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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp;  not <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  holds <br/> for <font color="Olive" title="b3">S</font> being    <a href="chord.html#M2" title="CHORD:mode.2">VertexSeparator</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>  st <font color="Olive" title="b3">S</font> is  <a href="chord.html#V4" title="CHORD:attr.4">minimal</a>  &amp;  not <font color="Olive" title="b3">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  holds <br/> for <font color="Olive" title="b4">H</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Olive" title="b3">S</font> holds  <font color="Olive" title="b4">H</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">tVG</font> =  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp;  not <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  implies  for <font color="Olive" title="b1">S</font> being    <a href="chord.html#M2" title="CHORD:mode.2">VertexSeparator</a> of <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>  st <font color="Olive" title="b1">S</font> is  <a href="chord.html#V4" title="CHORD:attr.4">minimal</a>  &amp;  not <font color="Olive" title="b1">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  holds <br/> for <font color="Olive" title="b2">H</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">S</font> holds  <font color="Olive" title="b2">H</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:117"/><span class="lab"><font color="Green" title="E91">A1</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">and </span><br/><a NAME="E2:117"/><span class="lab"><font color="Green" title="E92">A2</font></span>: 
 not <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being    <a href="chord.html#M2" title="CHORD:mode.2">VertexSeparator</a> of <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>  st <font color="Olive" title="b1">S</font> is  <a href="chord.html#V4" title="CHORD:attr.4">minimal</a>  &amp;  not <font color="Olive" title="b1">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  holds <br/> for <font color="Olive" title="b2">H</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">S</font> holds  <font color="Olive" title="b2">H</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">S</font> be    <a href="chord.html#M2" title="CHORD:mode.2">VertexSeparator</a> of <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">S</font> is  <a href="chord.html#V4" title="CHORD:attr.4">minimal</a>  &amp;  not <font color="Maroon" title="c5">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  for <font color="Olive" title="b1">H</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> holds  <font color="Olive" title="b1">H</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:117"/><span class="lab"><font color="Green" title="E93">A3</font></span>: 
<font color="Maroon" title="c5">S</font> is  <a href="chord.html#V4" title="CHORD:attr.4">minimal</a> 
 <span class="kw">and </span><br/><a NAME="E4:117"/><span class="lab"><font color="Green" title="E94">A4</font></span>: 
 not <font color="Maroon" title="c5">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> holds  <font color="Olive" title="b1">H</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">Gns</font> =  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">sa</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c8">sb</font> = <font color="Maroon" title="c4">b</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E91">A1</font></span></a>, <a class="txt" href="#E2:117"><span class="lab"><font color="Green" title="E92">A2</font></span></a>, <a class="ref" href="chord.html#T77" target="_self" title="CHORD:th.77">Th77</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">A</font> =  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c7">sa</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">B</font> =  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sb</font>;<br/>
<a NAME="E6:117"/><span class="lab"><font color="Green" title="E95">A5</font></span>: 
<span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c7">sa</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sb</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E91">A1</font></span></a>, <a class="txt" href="#E2:117"><span class="lab"><font color="Green" title="E92">A2</font></span></a>, <a class="ref" href="chord.html#T76" target="_self" title="CHORD:th.76">Th76</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">Gb</font> =  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font>,<span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sb</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">Ga</font> =  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font>,<span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c7">sa</font></span>)</span>;<br/>
<a NAME="E7:117"/><span class="lab"><font color="Green" title="E96">A6</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font>,<span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c7">sa</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c7">sa</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/>
<a NAME="E8:117"/><span class="lab"><font color="Green" title="E97">A7</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font>,<span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sb</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sb</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/>
<a NAME="E9:117"/><span class="lab"><font color="Green" title="E98">A8</font></span>: 
<span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sb</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E91">A1</font></span></a>, <a class="txt" href="#E2:117"><span class="lab"><font color="Green" title="E92">A2</font></span></a>, <a class="ref" href="chord.html#T75" target="_self" title="CHORD:th.75">Th75</a></span>;<br/>
<a NAME="E10:117"/><span class="lab"><font color="Green" title="E99">A9</font></span>: 
<span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c7">sa</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E91">A1</font></span></a>, <a class="txt" href="#E2:117"><span class="lab"><font color="Green" title="E92">A2</font></span></a>, <a class="ref" href="chord.html#T75" target="_self" title="CHORD:th.75">Th75</a></span>;<br/>
<a NAME="E11:117"/>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">A</font> =  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c7">sa</font>, <font color="Maroon" title="c14">B</font> =  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sb</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c15">Gs</font> be   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">Gs</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c16">x</font>, <font color="Maroon" title="c17">y</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c15">Gs</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="chord.html#D6" target="_self" title="CHORD:def.6">CHORD:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">y</font> implies <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c17">y</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E13:117"/><span class="lab"><font color="Green" title="E100">A10</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">y</font>
 <span class="kw">and </span><br/><a NAME="E14:117"/><span class="lab"><font color="Green" title="E101">A11</font></span>: 
 not <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c17">y</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c18">xg</font> = <font color="Maroon" title="c16">x</font>, <font color="Maroon" title="c19">yg</font> = <font color="Maroon" title="c17">y</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="ref" href="glib_000.html#T42" title="GLIB_000:th.42">GLIB_000:42</a></span>;<br/>
<a NAME="E16:117"/><span class="lab"><font color="Green" title="E102">A12</font></span>: 
<font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c15">Gs</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:117"><span class="lab"><font color="Green" title="E94">A4</font></span></a>, <a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/>
<a NAME="E17:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A13</font></span>: 
 ex <font color="Olive" title="b1">xag</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">xag</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font> &amp; <font color="Maroon" title="c18">xg</font>,<font color="Olive" title="b1">xag</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="#E1:117"><span class="lab"><font color="Green" title="E91">A1</font></span></a>, <a class="txt" href="#E2:117"><span class="lab"><font color="Green" title="E92">A2</font></span></a>, <a class="txt" href="#E3:117"><span class="lab"><font color="Green" title="E93">A3</font></span></a>, <a class="ref" href="chord.html#T82" target="_self" title="CHORD:th.82">Th82</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">Bx</font> = <font color="Maroon" title="c14">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">xg</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">Ax</font> = <font color="Maroon" title="c13">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">xg</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c22">Gbx</font> =  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">xg</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c23">Gax</font> =  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">xg</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c24">xy</font> = <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>;<br/>
<a NAME="E18:117"/><span class="lab"><font color="Green" title="E104">A14</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">xg</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">xg</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font> implies <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">S</font> )</span><br/><span class="kw">assume </span><a NAME="E1:117_1"/><span class="lab"><font color="Green" title="E105">A15</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">S</font></span><br/><a NAME="E2:117_1"/>
 not <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:117"><span class="lab"><font color="Green" title="E99">A9</font></span></a>, <a class="txt" href="#E1:117_1"><span class="lab"><font color="Green" title="E105">A15</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:117_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117_1"><span class="lab"><font color="Green" title="E105">A15</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" 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="E20:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A16</font></span>: 
<font color="Maroon" title="c13">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c25">yag</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="E22:117"/><span class="lab"><font color="Green" title="E106">A17</font></span>: 
<font color="Maroon" title="c25">yag</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font>
 <span class="kw">and </span><br/><a NAME="E23:117"/><span class="lab"><font color="Green" title="E107">A18</font></span>: 
<font color="Maroon" title="c19">yg</font>,<font color="Maroon" title="c25">yag</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="#E1:117"><span class="lab"><font color="Green" title="E91">A1</font></span></a>, <a class="txt" href="#E2:117"><span class="lab"><font color="Green" title="E92">A2</font></span></a>, <a class="txt" href="#E3:117"><span class="lab"><font color="Green" title="E93">A3</font></span></a>, <a class="txt" href="#E16:117"><span class="lab"><font color="Green" title="E102">A12</font></span></a>, <a class="ref" href="chord.html#T82" target="_self" title="CHORD:th.82">Th82</a></span>;<br/>
<a NAME="E24:117"/><span class="lab"><font color="Green" title="E108">A19</font></span>: 
<font color="Maroon" title="c25">yag</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">xg</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:117"><span class="lab"><font color="Green" title="E106">A17</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c26">Gb1</font> =  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c27">Ga1</font> =  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/>
<a NAME="E25:117"/><span class="lab"><font color="Green" title="E109">A20</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">xg</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">xg</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/>
<a NAME="E26:117"/>
 not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E91">A1</font></span></a>, <a class="txt" href="#E2:117"><span class="lab"><font color="Green" title="E92">A2</font></span></a>, <a class="ref" href="chord.html#D8" target="_self" title="CHORD:def.8">Def8</a></span>;<br/>
<a NAME="E27:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A21</font></span>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">S</font> is   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c28">Ga</font> =  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font>,<span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c7">sa</font></span>)</span> as   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c13">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:117"><span class="lab"><font color="Green" title="E105">A16</font></span></a>, <a class="ref" href="chord.html#T29" target="_self" title="CHORD:th.29">Th29</a></span>;<br/>
<a NAME="E29:117"/><span class="lab"><font color="Green" title="E111">A22</font></span>: 
 not <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:117"><span class="lab"><font color="Green" title="E100">A10</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c29">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> implies <font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">S</font> )</span><br/><span class="kw">assume </span><a NAME="E1:117_2"/><span class="lab"><font color="Green" title="E112">A23</font></span>: 
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">S</font></span><br/><a NAME="E2:117_2"/>
 not <font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:117"><span class="lab"><font color="Green" title="E98">A8</font></span></a>, <a class="txt" href="#E1:117_2"><span class="lab"><font color="Green" title="E112">A23</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:117_2"/>
