1.html#V1" target="_self">binary</a> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="trees_1.html#M1">Element</a> of  <a href="trees_3.html#K15">tree</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:13_1_1"/><i><font color="Green">E8</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="trees_3.html#K15">tree</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ex <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></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="Olive">b<sub>1</sub></font> ) or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></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="Olive">b<sub>1</sub></font> ) ) )
 </span><b>by </b><i><a class="ref" href="trees_3.html#T71">TREES_3:71</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1_1"><b>suppose </b></a><a NAME="E1:13_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:13_1_1_1_1"/>
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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="bintree1.html#T9" target="_self">Th9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1_2"><b>suppose </b></a><a NAME="E1:13_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></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="Olive">b<sub>1</sub></font> ) or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></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="Olive">b<sub>1</sub></font> ) )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:13_1_1_1_2"/><i><font color="Green">E9</font></i>: 
( ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></font> ) )
 ;<br/><div><i><font color="Green">E10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1_1_1_2_1"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></font> )
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_1_1_1_2_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">c<sub>1</sub></font> or  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:13_1_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:13_1_1_1_2_1_1_1"/>
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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="txt" href="bintree1.html#E1:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="bintree1.html#E1:13_1_1_1_2_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="bintree1.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:13_1_1_1_2_1_1_2"/>
not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:13_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E12</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</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="txt" href="bintree1.html#E1:13_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="bintree1.html#D2" target="_self">Def2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:13_1_1_1_2_1_1_2_1_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:13_1_1_1_2_1_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:13_1_1_1_2_1_1_2_1_1"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 ;<br/><a NAME="E5:13_1_1_1_2_1_1_2_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>7</sub></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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="bintree1.html#E1:13_1_1_1_2_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="bintree1.html#E4:13_1_1_1_2_1_1_2_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="bintree1.html#E4:13_1_1_1_2_1_1_2_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="trees_3.html#T72">TREES_3:72</a></i>;<br/><a NAME="E7:13_1_1_1_2_1_1_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="bintree1.html#E1:13_1_1_1_2_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="bintree1.html#E1:13_1_1_1_2_1_1_2_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="bintree1.html#E5:13_1_1_1_2_1_1_2_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="bintree1.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E8:13_1_1_1_2_1_1_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</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="txt" href="bintree1.html#E2:13_1_1_1_2_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E9:13_1_1_1_2_1_1_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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="txt" href="bintree1.html#E4:13_1_1_1_2_1_1_2_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/><b>hence </b><a NAME="E10:13_1_1_1_2_1_1_2_1_1"/>
<font color="Maroon">c<sub>7</sub></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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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="E2:13_1_1_1_2_1_1_2_1"/>
<font color="Maroon">c<sub>7</sub></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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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="E3:13_1_1_1_2_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <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#K7">^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_1.html#K7">^</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">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <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#K7">^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_1.html#K7">^</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="txt" href="bintree1.html#E1:13_1_1_1_2_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E4:13_1_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>7</sub></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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> or <font color="Maroon">c<sub>7</sub></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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default">1</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/><a NAME="E5:13_1_1_1_2_1_1_2_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</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_2.html#K1">succ</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</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_2.html#K1">succ</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="bintree1.html#E2:13_1_1_1_2_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E6:13_1_1_1_2_1_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="bintree1.html#E1:13_1_1_1_2_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="bintree1.html#E4:13_1_1_1_2_1_1_2_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="bintree1.html#T9" target="_self">Th9</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:13_1_1_1_2_1_1_2"/>
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1_1_1_2_2"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></font> )
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_1_1_1_2_2_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">c<sub>2</sub></font> or  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:13_1_1_1_2_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:13_1_1_1_2_2_1_1"/>
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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="txt" href="bintree1.html#E1:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="bintree1.html#E1:13_1_1_1_2_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="bintree1.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:13_1_1_1_2_2_1_2"/>
not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:13_1_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E12</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</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="txt" href="bintree1.html#E1:13_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="bintree1.html#D2" target="_self">Def2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1_2_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:13_1_1_1_2_2_1_2_1_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:13_1_1_1_2_2_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:13_1_1_1_2_2_1_2_1_1"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 ;<br/><a NAME="E5:13_1_1_1_2_2_1_2_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>7</sub></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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="bintree1.html#E1:13_1_1_1_2_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="bintree1.html#E4:13_1_1_1_2_2_1_2_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="bintree1.html#E4:13_1_1_1_2_2_1_2_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="trees_3.html#T73">TREES_3:73</a></i>;<br/><a NAME="E7:13_1_1_1_2_2_1_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="bintree1.html#E1:13_1_1_1_2_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="bintree1.html#E1:13_1_1_1_2_2_1_2_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="bintree1.html#E5:13_1_1_1_2_2_1_2_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="bintree1.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E8:13_1_1_1_2_2_1_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</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="txt" href="bintree1.html#E2:13_1_1_1_2_2_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E9:13_1_1_1_2_2_1_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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="txt" href="bintree1.html#E4:13_1_1_1_2_2_1_2_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/><b>hence </b><a NAME="E10:13_1_1_1_2_2_1_2_1_1"/>
<font color="Maroon">c<sub>7</sub></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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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="E2:13_1_1_1_2_2_1_2_1"/>
<font color="Maroon">c<sub>7</sub></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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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="E3:13_1_1_1_2_2_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <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#K7">^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_1.html#K7">^</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">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <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#K7">^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_1.html#K7">^</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="txt" href="bintree1.html#E1:13_1_1_1_2_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E4:13_1_1_1_2_2_1_2_1"/><b>then </b><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>7</sub></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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> or <font color="Maroon">c<sub>7</sub></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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default">1</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/><a NAME="E5:13_1_1_1_2_2_1_2_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</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_2.html#K1">succ</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</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_2.html#K1">succ</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="bintree1.html#E2:13_1_1_1_2_2_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E6:13_1_1_1_2_2_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="bintree1.html#E1:13_1_1_1_2_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="bintree1.html#E4:13_1_1_1_2_2_1_2_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="bintree1.html#T9" target="_self">Th9</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:13_1_1_1_2_2_1_2"/>
