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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a>;<br/>

<b>assume </b><a NAME="E1:21"/>
 <a href="trees_2.html#K10">branchdeg</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E2:21"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="trees_2.html#K1">succ</a> <span class="p2">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="trees_2.html#D13">TREES_2:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:21"/><i><font color="Green">E20</font></i>: 
 <a href="trees_2.html#K1">succ</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="card_2.html#T60">CARD_2:60</a></i>;<br/>
<a NAME="E5:21"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="modal_1.html#E4:21"><i><font color="Green">E20</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E7:21"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="modal_1.html#E5:21"><i><font color="Green">E21</font></i></a>, <a class="ref" href="trees_2.html#D5">TREES_2:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:21"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>
<i><font color="Green">E23</font></i>: <font color="Maroon">c<sub>3</sub></font> = 
<a href="xboole_0.html#K1">{}</a>  <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="txt" href="modal_1.html#E9:21"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:21_2"/>
( <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="modal_1.html#E10:21"><i><font color="Green">E23</font></i></a>, <a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E3:21_2"/><b>then </b><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="modal_1.html#T11" target="_self">Th11</a></i>;<br/><i><font color="Green">E26</font></i>: <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> = 
<a href="xboole_0.html#K1">{}</a>  <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>

;<br/><a NAME="E5:21_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="modal_1.html#E3:21_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="trees_2.html#T14">TREES_2:14</a></i>;<br/><a NAME="E6:21_2"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="modal_1.html#E4:21"><i><font color="Green">E20</font></i></a>, <a class="txt" href="modal_1.html#E4:21_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:21_2"/>
contradiction
 <b>by </b><i><a class="txt" href="modal_1.html#E10:21"><i><font color="Green">E23</font></i></a>, <a class="txt" href="modal_1.html#E1:21_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="trees_1.html#T16">TREES_1:16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:21"/>
 <a href="trees_2.html#K1">succ</a> <span class="p1">(<span class="default"><a href="modal_1.html#K1" target="_self">Root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="modal_1.html#E4:21"><i><font color="Green">E20</font></i></a>, <a class="txt" href="modal_1.html#E10:21"><i><font color="Green">E23</font></i></a></i>;<br/>


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