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117_2"><span class="lab"><font color="Green" title="E112">A23</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" 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="E31:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A24</font></span>: 
<font color="Maroon" title="c14">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c29">Gb</font> =  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font>,<span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">sb</font></span>)</span> as   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c14">B</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:117"><span class="lab"><font color="Green" title="E110">A21</font></span></a>, <a class="ref" href="chord.html#T29" target="_self" title="CHORD:th.29">Th29</a></span>;<br/>
<a NAME="E33:117"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E34:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A25</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E35:117"/><span class="lab"><font color="Green" title="E114">A26</font></span>: 
 ex <font color="Olive" title="b1">xbg</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">xbg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> &amp; <font color="Maroon" title="c18">xg</font>,<font color="Olive" title="b1">xbg</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="#E1:117"><span class="lab"><font color="Green" title="E91">A1</font></span></a>, <a class="txt" href="#E2:117"><span class="lab"><font color="Green" title="E92">A2</font></span></a>, <a class="txt" href="#E3:117"><span class="lab"><font color="Green" title="E93">A3</font></span></a>, <a class="txt" href="#E16:117"><span class="lab"><font color="Green" title="E102">A12</font></span></a>, <a class="ref" href="chord.html#T83" target="_self" title="CHORD:th.83">Th83</a></span>;<br/>
<a NAME="E36:117"/>
 not <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:117"><span class="lab"><font color="Green" title="E112">A24</font></span></a>, <a class="txt" href="#E16:117"><span class="lab"><font color="Green" title="E102">A12</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E37:117"/><span class="kw">then </span>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1">(<span class="default"><a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c29">Gb</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:117"><span class="lab"><font color="Green" title="E97">A7</font></span></a>, <a class="txt" href="#E35:117"><span class="lab"><font color="Green" title="E114">A26</font></span></a></span>;<br/>
<a NAME="E38:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A27</font></span>: 
 the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">xg</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T57" target="_self" title="CHORD:th.57">Th57</a></span>;<br/>
<a NAME="E39:117"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E40:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A28</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c30">ybg</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="E42:117"/><span class="lab"><font color="Green" title="E117">A29</font></span>: 
<font color="Maroon" title="c30">ybg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font>
 <span class="kw">and </span><br/><a NAME="E43:117"/><span class="lab"><font color="Green" title="E118">A30</font></span>: 
<font color="Maroon" title="c19">yg</font>,<font color="Maroon" title="c30">ybg</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="#E1:117"><span class="lab"><font color="Green" title="E91">A1</font></span></a>, <a class="txt" href="#E2:117"><span class="lab"><font color="Green" title="E92">A2</font></span></a>, <a class="txt" href="#E3:117"><span class="lab"><font color="Green" title="E93">A3</font></span></a>, <a class="txt" href="#E16:117"><span class="lab"><font color="Green" title="E102">A12</font></span></a>, <a class="ref" href="chord.html#T83" target="_self" title="CHORD:th.83">Th83</a></span>;<br/>
<span class="lab"><font color="Green" title="E119">A31</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> = 
<font color="Maroon" title="c14">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></span>
<br/>.= 
<font color="Maroon" title="c14">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T1" title="ENUMSET1:th.1">ENUMSET1:1</a></span>
;<br/>
<a NAME="E45:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">A32</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/>
<a NAME="E46:117"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E47:117"/><span class="kw">then </span>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c31">xb</font> = <font color="Maroon" title="c16">x</font>, <font color="Maroon" title="c32">yb</font> = <font color="Maroon" title="c17">y</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:117"><span class="lab"><font color="Green" title="E119">A31</font></span></a>, <a class="txt" href="#E40:117"><span class="lab"><font color="Green" title="E116">A28</font></span></a>, <a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/>
<a NAME="E49:117"/><span class="lab"><font color="Green" title="E121">A33</font></span>: 
<font color="Maroon" title="c30">ybg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">xg</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:117"><span class="lab"><font color="Green" title="E117">A29</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E50:117"/>
 not <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:117"><span class="lab"><font color="Green" title="E112">A24</font></span></a>, <a class="txt" href="#E16:117"><span class="lab"><font color="Green" title="E102">A12</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E51:117"/><span class="kw">then </span>
 not <font color="Maroon" title="c19">yg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">xg</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:117"><span class="lab"><font color="Green" title="E104">A14</font></span></a>, <a class="txt" href="#E29:117"><span class="lab"><font color="Green" title="E111">A22</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E52:117"/><span class="kw">then </span>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1">(<span class="default"><a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">xg</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:117"><span class="lab"><font color="Green" title="E104">A14</font></span></a>, <a class="txt" href="#E43:117"><span class="lab"><font color="Green" title="E118">A30</font></span></a>, <a class="txt" href="#E49:117"><span class="lab"><font color="Green" title="E121">A33</font></span></a></span>;<br/>
<a NAME="E53:117"/><span class="kw">then </span>
 the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:117"><span class="lab"><font color="Green" title="E115">A27</font></span></a>, <a class="ref" href="chord.html#T57" target="_self" title="CHORD:th.57">Th57</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c33">Wb</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span><span class="kw"> such that </span><br/><a NAME="E55:117"/><span class="lab"><font color="Green" title="E122">A34</font></span>: 
<font color="Maroon" title="c33">Wb</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c32">yb</font>,<font color="Maroon" title="c31">xb</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D1" title="GLIB_002:def.1">GLIB_002:def 1</a></span>;<br/>
<a NAME="E56:117"/>
 not <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:117"><span class="lab"><font color="Green" title="E105">A16</font></span></a>, <a class="txt" href="#E16:117"><span class="lab"><font color="Green" title="E102">A12</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E57:117"/><span class="kw">then </span>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1">(<span class="default"><a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c28">Ga</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:117"><span class="lab"><font color="Green" title="E96">A6</font></span></a>, <a class="txt" href="#E17:117"><span class="lab"><font color="Green" title="E103">A13</font></span></a></span>;<br/>
<a NAME="E58:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A35</font></span>: 
 the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">xg</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T57" target="_self" title="CHORD:th.57">Th57</a></span>;<br/>
<span class="lab"><font color="Green" title="E124">A36</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c13">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> = 
<font color="Maroon" title="c13">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></span>
<br/>.= 
<font color="Maroon" title="c13">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T1" title="ENUMSET1:th.1">ENUMSET1:1</a></span>
;<br/>
<a NAME="E60:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A37</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/>
<a NAME="E61:117"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E62:117"/><span class="kw">then </span>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c34">xa</font> = <font color="Maroon" title="c16">x</font>, <font color="Maroon" title="c35">ya</font> = <font color="Maroon" title="c17">y</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:117"><span class="lab"><font color="Green" title="E124">A36</font></span></a>, <a class="txt" href="#E34:117"><span class="lab"><font color="Green" title="E113">A25</font></span></a>, <a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/>
<a NAME="E64:117"/><span class="lab"><font color="Green" title="E126">A38</font></span>: 
 not <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:117"><span class="lab"><font color="Green" title="E100">A10</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E65:117"/>
 not <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:117"><span class="lab"><font color="Green" title="E105">A16</font></span></a>, <a class="txt" href="#E16:117"><span class="lab"><font color="Green" title="E102">A12</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E66:117"/><span class="kw">then </span>
 not <font color="Maroon" title="c19">yg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">xg</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:117"><span class="lab"><font color="Green" title="E109">A20</font></span></a>, <a class="txt" href="#E64:117"><span class="lab"><font color="Green" title="E126">A38</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E67:117"/><span class="kw">then </span>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1">(<span class="default"><a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">xg</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:117"><span class="lab"><font color="Green" title="E109">A20</font></span></a>, <a class="txt" href="#E23:117"><span class="lab"><font color="Green" title="E107">A18</font></span></a>, <a class="txt" href="#E24:117"><span class="lab"><font color="Green" title="E108">A19</font></span></a></span>;<br/>
<a NAME="E68:117"/><span class="kw">then </span>
 the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:117"><span class="lab"><font color="Green" title="E123">A35</font></span></a>, <a class="ref" href="chord.html#T57" target="_self" title="CHORD:th.57">Th57</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c36">Wa</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span><span class="kw"> such that </span><br/><a NAME="E70:117"/><span class="lab"><font color="Green" title="E127">A39</font></span>: 
<font color="Maroon" title="c36">Wa</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c34">xa</font>,<font color="Maroon" title="c35">ya</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D1" title="GLIB_002:def.1">GLIB_002:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E128">A40</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c13">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c14">B</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T24" title="XBOOLE_1:th.24">XBOOLE_1:24</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:117"><span class="lab"><font color="Green" title="E95">A5</font></span></a></span>
;<br/>
<a NAME="E72:117"/><span class="lab"><font color="Green" title="E129">A41</font></span>: 
<font color="Maroon" title="c36">Wa</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">ya</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E70:117"><span class="lab"><font color="Green" title="E127">A39</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>;<br/>
<a NAME="E73:117"/><span class="lab"><font color="Green" title="E130">A42</font></span>: 
<font color="Maroon" title="c36">Wa</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="c34">xa</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E70:117"><span class="lab"><font color="Green" title="E127">A39</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>;<br/>
<a NAME="E74:117"/><span class="lab"><font color="Green" title="E131">A43</font></span>: 
<font color="Maroon" title="c33">Wb</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">xa</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:117"><span class="lab"><font color="Green" title="E122">A34</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>;<br/>
<a NAME="E75:117"/><span class="lab"><font color="Green" title="E132">A44</font></span>: 
<font color="Maroon" title="c33">Wb</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="c35">ya</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:117"><span class="lab"><font color="Green" title="E122">A34</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c37">Pb</font> being   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span><span class="kw"> such that </span><br/><a NAME="E77:117"/><span class="lab"><font color="Green" title="E133">A45</font></span>: 
<font color="Maroon" title="c37">Pb</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c33">Wb</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c33">Wb</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <span class="kw">and </span><br/><a NAME="E78:117"/><span class="lab"><font color="Green" title="E134">A46</font></span>: 
<font color="Maroon" title="c37">Pb</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T39" target="_self" title="CHORD:th.39">Th39</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c38">Pa</font> being   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span><span class="kw"> such that </span><br/><a NAME="E80:117"/><span class="lab"><font color="Green" title="E135">A47</font></span>: 
