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<b>let </b><font color="Maroon">T</font> be   <a href="trees_1.html#NM1">Tree</a>;<br/><b>let </b><font color="Maroon">T1</font> be   <a href="trees_1.html#NM1">Tree</a>;<br/>



<a NAME="E1:39"/>
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">D</font> = <font color="Maroon">T</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">T1</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:39"/>
<font color="Maroon">D</font> is <a href="trees_1.html#V1" target="_self">Tree-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<a NAME="E1:39_1"/>
( <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a>  &amp; <font color="Maroon">T</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">T1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">T</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E2:39_1"/>
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_1.html#D5" target="_self">TREES_1:def 5</a><br/>

<b>thus </b><a NAME="E3:39_1"/>
for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> holds <br/> <a href="trees_1.html#K1" target="_self">ProperPrefixes</a> <font color="Olive">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>proof </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="E1:39_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:39_1_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:39_1_1"/><b>then </b>
(  <a href="trees_1.html#K1" target="_self">ProperPrefixes</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">T</font> &amp;  <a href="trees_1.html#K1" target="_self">ProperPrefixes</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">T1</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:39_1_1"/>
 <a href="trees_1.html#K1" target="_self">ProperPrefixes</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>


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

<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>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E4:39_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 ;<br/>

<a NAME="E5:39_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp; <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">k</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:39_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp; <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T1</font> )
 
<b>by </b><i><a class="txt" href="trees_1.html#E4:39_1"><i><font color="Green">E28</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E7:39_1"/>
<font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:39"/>
<font color="Maroon">T</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">T1</font> is   <a href="trees_1.html#NM1">Tree</a>
 ;<br/>


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