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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P1</font>, <font color="Olive" title="b2">P2</font>, <font color="Olive" title="b3">P3</font>, <font color="Olive" title="b4">P4</font> being  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b5">a</font>, <font color="Olive" title="b6">b</font>, <font color="Olive" title="b7">c</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Olive" title="b5">a</font>,<font color="Olive" title="b6">b</font> &amp; <font color="Olive" title="b2">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Olive" title="b5">a</font>,<font color="Olive" title="b7">c</font> &amp; <font color="Olive" title="b3">P3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Olive" title="b6">b</font>,<font color="Olive" title="b5">a</font> &amp; <font color="Olive" title="b4">P4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Olive" title="b6">b</font>,<font color="Olive" title="b7">c</font> &amp; not <font color="Olive" title="b6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">P2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  &amp; not <font color="Olive" title="b7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  &amp; not <font color="Olive" title="b5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">P4</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  holds <br/><font color="Olive" title="b1">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Olive" title="b1">P1</font>,<font color="Olive" title="b2">P2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">P3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Olive" title="b3">P3</font>,<font color="Olive" title="b4">P4</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">P1</font>, <font color="Maroon" title="c3">P2</font>, <font color="Maroon" title="c4">P3</font>, <font color="Maroon" title="c5">P4</font> be   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">T</font>  st <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> &amp; <font color="Maroon" title="c4">P3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c5">P4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> &amp; not <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  &amp; not <font color="Olive" title="b3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  &amp; not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P4</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  holds <br/><font color="Maroon" title="c2">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Maroon" title="c2">P1</font>,<font color="Maroon" title="c3">P2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Maroon" title="c4">P3</font>,<font color="Maroon" title="c5">P4</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">c</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c4">P3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">P4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> &amp; not <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  &amp; not <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  &amp; not <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P4</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  implies <font color="Maroon" title="c2">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Maroon" title="c2">P1</font>,<font color="Maroon" title="c3">P2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Maroon" title="c4">P3</font>,<font color="Maroon" title="c5">P4</font></span>)</span></span>)</span> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:63"/><i><font color="Green" title="E48">A1</font></i>: 
<font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>
 <b>and </b><br/><a NAME="E2:63"/><i><font color="Green" title="E49">A2</font></i>: 
<font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>
 <b>and </b><br/><a NAME="E3:63"/><i><font color="Green" title="E50">A3</font></i>: 
<font color="Maroon" title="c4">P3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font>
 <b>and </b><br/><a NAME="E4:63"/><i><font color="Green" title="E51">A4</font></i>: 
<font color="Maroon" title="c5">P4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>
 <b>and </b><br/><a NAME="E5:63"/><i><font color="Green" title="E52">A5</font></i>: 
not <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>and </b><br/><a NAME="E6:63"/><i><font color="Green" title="E53">A6</font></i>: 
not <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>and </b><br/><a NAME="E7:63"/><i><font color="Green" title="E54">A7</font></i>: 
not <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P4</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Maroon" title="c2">P1</font>,<font color="Maroon" title="c3">P2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Maroon" title="c4">P3</font>,<font color="Maroon" title="c5">P4</font></span>)</span></span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_1"/>
<font color="Maroon" title="c5">P4</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">P3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:63_1"/><b>then </b><i><font color="Green" title="E55">A8</font></i>: 
<font color="Maroon" title="c5">P4</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">P3</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="ref" href="helly.html#T13" target="_self" title="HELLY:th.13">Th13</a></i>;<br/><a NAME="E3:63_1"/>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P4</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="#E4:63"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="helly.html#T29" target="_self" title="HELLY:th.29">Th28</a></i>;<br/><a NAME="E4:63_1"/><b>then </b>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P3</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="#E2:63_1"><i><font color="Green" title="E55">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E5:63_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:63"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="#E3:63"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="#E6:63"><i><font color="Green" title="E53">A6</font></i></a>, <a class="ref" href="helly.html#T32" target="_self" title="HELLY:th.32">Th31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E9:63"/><b>then </b><i><font color="Green" title="E55">A9</font></i>: 
not <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P3</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<b>by </b><i><a class="txt" href="#E3:63"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="#E4:63"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="helly.html#T37" target="_self" title="HELLY:th.37">Th36</a></i>;<br/>
<a NAME="E10:63"/><i><font color="Green" title="E56">A10</font></i>: 
 <a href="helly.html#K3" title="HELLY:func.3">MiddleVertex</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Maroon" title="c2">P1</font>,<font color="Maroon" title="c3">P2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:63"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="#E2:63"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="#E5:63"><i><font color="Green" title="E52">A5</font></i></a>, <a class="txt" href="#E6:63"><i><font color="Green" title="E53">A6</font></i></a>, <a class="ref" href="helly.html#T51" target="_self" title="HELLY:th.51">Th50</a></i>;<br/>
<a NAME="E11:63"/>
 <a href="helly.html#K3" title="HELLY:func.3">MiddleVertex</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Maroon" title="c4">P3</font>,<font color="Maroon" title="c5">P4</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:63"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="#E4:63"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="#E7:63"><i><font color="Green" title="E54">A7</font></i></a>, <a class="txt" href="#E9:63"><i><font color="Green" title="E55">A9</font></i></a>, <a class="ref" href="helly.html#T51" target="_self" title="HELLY:th.51">Th50</a></i>;<br/>
<b>hence </b><a NAME="E12:63"/>
<font color="Maroon" title="c2">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Maroon" title="c2">P1</font>,<font color="Maroon" title="c3">P2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Maroon" title="c4">P3</font>,<font color="Maroon" title="c5">P4</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:63"><i><font color="Green" title="E56">A10</font></i></a>, <a class="ref" href="helly.html#T42" target="_self" title="HELLY:th.42">Th41</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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