<font color="Maroon" title="c38">Pa</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c36">Wa</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c36">Wa</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <span class="kw">and </span><br/><a NAME="E81:117"/><span class="lab"><font color="Green" title="E136">A48</font></span>: 
<font color="Maroon" title="c38">Pa</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T39" target="_self" title="CHORD:th.39">Th39</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c39">Pag</font> = <font color="Maroon" title="c38">Pa</font>, <font color="Maroon" title="c40">Pbg</font> = <font color="Maroon" title="c37">Pb</font> as   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T175" title="GLIB_001:th.175">GLIB_001:175</a></span>;<br/>
<a NAME="E83:117"/>
<font color="Maroon" title="c37">Pb</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="c37">Pb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
;<br/>
<a NAME="E84:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E137">A49</font></span>: 
<font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">yg</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:117"><span class="lab"><font color="Green" title="E132">A44</font></span></a>, <a class="txt" href="#E77:117"><span class="lab"><font color="Green" title="E133">A45</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>;<br/>
<a NAME="E85:117"/>
<font color="Maroon" title="c40">Pbg</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">Pb</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T98" title="GLIB_001:th.98">GLIB_001:98</a></span>;<br/>
<a NAME="E86:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E138">A50</font></span>: 
<font color="Maroon" title="c40">Pbg</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="c14">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:117"><span class="lab"><font color="Green" title="E120">A32</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E87:117"/>
<font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">Pa</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T98" title="GLIB_001:th.98">GLIB_001:98</a></span>;<br/>
<a NAME="E88:117"/><span class="kw">then </span>
<font color="Maroon" title="c39">Pag</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="c13">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:117"><span class="lab"><font color="Green" title="E125">A37</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E89:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E139">A51</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c40">Pbg</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E86:117"><span class="lab"><font color="Green" title="E138">A50</font></span></a>, <a class="txt" href="#E71:117"><span class="lab"><font color="Green" title="E128">A40</font></span></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c41">P</font> = <font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font>;<br/>
<a NAME="E90:117"/>
<font color="Maroon" title="c38">Pa</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">Pa</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> <font color="Maroon" title="c38">Pa</font></span>)</span>
 
;<br/>
<a NAME="E91:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">A52</font></span>: 
<font color="Maroon" title="c39">Pag</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> <font color="Maroon" title="c39">Pag</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">yg</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E72:117"><span class="lab"><font color="Green" title="E129">A41</font></span></a>, <a class="txt" href="#E80:117"><span class="lab"><font color="Green" title="E135">A47</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>;<br/>
<a NAME="E92:117"/><span class="lab"><font color="Green" title="E141">A53</font></span>: 
<font color="Maroon" title="c37">Pb</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">Pb</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> <font color="Maroon" title="c37">Pb</font></span>)</span>
 
;<br/>
<a NAME="E93:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E142">A54</font></span>: 
<font color="Maroon" title="c40">Pbg</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> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">xg</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:117"><span class="lab"><font color="Green" title="E131">A43</font></span></a>, <a class="txt" href="#E77:117"><span class="lab"><font color="Green" title="E133">A45</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>;<br/>
<a NAME="E94:117"/><span class="lab"><font color="Green" title="E143">A55</font></span>: 
<font color="Maroon" title="c40">Pbg</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">Pbg</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> <font color="Maroon" title="c40">Pbg</font></span>)</span>
 
;<br/>
<a NAME="E95:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E144">A56</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c40">Pbg</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E93:117"><span class="lab"><font color="Green" title="E142">A54</font></span></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></span>;<br/>
<a NAME="E96:117"/><span class="lab"><font color="Green" title="E145">A57</font></span>: 
<font color="Maroon" title="c40">Pbg</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="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
;<br/>
<a NAME="E97:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E146">A58</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c40">Pbg</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E84:117"><span class="lab"><font color="Green" title="E137">A49</font></span></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></span>;<br/>
<a NAME="E98:117"/><span class="lab"><font color="Green" title="E147">A59</font></span>: 
 not <font color="Maroon" title="c40">Pbg</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:117"><span class="lab"><font color="Green" title="E100">A10</font></span></a>, <a class="txt" href="#E96:117"><span class="lab"><font color="Green" title="E145">A57</font></span></a>, <a class="txt" href="#E94:117"><span class="lab"><font color="Green" title="E143">A55</font></span></a>, <a class="txt" href="#E84:117"><span class="lab"><font color="Green" title="E137">A49</font></span></a>, <a class="txt" href="#E93:117"><span class="lab"><font color="Green" title="E142">A54</font></span></a>, <a class="ref" href="glib_001.html#T127" title="GLIB_001:th.127">GLIB_001:127</a></span>;<br/>
<a NAME="E99:117"/><span class="lab"><font color="Green" title="E148">A60</font></span>: 
 not <font color="Maroon" title="c40">Pbg</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:117"><span class="lab"><font color="Green" title="E100">A10</font></span></a>, <a class="txt" href="#E96:117"><span class="lab"><font color="Green" title="E145">A57</font></span></a>, <a class="txt" href="#E94:117"><span class="lab"><font color="Green" title="E143">A55</font></span></a>, <a class="txt" href="#E84:117"><span class="lab"><font color="Green" title="E137">A49</font></span></a>, <a class="txt" href="#E93:117"><span class="lab"><font color="Green" title="E142">A54</font></span></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></span>;<br/>
<a NAME="E100:117"/><span class="lab"><font color="Green" title="E149">A61</font></span>: 
 not <font color="Maroon" title="c40">Pbg</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:117"><span class="lab"><font color="Green" title="E100">A10</font></span></a>, <a class="txt" href="#E96:117"><span class="lab"><font color="Green" title="E145">A57</font></span></a>, <a class="txt" href="#E94:117"><span class="lab"><font color="Green" title="E143">A55</font></span></a>, <a class="txt" href="#E84:117"><span class="lab"><font color="Green" title="E137">A49</font></span></a>, <a class="txt" href="#E93:117"><span class="lab"><font color="Green" title="E142">A54</font></span></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></span>;<br/>
<a NAME="E101:117"/><span class="lab"><font color="Green" title="E150">A62</font></span>: 
 not <font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</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="#E4:117"><span class="lab"><font color="Green" title="E94">A4</font></span></a>, <a class="txt" href="#E14:117"><span class="lab"><font color="Green" title="E101">A11</font></span></a>, <a class="ref" href="chord.html#T45" target="_self" title="CHORD:th.45">Th45</a></span>;<br/>
<a NAME="E102:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E151">A63</font></span>: 
<font color="Maroon" title="c40">Pbg</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:117"><span class="lab"><font color="Green" title="E100">A10</font></span></a>, <a class="txt" href="#E96:117"><span class="lab"><font color="Green" title="E145">A57</font></span></a>, <a class="txt" href="#E94:117"><span class="lab"><font color="Green" title="E143">A55</font></span></a>, <a class="txt" href="#E84:117"><span class="lab"><font color="Green" title="E137">A49</font></span></a>, <a class="txt" href="#E93:117"><span class="lab"><font color="Green" title="E142">A54</font></span></a>, <a class="ref" href="chord.html#T46" target="_self" title="CHORD:th.46">Th46</a></span>;<br/>
<a NAME="E103:117"/>
<font color="Maroon" title="c38">Pa</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="c38">Pa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
;<br/>
<a NAME="E104:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E152">A64</font></span>: 
<font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">xg</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E73:117"><span class="lab"><font color="Green" title="E130">A42</font></span></a>, <a class="txt" href="#E80:117"><span class="lab"><font color="Green" title="E135">A47</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>;<br/>
<a NAME="E105:117"/><span class="lab"><font color="Green" title="E153">A65</font></span>: 
<font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c40">Pbg</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:117_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c40">Pbg</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c42">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:117_6"/><span class="lab"><font color="Green" title="E154">A66</font></span>: 
<font color="Maroon" title="c42">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c40">Pbg</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E4:117_6"/>
<font color="Maroon" title="c42">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:117_6"><span class="lab"><font color="Green" title="E154">A66</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E5:117_6"/><span class="kw">then </span>
<font color="Maroon" title="c42">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c38">Pa</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T110" title="GLIB_001:th.110">GLIB_001:110</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c43">a1</font>, <font color="Maroon" title="c44">a2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>, <font color="Maroon" title="c45">na</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:117_6"/><span class="lab"><font color="Green" title="E155">A67</font></span>: 
<font color="Maroon" title="c45">na</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <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="c39">Pag</font>
 <span class="kw">and </span><br/><a NAME="E8:117_6"/><span class="lab"><font color="Green" title="E156">A68</font></span>: 
<font color="Maroon" title="c43">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">na</font>
 <span class="kw">and </span><br/><a NAME="E9:117_6"/>
<font color="Maroon" title="c42">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">na</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:117_6"/><span class="lab"><font color="Green" title="E157">A69</font></span>: 
<font color="Maroon" title="c44">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">na</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:117_6"/><span class="lab"><font color="Green" title="E158">A70</font></span>: 
<font color="Maroon" title="c42">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c43">a1</font>,<font color="Maroon" title="c44">a2</font>, the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T103" title="GLIB_001:th.103">GLIB_001:103</a></span>;<br/>
<a NAME="E12:117_6"/><span class="lab"><font color="Green" title="E159">A71</font></span>: 
<font color="Maroon" title="c42">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c43">a1</font>,<font color="Maroon" title="c44">a2</font>,<font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:117_6"><span class="lab"><font color="Green" title="E158">A70</font></span></a>, <a class="ref" href="glib_000.html#T72" title="GLIB_000:th.72">GLIB_000:72</a></span>;<br/>
<a NAME="E13:117_6"/>
<font color="Maroon" title="c42">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c40">Pbg</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:117_6"><span class="lab"><font color="Green" title="E154">A66</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E14:117_6"/><span class="kw">then </span>
<font color="Maroon" title="c42">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c37">Pb</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T110" title="GLIB_001:th.110">GLIB_001:110</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c46">b1</font>, <font color="Maroon" title="c47">b2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>, <font color="Maroon" title="c48">nb</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E16:117_6"/>
<font color="Maroon" title="c48">nb</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <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="c40">Pbg</font>
 <span class="kw">and </span><br/><a NAME="E17:117_6"/>
<font color="Maroon" title="c46">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c48">nb</font>
 <span class="kw">and </span><br/><a NAME="E18:117_6"/>
<font color="Maroon" title="c42">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c48">nb</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E19:117_6"/>
