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<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">v</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>



<b>thus </b><a NAME="E1:5"/>
 <a href="trees_1.html#K1">ProperPrefixes</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">v</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</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">y</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">y</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K1">ProperPrefixes</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">v1</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:5_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">v1</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">v</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="trees_1.html#D4">TREES_1:def 4</a></i>;<br/>
<a NAME="E4:5_1"/>
( ( <font color="Maroon">v1</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">v</font> &amp; <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> ) or <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 
<b>by </b><i>, <a class="ref" href="trees_1.html#T32">TREES_1:32</a></i>;<br/>
<a NAME="E5:5_1"/><b>then </b>
( <font color="Maroon">v1</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">v</font> or <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E6:5_1"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">v</font> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="trees_1.html#D4">TREES_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:5_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">v</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>let </b><font color="Maroon">y</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="E2:5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">v</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:5"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">v</font> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><i><font color="Green">E58</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">v</font>
 ;<br/><b>then consider </b><font color="Maroon">v1</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:5_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">v1</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="trees_1.html#D4">TREES_1:def 4</a></i>;<br/><a NAME="E4:5_2"/>
<font color="Maroon">v</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">v</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="trees_1.html#T8">TREES_1:8</a></i>;<br/><a NAME="E5:5_2"/><b>then </b>
<font color="Maroon">v1</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">v</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T58">XBOOLE_1:58</a></i>;<br/><b>hence </b><a NAME="E6:5_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K1">ProperPrefixes</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="trees_1.html#D4">TREES_1:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:5"/>
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span> &amp; <font color="Maroon">v</font> <a href="finseq_1.html#K7">^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/>
<a NAME="E6:5"/><b>then </b>
( <font color="Maroon">v</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">v</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span> &amp; <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a>, <a class="ref" href="trees_1.html#T8">TREES_1:8</a></i>;<br/>
<a NAME="E7:5"/><b>then </b>
<font color="Maroon">v</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">v</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E8:5"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K1">ProperPrefixes</a> <font color="Maroon">v</font> or ( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K1">ProperPrefixes</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> ) )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="trees_1.html#D4">TREES_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E9:5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K1">ProperPrefixes</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
 <b>by </b><i/>;<br/>


</div>
