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<b>let </b><font color="Maroon">fT</font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">t</font> be    <a href="trees_1.html#M1" target="_self">Element</a> of <font color="Maroon">fT</font>;<br/>



<b>assume </b><a NAME="E1:75"/>
<font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:75"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E3:75"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font>
 
;<br/>
<a NAME="E4:75"/><b>then </b><i><font color="Green">E28</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>consider </b><font color="Maroon">p</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:75"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fT</font> <a href="trees_1.html#K4" target="_self">|</a> <font color="Maroon">t</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K6" target="_self">height</a> <span class="p1">(<span class="default"><font color="Maroon">fT</font> <a href="trees_1.html#K4" target="_self">|</a> <font color="Maroon">t</font></span>)</span> )
 <b>by </b><i><a class="ref" href="trees_1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E7:75"/>
<font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fT</font>
 
<b>by </b><i><a class="txt" href="trees_1.html#E18"><i><font color="Green">Lemma43</font></i></a>, </i>;<br/>
<a NAME="E8:75"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="trees_1.html#K6" target="_self">height</a> <font color="Maroon">fT</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="trees_1.html#T15" target="_self">Th15</a>, <a class="ref" href="trees_1.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E9:75"/><b>then </b>
<span class="p1">(<span class="default"><a href="trees_1.html#K6" target="_self">height</a> <span class="p2">(<span class="default"><font color="Maroon">fT</font> <a href="trees_1.html#K4" target="_self">|</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="trees_1.html#K6" target="_self">height</a> <font color="Maroon">fT</font>
 
<b>by </b><i><a class="txt" href="trees_1.html#E18"><i><font color="Green">Lemma43</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E10:75"/>
 <a href="trees_1.html#K6" target="_self">height</a> <span class="p1">(<span class="default"><font color="Maroon">fT</font> <a href="trees_1.html#K4" target="_self">|</a> <font color="Maroon">t</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="trees_1.html#K6" target="_self">height</a> <font color="Maroon">fT</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


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