<font color="Maroon" title="c47">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c48">nb</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">and </span><br/><a NAME="E20:117_6"/><span class="lab"><font color="Green" title="E160">A72</font></span>: 
<font color="Maroon" title="c42">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c46">b1</font>,<font color="Maroon" title="c47">b2</font>, the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T103" title="GLIB_001:th.103">GLIB_001:103</a></span>;<br/>
<a NAME="E21:117_6"/><span class="lab"><font color="Green" title="E161">A73</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:117"><span class="lab"><font color="Green" title="E119">A31</font></span></a>, <a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/>
<a NAME="E22:117_6"/>
<font color="Maroon" title="c42">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c46">b1</font>,<font color="Maroon" title="c47">b2</font>,<font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:117_6"><span class="lab"><font color="Green" title="E160">A72</font></span></a>, <a class="ref" href="glib_000.html#T72" title="GLIB_000:th.72">GLIB_000:72</a></span>;<br/>
<a NAME="E23:117_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E162">A74</font></span>: 
( ( <font color="Maroon" title="c43">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c46">b1</font> &amp; <font color="Maroon" title="c44">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">b2</font> ) or ( <font color="Maroon" title="c43">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">b2</font> &amp; <font color="Maroon" title="c44">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c46">b1</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:117_6"><span class="lab"><font color="Green" title="E159">A71</font></span></a>, <a class="ref" href="glib_000.html#T15" title="GLIB_000:th.15">GLIB_000:15</a></span>;<br/>
<a NAME="E24:117_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E163">A75</font></span>: 
( <font color="Maroon" title="c43">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> or <font color="Maroon" title="c43">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:117_6"><span class="lab"><font color="Green" title="E161">A73</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E25:117_6"/><span class="lab"><font color="Green" title="E164">A76</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:117"><span class="lab"><font color="Green" title="E124">A36</font></span></a>, <a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/>
<a NAME="E26:117_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E165">A77</font></span>: 
( <font color="Maroon" title="c43">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font> or <font color="Maroon" title="c43">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E27:117_6"/><span class="lab"><font color="Green" title="E166">A78</font></span>: 
( <font color="Maroon" title="c44">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font> or <font color="Maroon" title="c44">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:117_6"><span class="lab"><font color="Green" title="E164">A76</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E28:117_6"/><span class="lab"><font color="Green" title="E167">A79</font></span>: 
( <font color="Maroon" title="c44">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> or <font color="Maroon" title="c44">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:117_6"><span class="lab"><font color="Green" title="E162">A74</font></span></a>, <a class="txt" href="#E21:117_6"><span class="lab"><font color="Green" title="E161">A73</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_6_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:117_6_1"/>
( ( <font color="Maroon" title="c43">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> &amp; <font color="Maroon" title="c44">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> ) or ( <font color="Maroon" title="c43">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> &amp; <font color="Maroon" title="c44">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> ) or ( <font color="Maroon" title="c43">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> &amp; <font color="Maroon" title="c44">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> ) or ( <font color="Maroon" title="c43">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> &amp; <font color="Maroon" title="c44">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:117"><span class="lab"><font color="Green" title="E95">A5</font></span></a>, <a class="txt" href="#E26:117_6"><span class="lab"><font color="Green" title="E165">A77</font></span></a>, <a class="txt" href="#E27:117_6"><span class="lab"><font color="Green" title="E166">A78</font></span></a>, <a class="txt" href="#E24:117_6"><span class="lab"><font color="Green" title="E163">A75</font></span></a>, <a class="txt" href="#E28:117_6"><span class="lab"><font color="Green" title="E167">A79</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:117_6_1_1"/><span class="lab"><font color="Green" title="E168">A80</font></span>: 
( ( <font color="Maroon" title="c43">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> &amp; <font color="Maroon" title="c44">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> ) or ( <font color="Maroon" title="c43">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> &amp; <font color="Maroon" title="c44">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:117_6_1_1"/>
<font color="Maroon" title="c45">na</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c45">na</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T39" title="XREAL_1:th.39">XREAL_1:39</a></span>;<br/><span class="kw">hence </span><a NAME="E3:117_6_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:117"><span class="lab"><font color="Green" title="E100">A10</font></span></a>, <a class="txt" href="#E104:117"><span class="lab"><font color="Green" title="E152">A64</font></span></a>, <a class="txt" href="#E91:117"><span class="lab"><font color="Green" title="E140">A52</font></span></a>, <a class="txt" href="#E7:117_6"><span class="lab"><font color="Green" title="E155">A67</font></span></a>, <a class="txt" href="#E8:117_6"><span class="lab"><font color="Green" title="E156">A68</font></span></a>, <a class="txt" href="#E10:117_6"><span class="lab"><font color="Green" title="E157">A69</font></span></a>, <a class="txt" href="#E1:117_6_1_1"><span class="lab"><font color="Green" title="E168">A80</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>; <a class="txt" 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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:117_6_1_2"/>
( ( <font color="Maroon" title="c43">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> &amp; <font color="Maroon" title="c44">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> ) or ( <font color="Maroon" title="c43">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> &amp; <font color="Maroon" title="c44">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:117_6_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E101:117"><span class="lab"><font color="Green" title="E150">A62</font></span></a>, <a class="txt" href="#E12:117_6"><span class="lab"><font color="Green" title="E159">A71</font></span></a>, <a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></span>; <a class="txt" 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">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E106:117"/><span class="lab"><font color="Green" title="E154">A81</font></span>: 
<font color="Maroon" title="c39">Pag</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="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
;<br/>
<a NAME="E107:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E155">A82</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E104:117"><span class="lab"><font color="Green" title="E152">A64</font></span></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></span>;<br/>
<a NAME="E108:117"/><span class="lab"><font color="Green" title="E156">A83</font></span>: 
<font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">Pag</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> <font color="Maroon" title="c39">Pag</font></span>)</span>
 
;<br/>
<a NAME="E109:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E157">A84</font></span>: 
<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"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">Pbg</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E91:117"><span class="lab"><font color="Green" title="E140">A52</font></span></a>, <a class="txt" href="#E96:117"><span class="lab"><font color="Green" title="E145">A57</font></span></a>, <a class="txt" href="#E84:117"><span class="lab"><font color="Green" title="E137">A49</font></span></a>, <a class="ref" href="glib_001.html#T28" title="GLIB_001:th.28">GLIB_001:28</a></span>;<br/>
<a NAME="E110:117"/><span class="lab"><font color="Green" title="E158">A85</font></span>: 
<font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:117"><span class="lab"><font color="Green" title="E100">A10</font></span></a>, <a class="txt" href="#E101:117"><span class="lab"><font color="Green" title="E150">A62</font></span></a>, <a class="txt" href="#E106:117"><span class="lab"><font color="Green" title="E154">A81</font></span></a>, <a class="txt" href="#E108:117"><span class="lab"><font color="Green" title="E156">A83</font></span></a>, <a class="txt" href="#E104:117"><span class="lab"><font color="Green" title="E152">A64</font></span></a>, <a class="txt" href="#E91:117"><span class="lab"><font color="Green" title="E140">A52</font></span></a>, <a class="ref" href="chord.html#T46" target="_self" title="CHORD:th.46">Th46</a></span>;<br/>
<a NAME="E111:117"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c40">Pbg</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E108:117"><span class="lab"><font color="Green" title="E156">A83</font></span></a>, <a class="txt" href="#E91:117"><span class="lab"><font color="Green" title="E140">A52</font></span></a>, <a class="txt" href="#E96:117"><span class="lab"><font color="Green" title="E145">A57</font></span></a>, <a class="txt" href="#E84:117"><span class="lab"><font color="Green" title="E137">A49</font></span></a>, <a class="ref" href="chord.html#T28" target="_self" title="CHORD:th.28">Th28</a></span>;<br/>
<a NAME="E112:117"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E110:117"><span class="lab"><font color="Green" title="E158">A85</font></span></a>, <a class="txt" href="#E102:117"><span class="lab"><font color="Green" title="E151">A63</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E113:117"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E114:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E159">A86</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E115:117"/><span class="lab"><font color="Green" title="E160">A87</font></span>: 
 not <font color="Maroon" title="c39">Pag</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:117"><span class="lab"><font color="Green" title="E100">A10</font></span></a>, <a class="txt" href="#E106:117"><span class="lab"><font color="Green" title="E154">A81</font></span></a>, <a class="txt" href="#E108:117"><span class="lab"><font color="Green" title="E156">A83</font></span></a>, <a class="txt" href="#E104:117"><span class="lab"><font color="Green" title="E152">A64</font></span></a>, <a class="txt" href="#E91:117"><span class="lab"><font color="Green" title="E140">A52</font></span></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></span>;<br/>
<a NAME="E116:117"/><span class="lab"><font color="Green" title="E161">A88</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E108:117"><span class="lab"><font color="Green" title="E156">A83</font></span></a>, <a class="txt" href="#E91:117"><span class="lab"><font color="Green" title="E140">A52</font></span></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></span>;<br/>
<a NAME="E117:117"/>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c40">Pbg</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c42">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c42">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> or <font color="Maroon" title="c42">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c40">Pbg</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:117_7"/>
<font color="Maroon" title="c42">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c42">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c40">Pbg</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span></span><br/>

<a NAME="E2:117_7"/><span class="kw">then </span>
( <font color="Maroon" title="c42">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> or <font color="Maroon" title="c42">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:117_7"/>
<font color="Maroon" title="c42">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c40">Pbg</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E107:117"><span class="lab"><font color="Green" title="E155">A82</font></span></a>, <a class="txt" href="#E116:117"><span class="lab"><font color="Green" title="E161">A88</font></span></a>, <a class="txt" href="#E95:117"><span class="lab"><font color="Green" title="E144">A56</font></span></a>, <a class="txt" href="#E97:117"><span class="lab"><font color="Green" title="E146">A58</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" 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="E118:117"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c40">Pbg</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E89:117"><span class="lab"><font color="Green" title="E139">A51</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E119:117"/><span class="kw">then </span>
