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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>assume </b><a NAME="E1:5"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> <a href="finseq_2.html#K3">*</a> 
 ;<br/>

<b>assume </b><a NAME="E2:5"/>
 <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<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="E4:5"/><i><font color="Green">E8</font></i>: 
<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="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>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> <b>by </b><i><a class="txt" href="bintree2.html#E4:5"><i><font color="Green">E8</font></i></a></i>;<br/>
<a NAME="E6:5"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="bintree1.html#V1">binary</a>
 
<b>by </b><i><a class="txt" href="bintree2.html#E1:5"><i><font color="Green">E7</font></i></a>, <a class="ref" href="bintree2.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E7:5"/><b>then </b><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="margrel1.html#K6">BOOLEAN</a> 
 
<b>by </b><i><a class="ref" href="bintree2.html#T2" target="_self">Th2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">0</span><a href="binarith.html#K13">*&gt;</a></span>;<br/>
<a NAME="E9:5"/><i><font color="Green">E10</font></i>: 
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:5"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></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="E1:5_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E2:5_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:5_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="margrel1.html#K6">BOOLEAN</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#D13">MARGREL1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:5"/><b>then </b>
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">0</span><a href="binarith.html#K13">*&gt;</a></span> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="margrel1.html#K6">BOOLEAN</a> 
 
<b>by </b><i><a class="txt" href="bintree2.html#E9:5"><i><font color="Green">E10</font></i></a>, <a class="txt" href="bintree2.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<a NAME="E12:5"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">0</span><a href="binarith.html#K13">*&gt;</a></span> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="margrel1.html#K6">BOOLEAN</a> 
 
<b>by </b><i><a class="txt" href="bintree2.html#E7:5"><i><font color="Green">E9</font></i></a>, <a class="txt" href="bintree2.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<a NAME="E13:5"/><b>then </b><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">0</span><a href="binarith.html#K13">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="bintree2.html#E1:5"><i><font color="Green">E7</font></i></a>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a>, <a class="ref" href="margrel1.html#D12">MARGREL1:def 12</a></i>;<br/>
<a NAME="E14:5"/>
<font color="Maroon">c<sub>4</sub></font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">0</span><a href="binarith.html#K13">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="trees_1.html#T31">TREES_1:31</a></i>;<br/>
<b>hence </b><a NAME="E15:5"/>
contradiction
 <b>by </b><i><a class="txt" href="bintree2.html#E4:5"><i><font color="Green">E8</font></i></a>, <a class="txt" href="bintree2.html#E13:5"><i><font color="Green">E11</font></i></a>, <a class="ref" href="trees_1.html#D8">TREES_1:def 8</a></i>;<br/>


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