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<b>let </b><font color="Maroon" title="c1">T</font> be   <a href="helly.html#NM1" title="HELLY:NM.1">_Tree</a>;<br/><b>let </b><font color="Maroon" title="c2">P</font> be   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">j</font> be  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:30"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j</font>
 <b>and </b><br/><a NAME="E2:30"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font>
 <b>and </b><br/><a NAME="E3:30"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">i</font> = <font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c6">j</font> = <font color="Maroon" title="c4">j</font> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:30"/>
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j</font>
 
<b>by </b><i><a class="txt" href="helly.html#E1:30"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>
<a NAME="E6:30"/><b>then </b><i><font color="Green" title="E26">A4</font></i>: 
( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> )
 
<b>by </b><i><a class="txt" href="helly.html#E2:30"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="helly.html#E3:30"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/>
<a NAME="E7:30"/><b>then </b><i><font color="Green" title="E27">A5</font></i>: 
not <font color="Maroon" title="c2">P</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 
<b>by </b><i><a class="txt" href="helly.html#E1:30"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="glib_001.html#T127" title="GLIB_001:th.127">GLIB_001:127</a></i>;<br/>
<a NAME="E8:30"/>
<font color="Maroon" title="c2">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 
<b>by </b><i><a class="txt" href="helly.html#E3:30"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="helly.html#E6:30"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/>
<a NAME="E9:30"/><b>then </b>
<font color="Maroon" title="c2">P</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></i>;<br/>
<b>hence </b><a NAME="E10:30"/>
contradiction
 <b>by </b><i><a class="txt" href="helly.html#E7:30"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="helly.html#T24" target="_self" title="HELLY:th.24">Th24</a></i>;<br/>


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