<font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E106:117"><span class="lab"><font color="Green" title="E154">A81</font></span></a>, <a class="txt" href="#E108:117"><span class="lab"><font color="Green" title="E156">A83</font></span></a>, <a class="txt" href="#E104:117"><span class="lab"><font color="Green" title="E152">A64</font></span></a>, <a class="txt" href="#E91:117"><span class="lab"><font color="Green" title="E140">A52</font></span></a>, <a class="txt" href="#E115:117"><span class="lab"><font color="Green" title="E160">A87</font></span></a>, <a class="txt" href="#E96:117"><span class="lab"><font color="Green" title="E145">A57</font></span></a>, <a class="txt" href="#E94:117"><span class="lab"><font color="Green" title="E143">A55</font></span></a>, <a class="txt" href="#E84:117"><span class="lab"><font color="Green" title="E137">A49</font></span></a>, <a class="txt" href="#E93:117"><span class="lab"><font color="Green" title="E142">A54</font></span></a>, <a class="txt" href="#E99:117"><span class="lab"><font color="Green" title="E148">A60</font></span></a>, <a class="txt" href="#E98:117"><span class="lab"><font color="Green" title="E147">A59</font></span></a>, <a class="txt" href="#E105:117"><span class="lab"><font color="Green" title="E153">A65</font></span></a>, <a class="ref" href="chord.html#T27" target="_self" title="CHORD:th.27">Th27</a></span>;<br/>
<a NAME="E120:117"/><span class="kw">then </span>
<font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font> is  <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E114:117"><span class="lab"><font color="Green" title="E159">A86</font></span></a>, <a class="ref" href="chord.html#D11" target="_self" title="CHORD:def.11">Def11</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c42">m</font>, <font color="Maroon" title="c43">n</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E122:117"/><span class="lab"><font color="Green" title="E162">A89</font></span>: 
<font color="Maroon" title="c42">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c43">n</font>
 <span class="kw">and </span><br/><a NAME="E123:117"/><span class="lab"><font color="Green" title="E163">A90</font></span>: 
<font color="Maroon" title="c43">n</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> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E124:117"/><span class="lab"><font color="Green" title="E164">A91</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">n</font>
 <span class="kw">and </span><br/><a NAME="E125:117"/><span class="lab"><font color="Green" title="E165">A92</font></span>: 
 ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">n</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">and </span><br/><a NAME="E126:117"/><span class="lab"><font color="Green" title="E166">A93</font></span>: 
 for <font color="Olive" title="b1">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  holds <br/> not <font color="Olive" title="b1">f</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">n</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#D10" target="_self" title="CHORD:def.10">Def10</a></span>;<br/>
<a NAME="E127:117"/><span class="lab"><font color="Green" title="E167">A94</font></span>: 
<font color="Maroon" title="c42">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c43">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E122:117"><span class="lab"><font color="Green" title="E162">A89</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c44">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E129:117"/><span class="lab"><font color="Green" title="E168">A95</font></span>: 
<font color="Maroon" title="c44">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">n</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E125:117"><span class="lab"><font color="Green" title="E165">A92</font></span></a></span>;<br/>
<a NAME="E130:117"/><span class="lab"><font color="Green" title="E169">A96</font></span>: 
<font color="Maroon" title="c44">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font>,<font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E129:117"><span class="lab"><font color="Green" title="E168">A95</font></span></a>, <a class="ref" href="glib_000.html#T14" title="GLIB_000:th.14">GLIB_000:14</a></span>;<br/>
<a NAME="E131:117"/><span class="lab"><font color="Green" title="E170">A97</font></span>: 
<font color="Maroon" title="c42">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E132:117"/><span class="lab"><font color="Green" title="E171">A98</font></span>: 
 not <font color="Maroon" title="c39">Pag</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:117"><span class="lab"><font color="Green" title="E100">A10</font></span></a>, <a class="txt" href="#E106:117"><span class="lab"><font color="Green" title="E154">A81</font></span></a>, <a class="txt" href="#E108:117"><span class="lab"><font color="Green" title="E156">A83</font></span></a>, <a class="txt" href="#E104:117"><span class="lab"><font color="Green" title="E152">A64</font></span></a>, <a class="txt" href="#E91:117"><span class="lab"><font color="Green" title="E140">A52</font></span></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></span>;<br/>
<a NAME="E133:117"/><span class="lab"><font color="Green" title="E172">A99</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/>
<a NAME="E134:117"/><span class="lab"><font color="Green" title="E173">A100</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c42">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_8"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:117_8"/>
( ( <font color="Maroon" title="c42">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font> &amp; <font color="Maroon" title="c43">n</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="c39">Pag</font> ) or ( <font color="Maroon" title="c42">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c43">n</font> ) or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c42">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_8_1"><span class="kw">suppose </span></a><a NAME="E1:117_8_1"/><span class="lab"><font color="Green" title="E174">A101</font></span>: 
( <font color="Maroon" title="c42">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font> &amp; <font color="Maroon" title="c43">n</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="c39">Pag</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:117_8_1"/><span class="kw">then </span>
<font color="Maroon" title="c43">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c39">Pag</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E133:117"><span class="lab"><font color="Green" title="E172">A99</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E3:117_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E175">A102</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T32" title="GLIB_001:th.32">GLIB_001:32</a></span>;<br/><a NAME="E4:117_8_1"/>
<font color="Maroon" title="c42">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c39">Pag</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E134:117"><span class="lab"><font color="Green" title="E173">A100</font></span></a>, <a class="txt" href="#E1:117_8_1"><span class="lab"><font color="Green" title="E174">A101</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E5:117_8_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T32" title="GLIB_001:th.32">GLIB_001:32</a></span>;<br/><a NAME="E6:117_8_1"/><span class="kw">then </span>
<font color="Maroon" title="c39">Pag</font> is  <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E132:117"><span class="lab"><font color="Green" title="E171">A98</font></span></a>, <a class="txt" href="#E122:117"><span class="lab"><font color="Green" title="E162">A89</font></span></a>, <a class="txt" href="#E125:117"><span class="lab"><font color="Green" title="E165">A92</font></span></a>, <a class="txt" href="#E1:117_8_1"><span class="lab"><font color="Green" title="E174">A101</font></span></a>, <a class="txt" href="#E3:117_8_1"><span class="lab"><font color="Green" title="E175">A102</font></span></a>, <a class="ref" href="chord.html#T85" target="_self" title="CHORD:th.85">Th85</a></span>;<br/><a NAME="E7:117_8_1"/><span class="kw">then </span>
<font color="Maroon" title="c38">Pa</font> is  <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:117"><span class="lab"><font color="Green" title="E124">A36</font></span></a>, <a class="ref" href="chord.html#T87" target="_self" title="CHORD:th.87">Th87</a></span>;<br/><span class="kw">hence </span><a NAME="E8:117_8_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E81:117"><span class="lab"><font color="Green" title="E136">A48</font></span></a>, <a class="ref" href="chord.html#T89" target="_self" title="CHORD:th.89">Th89</a></span>; <a class="txt" 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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_8_2"><span class="kw">suppose </span></a><a NAME="E1:117_8_2"/><span class="lab"><font color="Green" title="E174">A103</font></span>: 
( <font color="Maroon" title="c42">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c43">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:117_8_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E175">A104</font></span>: 
<font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E131:117"><span class="lab"><font color="Green" title="E170">A97</font></span></a>, <a class="ref" href="glib_001.html#T7" title="GLIB_001:th.7">GLIB_001:7</a></span>;<br/><a NAME="E3:117_8_2"/>
<font color="Maroon" title="c42">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c39">Pag</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E134:117"><span class="lab"><font color="Green" title="E173">A100</font></span></a>, <a class="txt" href="#E1:117_8_2"><span class="lab"><font color="Green" title="E174">A103</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E4:117_8_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E176">A105</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T32" title="GLIB_001:th.32">GLIB_001:32</a></span>;<br/><a NAME="E5:117_8_2"/><span class="lab"><font color="Green" title="E177">A106</font></span>: 
 not <font color="Maroon" title="c43">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c39">Pag</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117_8_2"><span class="lab"><font color="Green" title="E174">A103</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E6:117_8_2"/>
<font color="Maroon" title="c43">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E123:117"><span class="lab"><font color="Green" title="E163">A90</font></span></a>, <a class="txt" href="#E133:117"><span class="lab"><font color="Green" title="E172">A99</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c45">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E8:117_8_2"/><span class="lab"><font color="Green" title="E178">A107</font></span>: 
<font color="Maroon" title="c45">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">Pbg</font>
 <span class="kw">and </span><br/><a NAME="E9:117_8_2"/><span class="lab"><font color="Green" title="E179">A108</font></span>: 
<font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c45">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:117_8_2"><span class="lab"><font color="Green" title="E177">A106</font></span></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">GLIB_001:34</a></span>;<br/><a NAME="E10:117_8_2"/><span class="lab"><font color="Green" title="E180">A109</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c45">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E108:117"><span class="lab"><font color="Green" title="E156">A83</font></span></a>, <a class="txt" href="#E91:117"><span class="lab"><font color="Green" title="E140">A52</font></span></a>, <a class="txt" href="#E96:117"><span class="lab"><font color="Green" title="E145">A57</font></span></a>, <a class="txt" href="#E84:117"><span class="lab"><font color="Green" title="E137">A49</font></span></a>, <a class="txt" href="#E8:117_8_2"><span class="lab"><font color="Green" title="E178">A107</font></span></a>, <a class="ref" href="glib_001.html#T33" title="GLIB_001:th.33">GLIB_001:33</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c46">n1</font> = <font color="Maroon" title="c45">n1</font> as   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:117_8_2"><span class="lab"><font color="Green" title="E179">A108</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c47">n11</font> = <font color="Maroon" title="c46">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E13:117_8_2"/><span class="lab"><font color="Green" title="E181">A110</font></span>: 
<font color="Maroon" title="c47">n11</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="c40">Pbg</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:117_8_2"><span class="lab"><font color="Green" title="E178">A107</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E14:117_8_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E182">A111</font></span>: 
<font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">n11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T7" title="GLIB_001:th.7">GLIB_001:7</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_8_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:117_8_2_1"/>
( ( <font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font> &amp; <font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">n11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) or ( <font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font> &amp; <font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">n11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> ) or ( <font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">n11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> ) or ( <font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">n11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:117"><span class="lab"><font color="Green" title="E125">A37</font></span></a>, <a class="txt" href="#E45:117"><span class="lab"><font color="Green" title="E120">A32</font></span></a>, <a class="txt" href="#E2:117_8_2"><span class="lab"><font color="Green" title="E175">A104</font></span></a>, <a class="txt" href="#E14:117_8_2"><span class="lab"><font color="Green" title="E182">A111</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_8_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:117_8_2_1_1"/><span class="lab"><font color="Green" title="E183">A112</font></span>: 
( <font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font> &amp; <font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">n11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_8_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:117_8_2_1_1_1"/>
