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

<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>let </b><font color="Maroon">p</font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:36"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a>  : <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> </span>}</span> ;<br/>
<a NAME="E3:36"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">T</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a>  : <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> </span>}</span>  <a href="tarski.html#R2">are_equipotent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font>) <b>-&gt; </b>   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  = <font color="Maroon">p'</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:36_1"/><i><font color="Green">E34</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E3:36_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">n</font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> holds  <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S4">FUNCT_1:sch 4</a>();<br/></i>
<b>take </b>
<font color="Maroon">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a><br/>

<b>thus </b><a NAME="E4:36_1"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:36_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:36_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E3:36_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">x</font>, <font color="Maroon">n</font> = <font color="Maroon">y</font> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> <b>by </b><i>, , </i>;<br/>
<font color="Maroon">p'</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">m</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">p'</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">n</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E6:36_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/>


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

<b>thus </b><a NAME="E5:36_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E6:36_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a>  : <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</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="E1:36_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:36_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:36_1_2"/>
<font color="Maroon">T</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="ref" href="trees_1.html#D5">TREES_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a>  <b>by </b><i>, </i>;<br/>
<a NAME="E7:36_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E8:36_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a>  : <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> </span>}</span> 
 <b>by </b><i>, , </i>;<br/>


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

<b>let </b><font color="Maroon">i</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="E7:36_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a>  : <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E9:36_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E10:36_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> <b>by </b><i/>;<br/>
<a NAME="E12:36_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E13:36_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">n</font></span>)</span> where <font color="Olive">n</font> is    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a>  : <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="card_1.html#T68">CARD_1:68</a></i>;<br/>
<a NAME="E5:36"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</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="E1:36_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E3:36_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> )
 ;<br/>
<b>thus </b><a NAME="E4:36_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="trees_1.html#D9">TREES_1:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:36"/>
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_3"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font>
 ;<br/>

<a NAME="E2:36_3"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/>
<a NAME="E3:36_3"/><b>then </b>
 ex <font color="Olive">n</font> being   <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a>  st <br/>(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font> )
 
;<br/>
<b>hence </b><a NAME="E4:36_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T48">FINSEQ_1:48</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:36"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">X</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E8:36"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_1.html#T81">CARD_1:81</a></i>;<br/>
<b>hence </b><a NAME="E9:36"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="trees_1.html#K4">|</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="card_2.html#T67">CARD_2:67</a></i>;<br/>


</div>
