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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  st 4 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  &amp; <font color="Olive" title="b1">v</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  holds <br/> ex <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font>, <font color="Olive" title="b4">z</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">x</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">y</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">z</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">y</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">z</font> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">z</font> &amp; <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp;  not <font color="Olive" title="b1">v</font>,<font color="Olive" title="b3">y</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp;  not <font color="Olive" title="b1">v</font>,<font color="Olive" title="b4">z</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">v</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 4 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  &amp; <font color="Maroon" title="c2">v</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies  ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">z</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">z</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">z</font> &amp; <font color="Maroon" title="c2">v</font>,<font color="Olive" title="b1">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp;  not <font color="Maroon" title="c2">v</font>,<font color="Olive" title="b2">y</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp;  not <font color="Maroon" title="c2">v</font>,<font color="Olive" title="b3">z</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:121"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
( 4 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  &amp; <font color="Maroon" title="c2">v</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">z</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">z</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">z</font> &amp; <font color="Maroon" title="c2">v</font>,<font color="Olive" title="b1">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp;  not <font color="Maroon" title="c2">v</font>,<font color="Olive" title="b2">y</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp;  not <font color="Maroon" title="c2">v</font>,<font color="Olive" title="b3">z</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )</span><br/>

<a NAME="E2:121"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> 3 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T39" onmouseover="rs('nat_1/T39')" onmouseout="rh()">NAT_1:39</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E4:121"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">v</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp;  not <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">y</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:121"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="glibpre0.html#T95" target="_self" onmouseover="rs('glibpre0/T95')" onmouseout="rh()">Th101</a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:121"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
( <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:121"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="glibpre0.html#T17" target="_self" onmouseover="rs('glibpre0/T17')" onmouseout="rh()">Th21</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">z</font> = <font color="Maroon" title="c5">z</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:121"><span class="lab"><font color="Green" title="E39">A3</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">x</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">z</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">y</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">z</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">y</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">z</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">z</font> &amp; <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp;  not <font color="Maroon" title="c2">v</font>,<font color="Olive" title="b1">y</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp;  not <font color="Maroon" title="c2">v</font>,<font color="Olive" title="b2">z</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c4">y</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">z</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">z</font> &amp; <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp;  not <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">y</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp;  not <font color="Maroon" title="c2">v</font>,<font color="Olive" title="b1">z</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c6">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">z</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">z</font> &amp; <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp;  not <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">y</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp;  not <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c6">z</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E8:121"/>
( <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">z</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">z</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:121"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="txt" href="#E6:121"><span class="lab"><font color="Green" title="E39">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp;  not <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">y</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp;  not <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c6">z</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E9:121"/>
<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:121"><span class="lab"><font color="Green" title="E38">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">y</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp;  not <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c6">z</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E10:121"/>
 not <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">y</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:121"><span class="lab"><font color="Green" title="E38">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c6">z</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> </span><br/>

<span class="kw">thus </span><a NAME="E11:121"/>
 not <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c6">z</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:121"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E4:121"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="txt" href="#E6:121"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="glibpre0.html#T94" target="_self" onmouseover="rs('glibpre0/T94')" onmouseout="rh()">Th100</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
