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<b>let </b><font color="Maroon" title="c1">T1</font>, <font color="Maroon" title="c2">T2</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>;<br/>

<b>set </b><font color="Maroon" title="c3">m</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c1">T1</font></span>)</span>,<span class="p1">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c2">T2</font></span>)</span>;<br/>
<a NAME="E1:150"/>
(  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c2">T2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c1">T1</font> or  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c2">T2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c1">T1</font> )
 
;<br/>
<a NAME="E2:150"/><b>then </b><i><font color="Green" title="E61">A1</font></i>: 
(  <a href="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c1">T1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c2">T2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; (  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c1">T1</font></span>)</span>,<span class="p1">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c2">T2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c1">T1</font> or  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c1">T1</font></span>)</span>,<span class="p1">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c2">T2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c2">T2</font> ) )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T147" title="FINSEQ_2:th.147">FINSEQ_2:147</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">t</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>;<br/><b>assume </b><a NAME="E1:150_1"/>
<font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 ;<br/><a NAME="E2:150_1"/><b>then </b>
( <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T1</font> or <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">T2</font> )
 <b>by </b><i><a class="txt" href="trees_3.html#E2:150"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:150_1"/>
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c4">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c1">T1</font></span>)</span>,<span class="p1">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c2">T2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:150"/>
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K15" title="TREES_3:func.15">tree</a> <font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c1">T1</font></span>)</span>,<span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c2">T2</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="trees_3.html#E2:150"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="trees_3.html#T82" target="_self" title="TREES_3:th.82">Th82</a></i>;<br/>


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