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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:3_1"/>
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:3_1"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="lang1.html#K3">*&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="ref" href="trees_2.html#D5">TREES_2:def 5</a></i>;<br/><a NAME="E3:3_1"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>hence </b><a NAME="E4:3_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="modal_1.html#T53">MODAL_1:53</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:3"/><b>then </b><i><font color="Green">E2</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&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#T53">MODAL_1:53</a></i>;<br/>
<b>assume </b><a NAME="E4:3"/><i><font color="Green">E3</font></i>: 
 <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E6:3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="bintree1.html#E4:3"><i><font color="Green">E3</font></i></a></i>;<br/>
<a NAME="E7:3"/><b>then </b>
<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"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="lang1.html#K3">*&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="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="E9:3"/><i><font color="Green">E4</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>
<a NAME="E10:3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<b>hence </b><a NAME="E11:3"/>
contradiction
 <b>by </b><i><a class="txt" href="bintree1.html#E3:3"><i><font color="Green">E2</font></i></a>, <a class="txt" href="bintree1.html#E9:3"><i><font color="Green">E4</font></i></a>, <a class="ref" href="trees_1.html#D5">TREES_1:def 5</a></i>;<br/>


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