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

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="lang1.html#NM1">Symbol</a> of <a href="dtconstr.html#K5" target="_self">PeanoNat</a> ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="dtconstr.html#K8" target="_self">root-tree</a> <font color="Maroon">c<sub>6</sub></font> as    <a href="dtconstr.html#M1" target="_self">Element</a> of  <a href="dtconstr.html#K4" target="_self">TS</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a>  ;<br/>
<a NAME="E2:60"/>
 <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="trees_4.html#K2">root-tree</a> 0</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="trees_1.html#T79">TREES_1:79</a>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/>
<a NAME="E3:60"/><b>then </b>
 <a href="dtconstr.html#K17" target="_self">PreTraversal</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">0 <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span> <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>6</sub></font></span><a href="lang1.html#K3">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="dtconstr.html#T16" target="_self">Th16</a></i>;<br/>
<b>then </b><i><font color="Green">E42</font></i>:  <a href="dtconstr.html#K17" target="_self">PreTraversal</a> <font color="Maroon">c<sub>9</sub></font> = 
<a href="xboole_0.html#K1">{}</a>  <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>6</sub></font></span><a href="lang1.html#K3">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T72">FINSEQ_2:72</a></i>
<br/>.= 
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>6</sub></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>thus </b><a NAME="E5:60"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0 implies  <a href="dtconstr.html#K20" target="_self">PreTraversalLanguage</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><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></span><a href="domain_1.html#K6">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:60_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:60_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K20" target="_self">PreTraversalLanguage</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="trees_3.html#M5">Element</a> of  <a href="trees_3.html#K5">FinTrees</a> the <a href="struct_0.html#U1">carrier</a> of <a href="dtconstr.html#K5" target="_self">PeanoNat</a> <b> such that </b><br/><a NAME="E3:60_2_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K17" target="_self">PreTraversal</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K4" target="_self">TS</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a>  &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:60_2"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E4:60_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K8" target="_self">root-tree</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E3:60_2_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="dtconstr.html#T9" target="_self">Th9</a></i>;<br/><b>hence </b><a NAME="E5:60_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><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></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="dtconstr.html#E4:60"><i><font color="Green">E42</font></i></a>, <a class="txt" href="dtconstr.html#E3:60_2_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:60_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><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></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E4:60_2"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="lang1.html#K3">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:60_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/>
<b>hence </b><a NAME="E6:60_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K20" target="_self">PreTraversalLanguage</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E4:60"><i><font color="Green">E42</font></i></a>, <a class="txt" href="dtconstr.html#E1:60_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="dtconstr.html#E4:60_2"><i><font color="Green">E44</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E6:60"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:60_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K20" target="_self">PreTraversalLanguage</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="trees_3.html#M5">Element</a> of  <a href="trees_3.html#K5">FinTrees</a> the <a href="struct_0.html#U1">carrier</a> of <a href="dtconstr.html#K5" target="_self">PeanoNat</a> <b> such that </b><br/><a NAME="E3:60_3"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K17" target="_self">PreTraversal</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K4" target="_self">TS</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a>  &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="dtconstr.html#E6:60"><i><font color="Green">E43</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4" target="_self">TS</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a> <b> such that </b><br/><a NAME="E5:60_3"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="dtconstr.html#E3:60_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="dtconstr.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E6:60_3"/>
(  <a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">c<sub>12</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> or  <a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">c<sub>12</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> )
 <b>by </b><i><a class="txt" href="dtconstr.html#E5:60_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="dtconstr.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E7:60_3"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="trees_3.html#M5">Element</a> of  <a href="trees_3.html#K5">FinTrees</a> the <a href="struct_0.html#U1">carrier</a> of <a href="dtconstr.html#K5" target="_self">PeanoNat</a> <b> such that </b><br/><a NAME="E9:60_3"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K4" target="_self">TS</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a>  )
 <b>by </b><i><a class="ref" href="dtconstr.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E10:60_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="trees_4.html#K9">-tree</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E5:60_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="dtconstr.html#E9:60_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="trees_4.html#D5">TREES_4:def 5</a></i>;<br/><a NAME="E11:60_3"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_3.html#K14">^</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="trees_4.html#T13">TREES_4:13</a></i>;<br/><a NAME="E12:60_3"/><b>then </b>
 <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="trees_3.html#T83">TREES_3:83</a></i>;<br/><a NAME="E13:60_3"/><b>then </b>
