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

<b>let </b><font color="Maroon">w</font> be  <a href="trees_3.html#V5" target="_self">FinTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">t</font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a>;<br/>



<b>assume </b><a NAME="E1:147"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">w</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:147"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E5:147"/><i><font color="Green">E40</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:147"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E9:147"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">t</font> &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 
<b>by </b><i>, <a class="ref" href="trees_3.html#D7" target="_self">Def7</a>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/>
<a NAME="E10:147"/><b>then </b>
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="trees_3.html#K13" target="_self">tree</a> <font color="Maroon">w</font></span>)</span> <a href="trees_1.html#K4">|</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K13" target="_self">tree</a> <font color="Maroon">w</font> &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, <a class="ref" href="trees_3.html#D7" target="_self">Def7</a>, <a class="ref" href="trees_3.html#D5" target="_self">Def5</a>, <a class="ref" href="trees_3.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E11:147"/>
 <a href="trees_1.html#K6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K13" target="_self">tree</a> <font color="Maroon">w</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="trees_1.html#K6">height</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="trees_1.html#T85">TREES_1:85</a></i>;<br/>


</div>