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:13_1_1_1_2"/>
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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="txt" href="bintree1.html#E3:13_1_1_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="bintree1.html#E4:13_1_1_1_2"><i><font color="Green">E10</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:13_1"/>
 <a href="trees_3.html#K15">tree</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="bintree1.html#V1" target="_self">binary</a>
 <b>by </b><i><a class="ref" href="bintree1.html#D2" target="_self">Def2</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:13"/><i><font color="Green">E7</font></i>: 
 <a href="trees_3.html#K15">tree</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="bintree1.html#V1" target="_self">binary</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></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">c<sub>4</sub></font> as    <a href="trees_1.html#M1">Element</a> of  <a href="trees_3.html#K15">tree</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="trees_3.html#T72">TREES_3:72</a></i>;<br/><b>assume </b><a NAME="E2:13_2"/>
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E3:13_2"/><b>then </b>
not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="trees_3.html#K15">tree</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="bintree1.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E4:13_2"/><b>then </b><i><font color="Green">E8</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>5</sub></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"><span class="p3">(<span class="default"><span class="p4"><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">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K7">^</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"><span class="p3">(<span class="default"><span class="p4"><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">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K7">^</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="txt" href="bintree1.html#E2:13"><i><font color="Green">E7</font></i></a>, <a class="ref" href="bintree1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E5:13_2"/><i><font color="Green">E9</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 <b>by </b><i><a class="ref" href="trees_2.html#D5">TREES_2:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:13_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:13_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></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">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:13_2_1_1"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><a NAME="E5:13_2_1_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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="bintree1.html#E4:13_2_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="trees_3.html#T72">TREES_3:72</a></i>;<br/><a NAME="E6:13_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><a NAME="E7:13_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>5</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 ;<br/><a NAME="E8:13_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>5</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K7">^</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="txt" href="bintree1.html#E4:13_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="bintree1.html#E5:13_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E9:13_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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="txt" href="bintree1.html#E4:13_2_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/><b>hence </b><a NAME="E10:13_2_1_1"/>
<font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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="E2:13_2_1"/>
<font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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="E3:13_2_1"/><b>then </b><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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/><a NAME="E4:13_2_1"/>
( <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#K7">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K7">^</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_2.html#K1">succ</a> <font color="Maroon">c<sub>5</sub></font> &amp; <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#K7">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K7">^</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_2.html#K1">succ</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="bintree1.html#E4:13_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:13_2_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="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>5</sub></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="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><b>hence </b><a NAME="E6:13_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="bintree1.html#E3:13_2_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="bintree1.html#T9" target="_self">Th9</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:13_2"/>
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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>
<b>hence </b><a NAME="E4:13"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="bintree1.html#V1" target="_self">binary</a>
 <b>by </b><i><a class="ref" href="bintree1.html#D2" target="_self">Def2</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></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">c<sub>4</sub></font> as    <a href="trees_1.html#M1">Element</a> of  <a href="trees_3.html#K15">tree</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="trees_3.html#T73">TREES_3:73</a></i>;<br/><b>assume </b><a NAME="E2:13_3"/>
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E3:13_3"/><b>then </b>
not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="trees_3.html#K15">tree</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="bintree1.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E4:13_3"/><b>then </b><i><font color="Green">E8</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>5</sub></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"><span class="p3">(<span class="default"><span class="p4"><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">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K7">^</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"><span class="p3">(<span class="default"><span class="p4"><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">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K7">^</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="txt" href="bintree1.html#E2:13"><i><font color="Green">E7</font></i></a>, <a class="ref" href="bintree1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E5:13_3"/><i><font color="Green">E9</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 <b>by </b><i><a class="ref" href="trees_2.html#D5">TREES_2:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:13_3_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:13_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></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">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:13_3_1_1"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><a NAME="E5:13_3_1_1"/>
<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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="bintree1.html#E4:13_3_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="trees_3.html#T73">TREES_3:73</a></i>;<br/><a NAME="E6:13_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><a NAME="E7:13_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>5</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 ;<br/><a NAME="E8:13_3_1_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>5</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K7">^</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="txt" href="bintree1.html#E4:13_3"><i><font color="Green">E8</font></i></a>, <a class="txt" href="bintree1.html#E5:13_3"><i><font color="Green">E9</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E9:13_3_1_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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="txt" href="bintree1.html#E4:13_3_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/><b>hence </b><a NAME="E10:13_3_1_1"/>
<font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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="E2:13_3_1"/>
<font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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="E3:13_3_1"/><b>then </b><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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/><a NAME="E4:13_3_1"/>
( <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#K7">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K7">^</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_2.html#K1">succ</a> <font color="Maroon">c<sub>5</sub></font> &amp; <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#K7">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K7">^</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_2.html#K1">succ</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="bintree1.html#E4:13_3"><i><font color="Green">E8</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:13_3_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="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>5</sub></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="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><b>hence </b><a NAME="E6:13_3_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="bintree1.html#E3:13_3_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="bintree1.html#T9" target="_self">Th9</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:13_3"/>
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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">c<sub>4</sub></font> <a href="finseq_1.html#K7">^</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>
<b>hence </b><a NAME="E6:13"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="bintree1.html#V1" target="_self">binary</a>
 <b>by </b><i><a class="ref" href="bintree1.html#D2" target="_self">Def2</a></i>;<br/>


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