(  <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span> <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> )
 <b>by </b><i><a class="txt" href="dtconstr.html#E3:60_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="dtconstr.html#T16" target="_self">Th16</a></i>;<br/><b>hence </b><a NAME="E14:60_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span> <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> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 </span>}</span> 
 ;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E8:60"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span> <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> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:60"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span> <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> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="dtconstr.html#M1" target="_self">Element</a> of  <a href="dtconstr.html#K4" target="_self">TS</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="dtconstr.html#K17" target="_self">PreTraversal</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span> <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> ) );<br/>
<a NAME="E11:60"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:60_4"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>12</sub></font>]
 ;<br/><b>thus </b><a NAME="E2:60_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:60_4_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_4_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_4_1_1_1"><b>suppose </b></a><a NAME="E1:60_4_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="dtconstr.html#M1" target="_self">Element</a> of  <a href="dtconstr.html#K4" target="_self">TS</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a> <b> such that </b><br/><a NAME="E3:60_4_1_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="dtconstr.html#K17" target="_self">PreTraversal</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span> <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> )
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:60_4"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E4:60_4_1_1_1"/>
 <a href="dtconstr.html#K17" target="_self">PreTraversal</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span> <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>
 <b>by </b><i><a class="ref" href="dtconstr.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E5:60_4_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="finseqop.html#K4">|-&gt;</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="dtconstr.html#E3:60_4_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a>, <a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/><a NAME="E6:60_4_1_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/><b>take </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="dtconstr.html#K10" target="_self">-tree</a> <font color="Maroon">c<sub>13</sub></font>;<br/><a NAME="E7:60_4_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="trees_4.html#K4">-tree</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="lang1.html#K3">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="trees_4.html#D5">TREES_4:def 5</a></i>;<br/> <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> = 
 <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K14">^</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="trees_4.html#T13">TREES_4:13</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="dtconstr.html#E6:60_4_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="trees_3.html#T83">TREES_3:83</a></i>
;<br/><b>hence </b><a NAME="E9:60_4_1_1_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="dtconstr.html#K17" target="_self">PreTraversal</a> <font color="Maroon">c<sub>14</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>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span> <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> )
 <b>by </b><i><a class="txt" href="dtconstr.html#E7:60_4_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="dtconstr.html#T16" target="_self">Th16</a>, <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_4_1_1_2"><b>suppose </b></a><a NAME="E1:60_4_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="dtconstr.html#K10" target="_self">-tree</a> <font color="Maroon">c<sub>9</sub></font>;<br/><a NAME="E2:60_4_1_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="trees_4.html#K4">-tree</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="lang1.html#K3">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="trees_4.html#D5">TREES_4:def 5</a></i>;<br/> <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> = 
 <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K14">^</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="trees_4.html#T13">TREES_4:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="trees_1.html#K6">height</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="trees_3.html#T83">TREES_3:83</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="dtconstr.html#E1:60_4_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="trees_1.html#T79">TREES_1:79</a>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>
;<br/><b>hence </b><a NAME="E4:60_4_1_1_2"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="dtconstr.html#K17" target="_self">PreTraversal</a> <font color="Maroon">c<sub>13</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>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span> <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> )
 <b>by </b><i><a class="txt" href="dtconstr.html#E2:60_4_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="dtconstr.html#T16" target="_self">Th16</a>, <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E13:60"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="dtconstr.html#E11:60"><i><font color="Green">E45</font></i></a>, <a class="txt" href="dtconstr.html#E12:60"><i><font color="Green">E46</font></i></a>);<br/></i>
<a NAME="E14:60"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="dtconstr.html#M1" target="_self">Element</a> of  <a href="dtconstr.html#K4" target="_self">TS</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="dtconstr.html#K17" target="_self">PreTraversal</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="finseqop.html#K4">|-&gt;</a> 1</span>)</span> <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> )
 
<b>by </b><i><a class="txt" href="dtconstr.html#E10:60"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:60"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K20" target="_self">PreTraversalLanguage</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E6:60"><i><font color="Green">E43</font></i></a>, <a class="txt" href="dtconstr.html#E10:60"><i><font color="Green">E44</font></i></a></i>;<br/>


</div>
