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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">G1</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">G2</font> being    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c1">G1</font> holds <br/> ( ( <font color="Maroon" title="c1">G1</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a>  implies <font color="Olive" title="b1">G2</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) &amp; ( <font color="Olive" title="b1">G2</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  implies <font color="Maroon" title="c1">G1</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a>  ) &amp; (  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a>  implies <font color="Olive" title="b1">G2</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a>  ) &amp; ( <font color="Olive" title="b1">G2</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a>  implies  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">G2</font> be    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c1">G1</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="c1">G1</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) &amp; ( <font color="Maroon" title="c2">G2</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  implies <font color="Maroon" title="c1">G1</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a>  ) &amp; (  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a>  ) &amp; ( <font color="Maroon" title="c2">G2</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a>  implies  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a>  ) )</span><br/>



<div><span class="kw">hereby </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">G2</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  implies <font color="Maroon" title="c1">G1</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a>  ) &amp; (  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a>  ) &amp; ( <font color="Maroon" title="c2">G2</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a>  implies  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a>  ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:176_1"/><span class="lab"><font color="Green" title="E87">A1</font></span>: 
<font color="Maroon" title="c1">G1</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</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"> <font color="Maroon" title="c2">G2</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a> </span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="176_1_1"><span class="kw">now </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">  for <font color="Olive" title="b1">v2</font>, <font color="Olive" title="b2">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">w2</font> holds <br/> not <font color="Olive" title="b1">v2</font>,<font color="Olive" title="b2">w2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">v2</font>, <font color="Maroon" title="c4">w2</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</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">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">w2</font> implies  not <font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">w2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:176_1_1"/><span class="lab"><font color="Green" title="E88">A2</font></span>: 
<font color="Maroon" title="c3">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">w2</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="c3">v2</font>,<font color="Maroon" title="c4">w2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> </span><br/><span class="kw">reconsider </span><font color="Maroon" title="c5">v1</font> = <font color="Maroon" title="c3">v2</font>, <font color="Maroon" title="c6">w1</font> = <font color="Maroon" title="c4">w2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#T99" target="_self" onmouseover="rs('glib_012/T99')" onmouseout="rh()">Th99</a></span>;<br/><a NAME="E3:176_1_1"/>
<font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">w1</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="#E1:176_1"><span class="lab"><font color="Green" title="E87">A1</font></span></a>, <a class="txt" href="#E1:176_1_1"><span class="lab"><font color="Green" title="E88">A2</font></span></a>, <a class="ref" href="chord.html#D6" onmouseover="rs('chord/D6')" onmouseout="rh()">CHORD:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E4:176_1_1"/>
 not <font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">w2</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="#E1:176_1_1"><span class="lab"><font color="Green" title="E88">A2</font></span></a>, <a class="ref" href="glib_012.html#T99" target="_self" onmouseover="rs('glib_012/T99')" onmouseout="rh()">Th99</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="E3:176_1"/>
<font color="Maroon" title="c2">G2</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glibpre0.html#T60" onmouseover="rs('glibpre0/T60')" onmouseout="rh()">GLIBPRE0:60</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>

<div><span class="kw">hereby </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"> (  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a>  iff <font color="Maroon" title="c2">G2</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:176_2"/><span class="lab"><font color="Green" title="E87">A3</font></span>: 
<font color="Maroon" title="c2">G2</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</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"> <font color="Maroon" title="c1">G1</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> </span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="176_2_1"><span class="kw">now </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">  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">w1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font>  st <font color="Olive" title="b1">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">w1</font> holds <br/><font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">w1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">v1</font>, <font color="Maroon" title="c4">w1</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</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">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">w1</font> implies <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">w1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:176_2_1"/><span class="lab"><font color="Green" title="E88">A4</font></span>: 
<font color="Maroon" title="c3">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">w1</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">v1</font>,<font color="Maroon" title="c4">w1</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> </span><br/><span class="kw">reconsider </span><font color="Maroon" title="c5">v2</font> = <font color="Maroon" title="c3">v1</font>, <font color="Maroon" title="c6">w2</font> = <font color="Maroon" title="c4">w1</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#T99" target="_self" onmouseover="rs('glib_012/T99')" onmouseout="rh()">Th99</a></span>;<br/><a NAME="E3:176_2_1"/>
 not <font color="Maroon" title="c5">v2</font>,<font color="Maroon" title="c6">w2</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="#E1:176_2"><span class="lab"><font color="Green" title="E87">A3</font></span></a>, <a class="txt" href="#E1:176_2_1"><span class="lab"><font color="Green" title="E88">A4</font></span></a>, <a class="ref" href="glibpre0.html#T60" onmouseover="rs('glibpre0/T60')" onmouseout="rh()">GLIBPRE0:60</a></span>;<br/><span class="kw">hence </span><a NAME="E4:176_2_1"/>
<font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">w1</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="#E1:176_2_1"><span class="lab"><font color="Green" title="E88">A4</font></span></a>, <a class="ref" href="glib_012.html#T99" target="_self" onmouseover="rs('glib_012/T99')" onmouseout="rh()">Th99</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="E3:176_2"/>
<font color="Maroon" title="c1">G1</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#D6" onmouseover="rs('chord/D6')" onmouseout="rh()">CHORD:def 6</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>