( <font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">n11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 or <font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">n11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">Pag</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> <font color="Maroon" title="c39">Pag</font></span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E104:117"><span class="lab"><font color="Green" title="E152">A64</font></span></a>, <a class="txt" href="#E91:117"><span class="lab"><font color="Green" title="E140">A52</font></span></a>, <a class="txt" href="#E1:117_8_2_1_1"><span class="lab"><font color="Green" title="E183">A112</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_8_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:117_8_2_1_1_1_1"/><span class="lab"><font color="Green" title="E184">A113</font></span>: 
<font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">n11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_8_2_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:117_8_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E185">A114</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c42">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_8_2_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:117_8_2_1_1_1_1_1_1"/>
( 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c42">m</font> or 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">m</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117_8_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E185">A114</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_8_2_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:117_8_2_1_1_1_1_1_1_1"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c42">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:117_8_2_1_1_1_1_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c42">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a>, <a class="ref" href="jordan12.html#T2" title="JORDAN12:th.2">JORDAN12:2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:117_8_2_1_1_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E124:117"><span class="lab"><font color="Green" title="E164">A91</font></span></a>, <a class="txt" href="#E134:117"><span class="lab"><font color="Green" title="E173">A100</font></span></a>, <a class="txt" href="#E4:117_8_2"><span class="lab"><font color="Green" title="E176">A105</font></span></a>, <a class="txt" href="#E9:117_8_2"><span class="lab"><font color="Green" title="E179">A108</font></span></a>, <a class="txt" href="#E10:117_8_2"><span class="lab"><font color="Green" title="E180">A109</font></span></a>, <a class="txt" href="#E1:117_8_2_1_1_1_1"><span class="lab"><font color="Green" title="E184">A113</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" 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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_8_2_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:117_8_2_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E186">A115</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:117_8_2_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E187">A116</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117_8_2"><span class="lab"><font color="Green" title="E174">A103</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:117_8_2_1_1_1_1_1_1_2"/><span class="kw">then </span>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c39">Pag</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E4:117_8_2_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E188">A117</font></span>: 
<font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T32" title="GLIB_001:th.32">GLIB_001:32</a></span>;<br/><a NAME="E5:117_8_2_1_1_1_1_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:117_8_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E187">A116</font></span></a>, <a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/><a NAME="E6:117_8_2_1_1_1_1_1_1_2"/><span class="kw">then </span>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">Pbg</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E7:117_8_2_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E189">A118</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E109:117"><span class="lab"><font color="Green" title="E157">A84</font></span></a>, <a class="ref" href="glib_001.html#T100" title="GLIB_001:th.100">GLIB_001:100</a>, <a class="ref" href="jordan12.html#T2" title="JORDAN12:th.2">JORDAN12:2</a></span>;<br/><a NAME="E8:117_8_2_1_1_1_1_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E134:117"><span class="lab"><font color="Green" title="E173">A100</font></span></a>, <a class="txt" href="#E1:117_8_2"><span class="lab"><font color="Green" title="E174">A103</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:117_8_2_1_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1,<font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117_8_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E186">A115</font></span></a>, <a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a>, <a class="ref" href="jordan12.html#T2" title="JORDAN12:th.2">JORDAN12:2</a></span>;<br/><span class="kw">hence </span><a NAME="E10:117_8_2_1_1_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E126:117"><span class="lab"><font color="Green" title="E166">A93</font></span></a>, <a class="txt" href="#E4:117_8_2"><span class="lab"><font color="Green" title="E176">A105</font></span></a>, <a class="txt" href="#E9:117_8_2"><span class="lab"><font color="Green" title="E179">A108</font></span></a>, <a class="txt" href="#E10:117_8_2"><span class="lab"><font color="Green" title="E180">A109</font></span></a>, <a class="txt" href="#E1:117_8_2_1_1_1_1"><span class="lab"><font color="Green" title="E184">A113</font></span></a>, <a class="txt" href="#E4:117_8_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E188">A117</font></span></a>, <a class="txt" href="#E7:117_8_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E189">A118</font></span></a>, <a class="ref" href="glib_000.html#T14" title="GLIB_000:th.14">GLIB_000:14</a></span>; <a class="txt" 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">end;</span></div></div><span class="kw">end;</span></div><a NAME="E3:117_8_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c39">Pag</font> is  <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E132:117"><span class="lab"><font color="Green" title="E171">A98</font></span></a>, <a class="txt" href="#E130:117"><span class="lab"><font color="Green" title="E169">A96</font></span></a>, <a class="txt" href="#E1:117_8_2"><span class="lab"><font color="Green" title="E174">A103</font></span></a>, <a class="txt" href="#E4:117_8_2"><span class="lab"><font color="Green" title="E176">A105</font></span></a>, <a class="txt" href="#E9:117_8_2"><span class="lab"><font color="Green" title="E179">A108</font></span></a>, <a class="txt" href="#E10:117_8_2"><span class="lab"><font color="Green" title="E180">A109</font></span></a>, <a class="txt" href="#E1:117_8_2_1_1_1_1"><span class="lab"><font color="Green" title="E184">A113</font></span></a>, <a class="ref" href="chord.html#T85" target="_self" title="CHORD:th.85">Th85</a>, <a class="ref" href="jordan12.html#T2" title="JORDAN12:th.2">JORDAN12:2</a></span>;<br/><a NAME="E4:117_8_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c38">Pa</font> is  <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:117"><span class="lab"><font color="Green" title="E124">A36</font></span></a>, <a class="ref" href="chord.html#T87" target="_self" title="CHORD:th.87">Th87</a></span>;<br/><span class="kw">hence </span><a NAME="E5:117_8_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E81:117"><span class="lab"><font color="Green" title="E136">A48</font></span></a>, <a class="ref" href="chord.html#T89" target="_self" title="CHORD:th.89">Th89</a></span>; <a class="txt" 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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_8_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:117_8_2_1_1_1_2"/><span class="lab"><font color="Green" title="E184">A119</font></span>: 
<font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">n11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">Pag</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> <font color="Maroon" title="c39">Pag</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_8_2_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c48">L</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font>;<br/><span class="kw">assume </span><a NAME="E1:117_8_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E185">A120</font></span>: 
<font color="Maroon" title="c42">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_8_2_1_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:117_8_2_1_1_1_2_1_1"/>
( <font color="Maroon" title="c42">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <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="c39">Pag</font> or <font color="Maroon" title="c42">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117_8_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E185">A120</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_8_2_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:117_8_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E186">A121</font></span>: 
<font color="Maroon" title="c42">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <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="c39">Pag</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:117_8_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E187">A122</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E3:117_8_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E188">A123</font></span>: 
<font color="Maroon" title="c42">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:117_8_2_1_1_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c42">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E5:117_8_2_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c42">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c39">Pag</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:117_8_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E188">A123</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E6:117_8_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E189">A124</font></span>: 
<font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T32" title="GLIB_001:th.32">GLIB_001:32</a></span>;<br/><a NAME="E7:117_8_2_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c42">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:117_8_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E187">A122</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E8:117_8_2_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c42">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">Pbg</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E9:117_8_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E190">A125</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E109:117"><span class="lab"><font color="Green" title="E157">A84</font></span></a>, <a class="ref" href="glib_001.html#T100" title="GLIB_001:th.100">GLIB_001:100</a></span>;<br/><a NAME="E10:117_8_2_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c42">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:117_8_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E188">A123</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E11:117_8_2_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font>,<font color="Maroon" title="c39">Pag</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> <font color="Maroon" title="c39">Pag</font></span>)</span>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E131:117"><span class="lab"><font color="Green" title="E170">A97</font></span></a>, <a class="txt" href="#E1:117_8_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E186">A121</font></span></a>, <a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E12:117_8_2_1_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E126:117"><span class="lab"><font color="Green" title="E166">A93</font></span></a>, <a class="txt" href="#E4:117_8_2"><span class="lab"><font color="Green" title="E176">A105</font></span></a>, <a class="txt" href="#E9:117_8_2"><span class="lab"><font color="Green" title="E179">A108</font></span></a>, <a class="txt" href="#E10:117_8_2"><span class="lab"><font color="Green" title="E180">A109</font></span></a>, <a class="txt" href="#E1:117_8_2_1_1_1_2"><span class="lab"><font color="Green" title="E184">A119</font></span></a>, <a class="txt" href="#E6:117_8_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E189">A124</font></span></a>, <a class="txt" href="#E9:117_8_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E190">A125</font></span></a></span>; <a class="txt" 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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_8_2_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:117_8_2_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c42">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:117_8_2_1_1_1_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117_8_2"><span class="lab"><font color="Green" title="E174">A103</font></span></a>, <a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></span>; <a class="txt" 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">end;</span></div></div><span class="kw">end;</span></div><a NAME="E3:117_8_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c39">Pag</font> is  <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E132:117"><span class="lab"><font color="Green" title="E171">A98</font></span></a>, <a class="txt" href="#E125:117"><span class="lab"><font color="Green" title="E165">A92</font></span></a>, <a class="txt" href="#E4:117_8_2"><span class="lab"><font color="Green" title="E176">A105</font></span></a>, <a class="txt" href="#E9:117_8_2"><span class="lab"><font color="Green" title="E179">A108</font></span></a>, <a class="txt" href="#E10:117_8_2"><span class="lab"><font color="Green" title="E180">A109</font></span></a>, <a class="txt" href="#E1:117_8_2_1_1_1_2"><span class="lab"><font color="Green" title="E184">A119</font></span></a>, <a class="ref" href="chord.html#T85" target="_self" title="CHORD:th.85">Th85</a></span>;<br/><a NAME="E4:117_8_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c38">Pa</font> is  <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:117"><span class="lab"><font color="Green" title="E124">A36</font></span></a>, <a class="ref" href="chord.html#T87" target="_self" title="CHORD:th.87">Th87</a></span>;<br/><span class="kw">hence </span><a NAME="E5:117_8_2_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E81:117"><span class="lab"><font color="Green" title="E136">A48</font></span></a>, <a class="ref" href="chord.html#T89" target="_self" title="CHORD:th.89">Th89</a></span>; <a class="txt" 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">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_8_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:117_8_2_1_2"/><span class="lab"><font color="Green" title="E183">A126</font></span>: 
