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<a NAME="E1:95_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">T</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">t1</font>, <font color="Olive">t2</font> being   <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">t1</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">t2</font> &amp; <font color="Olive">t2</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">t1</font> );<br/>
<a NAME="E2:95_1_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>] holds <br/><font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E3:95_1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">z</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_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> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>given </b><font color="Maroon">t1</font> being    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font>, <font color="Maroon">t2</font> being    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font><b> such that </b><a NAME="E1:95_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> &amp; <font color="Maroon">t2</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">t1</font> )
 ;<br/>

<b>given </b><font color="Maroon">t3</font> being    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font>, <font color="Maroon">t4</font> being    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font><b> such that </b><a NAME="E2:95_1_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t3</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t4</font> &amp; <font color="Maroon">t4</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">t3</font> )
 ;<br/>

<b>take </b>
<font color="Maroon">t1</font>
;<br/>

<b>take </b>
<font color="Maroon">t4</font>
;<br/>

<b>thus </b><a NAME="E3:95_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t4</font> &amp; <font color="Maroon">t4</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">t1</font> )
 <b>by </b><i><a class="ref" href="trees_3.html#D5" target="_self">Def5</a>, <a class="ref" href="trees_3.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:95_1_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> holds <br/> not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Maroon">x</font>] ) )
 <b>from </b><i><a class="ref" href="card_3.html#S2">CARD_3:sch 2</a>(, <a class="ref" href="trees_3.html#D3" target="_self">Def3</a>, <a class="ref" href="trees_3.html#D4" target="_self">Def4</a>);<br/></i>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="trees_3.html#D5" target="_self">Def5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:95_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 ;<br/><b>then reconsider </b><font color="Maroon">t1</font> = <font color="Maroon">p</font> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">T</font> ;<br/><b>thus </b><a NAME="E3:95_1_1_2"/>
not <font color="Maroon">x</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:95_1_1_2_1"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">p</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a><br/>

<a NAME="E2:95_1_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font>
 
<b>by </b><i><a class="ref" href="trees_3.html#D5" target="_self">Def5</a></i>;<br/>
<b>hence </b><a NAME="E3:95_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:95_1_1"/>
(  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">T</font> is <a href="finset_1.html#V1">finite</a> &amp; not  <a href="trees_1.html#K3">Leaves</a> <font color="Maroon">T</font> is <a href="xboole_0.html#V1">empty</a> )
 <b>by </b><i><a class="ref" href="trees_1.html#D8">TREES_1:def 8</a></i>;<br/>


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