<div><span class="kw">hereby </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">G2</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a>  implies  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:176_3"/><span class="lab"><font color="Green" title="E87">A5</font></span>: 
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</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"> <font color="Maroon" title="c2">G2</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> </span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="176_3_1"><span class="kw">now </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">  for <font color="Olive" title="b1">v2</font>, <font color="Olive" title="b2">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">w2</font> holds <br/><font color="Olive" title="b1">v2</font>,<font color="Olive" title="b2">w2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">v2</font>, <font color="Maroon" title="c4">w2</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</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">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">w2</font> implies <font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">w2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:176_3_1"/><span class="lab"><font color="Green" title="E88">A6</font></span>: 
<font color="Maroon" title="c3">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">w2</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">v2</font>,<font color="Maroon" title="c4">w2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> </span><br/><span class="kw">reconsider </span><font color="Maroon" title="c5">v1</font> = <font color="Maroon" title="c3">v2</font>, <font color="Maroon" title="c6">w1</font> = <font color="Maroon" title="c4">w2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</a></span>;<br/><a NAME="E3:176_3_1"/>
 for <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">w1</font>,<font color="Maroon" title="c1">G1</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="176_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c7">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><a NAME="E2:176_3_1_1"/><span class="lab"><font color="Green" title="E89">A7</font></span>: 
<font color="Maroon" title="c7">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">w1</font>,<font color="Maroon" title="c1">G1</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="E3:176_3_1_1"/>
<font color="Maroon" title="c7">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:176_3"><span class="lab"><font color="Green" title="E87">A5</font></span></a>, <a class="txt" href="#E2:176_3_1_1"><span class="lab"><font color="Green" title="E89">A7</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">v9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:176_3_1_1"/><span class="lab"><font color="Green" title="E90">A8</font></span>: 
<font color="Maroon" title="c7">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">v9</font>,<font color="Maroon" title="c8">v9</font>,<font color="Maroon" title="c1">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_009.html#D2" onmouseover="rs('glib_009/D2')" onmouseout="rh()">GLIB_009:def 2</a></span>;<br/>
<a NAME="E6:176_3_1_1"/>
( <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v9</font> &amp; <font color="Maroon" title="c6">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:176_3_1_1"><span class="lab"><font color="Green" title="E89">A7</font></span></a>, <a class="txt" href="#E5:176_3_1_1"><span class="lab"><font color="Green" title="E90">A8</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="E7:176_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:176_3_1"><span class="lab"><font color="Green" title="E88">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><a NAME="E4:176_3_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">w1</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:176_3_1"><span class="lab"><font color="Green" title="E88">A6</font></span></a>, <a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</a></span>;<br/><span class="kw">hence </span><a NAME="E5:176_3_1"/>
<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">w2</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><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:176_3"/>
<font color="Maroon" title="c2">G2</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#D6" onmouseover="rs('chord/D6')" onmouseout="rh()">CHORD:def 6</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>

<div><span class="kw">hereby </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>
<div class="add"><span class="kw">assume </span><a NAME="E1:176_4"/><span class="lab"><font color="Green" title="E87">A9</font></span>: 
<font color="Maroon" title="c2">G2</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</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">  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a> </span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="176_4_1"><span class="kw">now </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">  for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> holds <br/><font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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"> ( <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> implies <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:176_4_1"/><span class="lab"><font color="Green" title="E88">A10</font></span>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</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">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a> </span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">v1</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e</font>, <font color="Maroon" title="c5">w1</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</a></span>;<br/><a NAME="E3:176_4_1"/><span class="lab"><font color="Green" title="E89">A11</font></span>: 
<font color="Maroon" title="c3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">w1</font>,<font color="Maroon" title="c1">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:176_4_1"><span class="lab"><font color="Green" title="E88">A10</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E4:176_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A12</font></span>: 
<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">w1</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>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">v2</font> = <font color="Maroon" title="c4">v1</font>, <font color="Maroon" title="c7">w2</font> = <font color="Maroon" title="c5">w1</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#T99" target="_self" onmouseover="rs('glib_012/T99')" onmouseout="rh()">Th99</a></span>;<br/><span class="kw">assume </span><a NAME="E6:176_4_1"/>
 not <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</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"> contradiction</span><br/><a NAME="E7:176_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A13</font></span>: 
<font color="Maroon" title="c4">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">w1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:176_4_1"><span class="lab"><font color="Green" title="E89">A11</font></span></a>, <a class="ref" href="glib_009.html#D2" onmouseover="rs('glib_009/D2')" onmouseout="rh()">GLIB_009:def 2</a></span>;<br/><a NAME="E8:176_4_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c7">w2</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="#E4:176_4_1"><span class="lab"><font color="Green" title="E90">A12</font></span></a>, <a class="ref" href="glib_012.html#T99" target="_self" onmouseover="rs('glib_012/T99')" onmouseout="rh()">Th99</a></span>;<br/><span class="kw">hence </span><a NAME="E9:176_4_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:176_4"><span class="lab"><font color="Green" title="E87">A9</font></span></a>, <a class="txt" href="#E7:176_4_1"><span class="lab"><font color="Green" title="E91">A13</font></span></a>, <a class="ref" href="chord.html#D6" onmouseover="rs('chord/D6')" onmouseout="rh()">CHORD:def 6</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><a NAME="E3:176_4"/><span class="kw">then </span>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:176_4"/>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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">thus </span><a NAME="E5:176"/>
verum
 ; <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>