( <font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font> &amp; <font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">n11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c48">bc</font> = <font color="Maroon" title="c37">Pb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">n11</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c29">Gb</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c49">ac</font> = <font color="Maroon" title="c38">Pa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c28">Ga</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117_8_2_1_2"><span class="lab"><font color="Green" title="E183">A126</font></span></a>, <a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c50">WAB</font> =  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c49">ac</font>,<font color="Maroon" title="c44">e</font>,<font color="Maroon" title="c48">bc</font>);<br/><a NAME="E4:117_8_2_1_2"/>
<font color="Maroon" title="c44">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c49">ac</font>,<font color="Maroon" title="c48">bc</font>, the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:117"><span class="lab"><font color="Green" title="E105">A16</font></span></a>, <a class="txt" href="#E31:117"><span class="lab"><font color="Green" title="E112">A24</font></span></a>, <a class="txt" href="#E129:117"><span class="lab"><font color="Green" title="E168">A95</font></span></a>, <a class="txt" href="#E4:117_8_2"><span class="lab"><font color="Green" title="E176">A105</font></span></a>, <a class="txt" href="#E9:117_8_2"><span class="lab"><font color="Green" title="E179">A108</font></span></a>, <a class="txt" href="#E10:117_8_2"><span class="lab"><font color="Green" title="E180">A109</font></span></a>, <a class="txt" href="#E1:117_8_2_1_2"><span class="lab"><font color="Green" title="E183">A126</font></span></a>, <a class="ref" href="chord.html#T19" target="_self" title="CHORD:th.19">Th19</a></span>;<br/><a NAME="E5:117_8_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E184">A127</font></span>: 
 the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c49">ac</font>,<font color="Maroon" title="c44">e</font>,<font color="Maroon" title="c48">bc</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c49">ac</font>,<font color="Maroon" title="c48">bc</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T15" title="GLIB_001:th.15">GLIB_001:15</a></span>;<br/><a NAME="E6:117_8_2_1_2"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#T9" title="GLIB_002:th.9">GLIB_002:9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c51">bb</font> = <font color="Maroon" title="c4">b</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c29">Gb</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/><a NAME="E8:117_8_2_1_2"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#T9" title="GLIB_002:th.9">GLIB_002:9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c52">aa</font> = <font color="Maroon" title="c3">a</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c28">Ga</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c53">WA</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c28">Ga</font><span class="kw"> such that </span><br/><a NAME="E11:117_8_2_1_2"/><span class="lab"><font color="Green" title="E185">A128</font></span>: 
<font color="Maroon" title="c53">WA</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c52">aa</font>,<font color="Maroon" title="c49">ac</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D1" title="GLIB_002:def.1">GLIB_002:def 1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c54">WB</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c29">Gb</font><span class="kw"> such that </span><br/><a NAME="E13:117_8_2_1_2"/><span class="lab"><font color="Green" title="E186">A129</font></span>: 
<font color="Maroon" title="c54">WB</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c48">bc</font>,<font color="Maroon" title="c51">bb</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D1" title="GLIB_002:def.1">GLIB_002:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c55">WA</font> = <font color="Maroon" title="c53">WA</font>, <font color="Maroon" title="c56">WB</font> = <font color="Maroon" title="c54">WB</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T167" title="GLIB_001:th.167">GLIB_001:167</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c57">WAs</font> = <font color="Maroon" title="c55">WA</font>, <font color="Maroon" title="c58">WBs</font> = <font color="Maroon" title="c56">WB</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> ;<br/><a NAME="E16:117_8_2_1_2"/><span class="lab"><font color="Green" title="E187">A130</font></span>: 
<font color="Maroon" title="c58">WBs</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c48">bc</font>,<font color="Maroon" title="c51">bb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:117_8_2_1_2"><span class="lab"><font color="Green" title="E186">A129</font></span></a>, <a class="ref" href="glib_001.html#T19" title="GLIB_001:th.19">GLIB_001:19</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c59">WaB</font> = <font color="Maroon" title="c57">WAs</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c49">ac</font>,<font color="Maroon" title="c44">e</font>,<font color="Maroon" title="c48">bc</font>)</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c60">Wab</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c57">WAs</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c49">ac</font>,<font color="Maroon" title="c44">e</font>,<font color="Maroon" title="c48">bc</font>)</span>)</span></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c58">WBs</font>;<br/><a NAME="E17:117_8_2_1_2"/>
<font color="Maroon" title="c57">WAs</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c52">aa</font>,<font color="Maroon" title="c49">ac</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:117_8_2_1_2"><span class="lab"><font color="Green" title="E185">A128</font></span></a>, <a class="ref" href="glib_001.html#T19" title="GLIB_001:th.19">GLIB_001:19</a></span>;<br/><a NAME="E18:117_8_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c57">WAs</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c49">ac</font>,<font color="Maroon" title="c44">e</font>,<font color="Maroon" title="c48">bc</font>)</span>)</span> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c52">aa</font>,<font color="Maroon" title="c48">bc</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:117_8_2_1_2"><span class="lab"><font color="Green" title="E184">A127</font></span></a>, <a class="ref" href="glib_001.html#T31" title="GLIB_001:th.31">GLIB_001:31</a></span>;<br/><a NAME="E19:117_8_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c57">WAs</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p2">(<span class="default"> the   <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c5">S</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c49">ac</font>,<font color="Maroon" title="c44">e</font>,<font color="Maroon" title="c48">bc</font>)</span>)</span></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c58">WBs</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:117_8_2_1_2"><span class="lab"><font color="Green" title="E187">A130</font></span></a>, <a class="ref" href="glib_001.html#T31" title="GLIB_001:th.31">GLIB_001:31</a></span>;<br/><span class="kw">hence </span><a NAME="E20:117_8_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E91">A1</font></span></a>, <a class="txt" href="#E2:117"><span class="lab"><font color="Green" title="E92">A2</font></span></a>, <a class="ref" href="chord.html#D8" target="_self" title="CHORD:def.8">Def8</a></span>; <a class="txt" 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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_8_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:117_8_2_1_3"/>
( <font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">n11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:117_8_2_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> or <font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E183">A131</font></span>: <font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font> = 
<font color="Maroon" title="c16">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E104:117"><span class="lab"><font color="Green" title="E152">A64</font></span></a>, <a class="txt" href="#E91:117"><span class="lab"><font color="Green" title="E140">A52</font></span></a>, <a class="txt" href="#E131:117"><span class="lab"><font color="Green" title="E170">A97</font></span></a>, <a class="txt" href="#E1:117_8_2"><span class="lab"><font color="Green" title="E174">A103</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>
<br/>.= 
<font color="Maroon" title="c40">Pbg</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> <font color="Maroon" title="c40">Pbg</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:117"><span class="lab"><font color="Green" title="E131">A43</font></span></a>, <a class="txt" href="#E77:117"><span class="lab"><font color="Green" title="E133">A45</font></span></a>, <a class="txt" href="#E92:117"><span class="lab"><font color="Green" title="E141">A53</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_8_2_1_3_2"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c48">L</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">Pbg</font>;<br/><span class="kw">assume </span><a NAME="E1:117_8_2_1_3_2"/><span class="lab"><font color="Green" title="E184">A132</font></span>: 
<font color="Maroon" title="c47">n11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">Pbg</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_8_2_1_3_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:117_8_2_1_3_2_1"/>
( <font color="Maroon" title="c47">n11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <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="c40">Pbg</font> or <font color="Maroon" title="c47">n11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">Pbg</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117_8_2_1_3_2"><span class="lab"><font color="Green" title="E184">A132</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_8_2_1_3_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:117_8_2_1_3_2_1_1"/><span class="lab"><font color="Green" title="E185">A133</font></span>: 
<font color="Maroon" title="c47">n11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <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="c40">Pbg</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:117_8_2_1_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E186">A134</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">Pbg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">n11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E3:117_8_2_1_3_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c47">n11</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:117_8_2_1_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E187">A135</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c47">n11</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E5:117_8_2_1_3_2_1_1"/>
<font color="Maroon" title="c47">n11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">Pbg</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:117_8_2_1_3_2_1_1"><span class="lab"><font color="Green" title="E186">A134</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:117_8_2_1_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E188">A136</font></span>: 
<font color="Maroon" title="c47">n11</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">Pbg</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E7:117_8_2_1_3_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">n11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">n11</font>,<font color="Maroon" title="c40">Pbg</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> <font color="Maroon" title="c40">Pbg</font></span>)</span>, the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117_8_2_1_3_2_1_1"><span class="lab"><font color="Green" title="E185">A133</font></span></a>, <a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></span>;<br/><a NAME="E8:117_8_2_1_3_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">n11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c40">Pbg</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> <font color="Maroon" title="c40">Pbg</font></span>)</span>,<font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">n11</font>, the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T14" title="GLIB_000:th.14">GLIB_000:14</a></span>;<br/><a NAME="E9:117_8_2_1_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E189">A137</font></span>: 
<font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">n11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c40">Pbg</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> <font color="Maroon" title="c40">Pbg</font></span>)</span>,<font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">n11</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T72" title="GLIB_000:th.72">GLIB_000:72</a></span>;<br/><a NAME="E10:117_8_2_1_3_2_1_1"/><span class="lab"><font color="Green" title="E190">A138</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c47">n11</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E11:117_8_2_1_3_2_1_1"/>
