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<b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">q</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">r</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<b>assume </b><a NAME="E1:28"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K1" target="_self">ProperPrefixes</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:28"/><b>then </b><i><font color="Green">E28</font></i>: 
 ex <font color="Olive">q1</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">q1</font> &amp; <font color="Olive">q1</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">p</font> )
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E3:28"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K1" target="_self">ProperPrefixes</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E4:28"/><b>then </b><i><font color="Green">E29</font></i>: 
 ex <font color="Olive">r1</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> &amp; <font color="Olive">r1</font> <a href="trees_1.html#NR3" target="_self">is_a_proper_prefix_of</a> <font color="Maroon">p</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:28"/>
<font color="Maroon">q</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:28"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E8:28"/>
<font color="Maroon">r</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:28"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E11:28"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> implies <font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="xboole_0.html#R3">are_c=-comparable</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:28_1"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E3:28_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>;<br/>
<a NAME="E4:28_1"/><b>then </b>
<font color="Maroon">q</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:28_1"/>
<font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:28"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> implies <font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="xboole_0.html#R3">are_c=-comparable</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:28_2"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E3:28_2"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>;<br/>
<a NAME="E4:28_2"/><b>then </b>
<font color="Maroon">r</font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">q</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:28_2"/>
<font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:28"/>
<font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 <b>by </b><i/>;<br/>


</div>
