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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">u</font>, <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b2">v</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  &amp; <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">w</font> &amp; <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  holds <br/> not <font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</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">u</font>, <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">w</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"> ( <font color="Maroon" title="c3">v</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  &amp; <font color="Maroon" title="c2">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">w</font> &amp; <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  implies  not <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:119"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
( <font color="Maroon" title="c3">v</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  &amp; <font color="Maroon" title="c2">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">w</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"> (  not <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  or  not <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:119"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</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">  not <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:119"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
( <font color="Maroon" title="c3">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  not <font color="Maroon" title="c5">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c1">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="glib_000.html#D51" onmouseover="rs('glib_000/D51')" onmouseout="rh()">GLIB_000:def 51</a></span>;<br/>
<a NAME="E5:119"/>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:119"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">v9</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="E7:119"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
<font color="Maroon" title="c5">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c6">v9</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T64" onmouseover="rs('glib_000/T64')" onmouseout="rh()">GLIB_000:64</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">e8</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E9:119"/><span class="lab"><font color="Green" title="E40">A5</font></span>: 
<font color="Maroon" title="c7">e8</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:119"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="chord.html#D3" onmouseover="rs('chord/D3')" onmouseout="rh()">CHORD:def 3</a></span>;<br/>
<a NAME="E10:119"/>
<font color="Maroon" title="c7">e8</font> is    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/>
<a NAME="E11:119"/><span class="kw">then </span>
<font color="Maroon" title="c7">e8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:119"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="ref" href="glib_000.html#T64" onmouseover="rs('glib_000/T64')" onmouseout="rh()">GLIB_000:64</a></span>;<br/>
<a NAME="E12:119"/><span class="kw">then </span>
<font color="Maroon" title="c7">e8</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:119"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E13:119"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A6</font></span>: 
( ( <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Maroon" title="c6">v9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> ) or ( <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> &amp; <font color="Maroon" title="c6">v9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:119"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E9:119"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="ref" href="glib_000.html#T15" onmouseover="rs('glib_000/T15')" onmouseout="rh()">GLIB_000:15</a></span>;<br/>
<a NAME="E14:119"/>
 for <font color="Olive" title="b1">e9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e9</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c1">G</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="119_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c8">e9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><a NAME="E2:119_1"/><span class="lab"><font color="Green" title="E42">A7</font></span>: 
<font color="Maroon" title="c8">e9</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font>,<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"> contradiction</span><br/>

<a NAME="E3:119_1"/>
<font color="Maroon" title="c8">e9</font> is    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/>
<a NAME="E4:119_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">e9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:119_1"><span class="lab"><font color="Green" title="E42">A7</font></span></a>, <a class="ref" href="glib_000.html#T64" onmouseover="rs('glib_000/T64')" onmouseout="rh()">GLIB_000:64</a></span>;<br/>
<a NAME="E5:119_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">e9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:119"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E6:119_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v9</font> ) or ( <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v9</font> &amp; <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:119"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E2:119_1"><span class="lab"><font color="Green" title="E42">A7</font></span></a>, <a class="ref" href="glib_000.html#T15" onmouseover="rs('glib_000/T15')" onmouseout="rh()">GLIB_000:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:119_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E13:119"><span class="lab"><font color="Green" title="E41">A6</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E15:119"/>
 not <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#D3" onmouseover="rs('chord/D3')" onmouseout="rh()">CHORD:def 3</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>
