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

<b>let </b><font color="Maroon" title="c1">t</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c1">t</font>;<br/><b>let </b><font color="Maroon" title="c3">q</font> be   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span>;<br/>





<a NAME="E1:68"/><i><font color="Green" title="E27">A1</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">t</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="ref" href="trees_2.html#D11" title="TREES_2:def.11">TREES_2:def 11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">pq</font> = <font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">q</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">t</font> <b>by </b><i><a class="ref" href="trees_1.html#D9" title="TREES_1:def.9">TREES_1:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">q</font> = <font color="Maroon" title="c3">q</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span> ;<br/>
<a NAME="E4:68"/><i><font color="Green" title="E28">A2</font></i>: 
(  ex <font color="Olive" title="b1">q</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">t</font> st <br/>( <font color="Olive" title="b1">q</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">pq</font> &amp;  <a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> <font color="Maroon" title="c1">t</font>,<font color="Maroon" title="c4">pq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">t</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a> </span>)</span> ) &amp;  ex <font color="Olive" title="b1">r</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span> st <br/>( <font color="Olive" title="b1">r</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">q</font> &amp;  <a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a> </span>)</span> ) )
 
<b>by </b><i><a class="ref" href="trees_9.html#D6" target="_self" title="TREES_9:def.6">Def6</a></i>;<br/>
<a NAME="E5:68"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">t</font></span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">t</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c2">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="trees_2.html#T8" title="TREES_2:th.8">TREES_2:8</a></i>;<br/>
<a NAME="E6:68"/><b>then </b>
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c4">pq</font>, <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c5">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="trees_9.html#E1:68"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="trees_2.html#T39" title="TREES_2:th.39">TREES_2:39</a></i>;<br/>
<a NAME="E7:68"/><b>then </b><i><font color="Green" title="E29">A3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pq</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c4">pq</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c5">q</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c4">pq</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="trees_9.html#D5" target="_self" title="TREES_9:def.5">Def5</a>, <a class="ref" href="card_1.html#T81" title="CARD_1:th.81">CARD_1:81</a></i>;<br/>
<a NAME="E8:68"/><b>then </b><i><font color="Green" title="E30">A4</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pq</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E9:68"/>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pq</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">t</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span> )
 
;<br/>
<a NAME="E10:68"/><b>then </b><i><font color="Green" title="E31">A5</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> <font color="Maroon" title="c1">t</font>,<font color="Maroon" title="c4">pq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pq</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a> </span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="trees_9.html#E4:68"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:68_1"/><i><font color="Green" title="E32">A6</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a> </span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:68_1"/><i><font color="Green" title="E33">A7</font></i>: 
( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="trees_9.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><a NAME="E4:68_1"/><i><font color="Green" title="E34">A8</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c5">q</font>
 <b>by </b><i><a class="txt" href="trees_9.html#E7:68"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="trees_9.html#E3:68_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="trees_9.html#T7" target="_self" title="TREES_9:th.7">Th7</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> <font color="Maroon" title="c1">t</font>,<font color="Maroon" title="c4">pq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> = 
<font color="Maroon" title="c1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">pq</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="txt" href="trees_9.html#E4:68"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="trees_9.html#E8:68"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="trees_9.html#E10:68"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="trees_9.html#E1:68_1"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pq</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="trees_9.html#E7:68"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="trees_9.html#E3:68_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="trees_9.html#D5" target="_self" title="TREES_9:def.5">Def5</a></i>
<br/>.= 
<font color="Maroon" title="c1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="trees_9.html#E1:68"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="trees_9.html#E4:68_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="trees_2.html#D11" title="TREES_2:def.11">TREES_2:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="trees_9.html#K1" title="TREES_9:func.1">succ</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="txt" href="trees_9.html#E3:68_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="trees_9.html#D5" target="_self" title="TREES_9:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
<b>by </b><i><a class="txt" href="trees_9.html#E4:68"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="trees_9.html#E10:68"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="trees_9.html#E1:68_1"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:68"/>
 <a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> <font color="Maroon" title="c1">t</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c3">q</font>
 <b>by </b><i><a class="txt" href="trees_9.html#E8:68"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="trees_9.html#E10:68"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/>


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