<font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">n11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c47">n11</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E108:117"><span class="lab"><font color="Green" title="E156">A83</font></span></a>, <a class="txt" href="#E91:117"><span class="lab"><font color="Green" title="E140">A52</font></span></a>, <a class="txt" href="#E96:117"><span class="lab"><font color="Green" title="E145">A57</font></span></a>, <a class="txt" href="#E84:117"><span class="lab"><font color="Green" title="E137">A49</font></span></a>, <a class="txt" href="#E6:117_8_2_1_3_2_1_1"><span class="lab"><font color="Green" title="E188">A136</font></span></a>, <a class="ref" href="glib_001.html#T33" title="GLIB_001:th.33">GLIB_001:33</a></span>;<br/><a NAME="E12:117_8_2_1_3_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">n11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E109:117"><span class="lab"><font color="Green" title="E157">A84</font></span></a>, <a class="txt" href="#E10:117_8_2_1_3_2_1_1"><span class="lab"><font color="Green" title="E190">A138</font></span></a>, <a class="txt" href="#E4:117_8_2_1_3_2_1_1"><span class="lab"><font color="Green" title="E187">A135</font></span></a>, <a class="ref" href="glib_001.html#T99" title="GLIB_001:th.99">GLIB_001:99</a></span>;<br/><span class="kw">hence </span><a NAME="E13:117_8_2_1_3_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E126:117"><span class="lab"><font color="Green" title="E166">A93</font></span></a>, <a class="txt" href="#E4:117_8_2"><span class="lab"><font color="Green" title="E176">A105</font></span></a>, <a class="txt" href="#E9:117_8_2"><span class="lab"><font color="Green" title="E179">A108</font></span></a>, <a class="txt" href="#E10:117_8_2"><span class="lab"><font color="Green" title="E180">A109</font></span></a>, <a class="txt" href="#E3:117_8_2_1_3"><span class="lab"><font color="Green" title="E183">A131</font></span></a>, <a class="txt" href="#E9:117_8_2_1_3_2_1_1"><span class="lab"><font color="Green" title="E189">A137</font></span></a></span>; <a class="txt" 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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_8_2_1_3_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:117_8_2_1_3_2_1_2"/>
<font color="Maroon" title="c47">n11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">Pbg</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:117_8_2_1_3_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c47">n11</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">Pbg</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:117_8_2_1_3_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E124:117"><span class="lab"><font color="Green" title="E164">A91</font></span></a>, <a class="txt" href="#E4:117_8_2"><span class="lab"><font color="Green" title="E176">A105</font></span></a>, <a class="txt" href="#E9:117_8_2"><span class="lab"><font color="Green" title="E179">A108</font></span></a>, <a class="txt" href="#E10:117_8_2"><span class="lab"><font color="Green" title="E180">A109</font></span></a>, <a class="txt" href="#E13:117_8_2"><span class="lab"><font color="Green" title="E181">A110</font></span></a>, <a class="txt" href="#E3:117_8_2_1_3"><span class="lab"><font color="Green" title="E183">A131</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" 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">end;</span></div></div><span class="kw">end;</span></div><a NAME="E5:117_8_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c40">Pbg</font> is  <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E100:117"><span class="lab"><font color="Green" title="E149">A61</font></span></a>, <a class="txt" href="#E130:117"><span class="lab"><font color="Green" title="E169">A96</font></span></a>, <a class="txt" href="#E4:117_8_2"><span class="lab"><font color="Green" title="E176">A105</font></span></a>, <a class="txt" href="#E9:117_8_2"><span class="lab"><font color="Green" title="E179">A108</font></span></a>, <a class="txt" href="#E10:117_8_2"><span class="lab"><font color="Green" title="E180">A109</font></span></a>, <a class="txt" href="#E3:117_8_2_1_3"><span class="lab"><font color="Green" title="E183">A131</font></span></a>, <a class="ref" href="chord.html#T85" target="_self" title="CHORD:th.85">Th85</a></span>;<br/><a NAME="E6:117_8_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c37">Pb</font> is  <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:117"><span class="lab"><font color="Green" title="E119">A31</font></span></a>, <a class="ref" href="chord.html#T87" target="_self" title="CHORD:th.87">Th87</a></span>;<br/><span class="kw">hence </span><a NAME="E7:117_8_2_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:117"><span class="lab"><font color="Green" title="E134">A46</font></span></a>, <a class="ref" href="chord.html#T89" target="_self" title="CHORD:th.89">Th89</a></span>; <a class="txt" 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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_8_2_1_4"><span class="kw">suppose </span></a><a NAME="E1:117_8_2_1_4"/><span class="lab"><font color="Green" title="E183">A139</font></span>: 
( <font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">n11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:117_8_2_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E184">A140</font></span>: 
( <font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">n11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> or <font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">n11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E3:117_8_2_1_4"/>
( <font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> or <font color="Maroon" title="c39">Pag</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117_8_2_1_4"><span class="lab"><font color="Green" title="E183">A139</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E4:117_8_2_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c18">xg</font>,<font color="Maroon" title="c19">yg</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="#E124:117"><span class="lab"><font color="Green" title="E164">A91</font></span></a>, <a class="txt" href="#E125:117"><span class="lab"><font color="Green" title="E165">A92</font></span></a>, <a class="txt" href="#E4:117_8_2"><span class="lab"><font color="Green" title="E176">A105</font></span></a>, <a class="txt" href="#E9:117_8_2"><span class="lab"><font color="Green" title="E179">A108</font></span></a>, <a class="txt" href="#E10:117_8_2"><span class="lab"><font color="Green" title="E180">A109</font></span></a>, <a class="txt" href="#E2:117_8_2_1_4"><span class="lab"><font color="Green" title="E184">A140</font></span></a>, <a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E5:117_8_2_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:117"><span class="lab"><font color="Green" title="E94">A4</font></span></a>, <a class="txt" href="#E14:117"><span class="lab"><font color="Green" title="E101">A11</font></span></a>, <a class="ref" href="chord.html#T45" target="_self" title="CHORD:th.45">Th45</a></span>; <a class="txt" 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">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_8_3"><span class="kw">suppose </span></a><a NAME="E1:117_8_3"/><span class="lab"><font color="Green" title="E174">A141</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c42">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c45">m1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:117_8_3"/><span class="lab"><font color="Green" title="E175">A142</font></span>: 
<font color="Maroon" title="c42">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c45">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><a NAME="E4:117_8_3"/>
<font color="Maroon" title="c43">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E127:117"><span class="lab"><font color="Green" title="E167">A94</font></span></a>, <a class="txt" href="#E1:117_8_3"><span class="lab"><font color="Green" title="E174">A141</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E5:117_8_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E176">A143</font></span>: 
 not <font color="Maroon" title="c43">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c39">Pag</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E6:117_8_3"/>
<font color="Maroon" title="c43">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E123:117"><span class="lab"><font color="Green" title="E163">A90</font></span></a>, <a class="txt" href="#E133:117"><span class="lab"><font color="Green" title="E172">A99</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c46">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E8:117_8_3"/><span class="lab"><font color="Green" title="E177">A144</font></span>: 
<font color="Maroon" title="c46">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">Pbg</font>
 <span class="kw">and </span><br/><a NAME="E9:117_8_3"/><span class="lab"><font color="Green" title="E178">A145</font></span>: 
<font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c46">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:117_8_3"><span class="lab"><font color="Green" title="E176">A143</font></span></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">GLIB_001:34</a></span>;<br/><a NAME="E10:117_8_3"/><span class="lab"><font color="Green" title="E179">A146</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c46">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c46">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E108:117"><span class="lab"><font color="Green" title="E156">A83</font></span></a>, <a class="txt" href="#E91:117"><span class="lab"><font color="Green" title="E140">A52</font></span></a>, <a class="txt" href="#E96:117"><span class="lab"><font color="Green" title="E145">A57</font></span></a>, <a class="txt" href="#E84:117"><span class="lab"><font color="Green" title="E137">A49</font></span></a>, <a class="txt" href="#E8:117_8_3"><span class="lab"><font color="Green" title="E177">A144</font></span></a>, <a class="ref" href="glib_001.html#T33" title="GLIB_001:th.33">GLIB_001:33</a></span>;<br/><a NAME="E11:117_8_3"/><span class="lab"><font color="Green" title="E180">A147</font></span>: 
<font color="Maroon" title="c45">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E12:117_8_3"/><span class="lab"><font color="Green" title="E181">A148</font></span>: 
<font color="Maroon" title="c45">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c46">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E127:117"><span class="lab"><font color="Green" title="E167">A94</font></span></a>, <a class="txt" href="#E3:117_8_3"><span class="lab"><font color="Green" title="E175">A142</font></span></a>, <a class="txt" href="#E9:117_8_3"><span class="lab"><font color="Green" title="E178">A145</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E13:117_8_3"/><span class="kw">then </span>
<font color="Maroon" title="c45">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">Pbg</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:117_8_3"><span class="lab"><font color="Green" title="E177">A144</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E14:117_8_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E182">A149</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c45">m1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E108:117"><span class="lab"><font color="Green" title="E156">A83</font></span></a>, <a class="txt" href="#E91:117"><span class="lab"><font color="Green" title="E140">A52</font></span></a>, <a class="txt" href="#E96:117"><span class="lab"><font color="Green" title="E145">A57</font></span></a>, <a class="txt" href="#E84:117"><span class="lab"><font color="Green" title="E137">A49</font></span></a>, <a class="txt" href="#E11:117_8_3"><span class="lab"><font color="Green" title="E180">A147</font></span></a>, <a class="ref" href="glib_001.html#T33" title="GLIB_001:th.33">GLIB_001:33</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c47">n1</font> = <font color="Maroon" title="c46">n1</font>, <font color="Maroon" title="c48">m1</font> = <font color="Maroon" title="c45">m1</font> as   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:117_8_3"><span class="lab"><font color="Green" title="E175">A142</font></span></a>, <a class="txt" href="#E9:117_8_3"><span class="lab"><font color="Green" title="E178">A145</font></span></a>, <a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c49">m11</font> = <font color="Maroon" title="c48">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1, <font color="Maroon" title="c50">n11</font> = <font color="Maroon" title="c47">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E17:117_8_3"/><span class="lab"><font color="Green" title="E183">A150</font></span>: 
<font color="Maroon" title="c49">m11</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c50">n11</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:117_8_3"><span class="lab"><font color="Green" title="E181">A148</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E18:117_8_3"/>
<font color="Maroon" title="c50">n11</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="c40">Pbg</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:117_8_3"><span class="lab"><font color="Green" title="E177">A144</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E19:117_8_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E184">A151</font></span>: 
<font color="Maroon" title="c49">m11</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">Pbg</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:117_8_3"><span class="lab"><font color="Green" title="E183">A150</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><div><span class="lab"><font color="Green" title="E185">A152</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_8_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:117_8_3_1"/><span class="lab"><font color="Green" title="E186">A153</font></span>: 
<font color="Maroon" title="c49">m11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c50">n11</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_8_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:117_8_3_1_1"/>
( <font color="Maroon" title="c49">m11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c50">n11</font> or <font color="Maroon" title="c49">m11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c50">n11</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117_8_3_1"><span class="lab"><font color="Green" title="E186">A153</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_8_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:117_8_3_1_1_1"/><span class="lab"><font color="Green" title="E187">A154</font></span>: 
<font color="Maroon" title="c49">m11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c50">n11</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:117_8_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c49">m11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">Pbg</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:117_8_3"><span class="lab"><font color="Green" title="E177">A144</font></span></a></span>;<br/><a NAME="E3:117_8_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c49">m11</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/><a NAME="E4:117_8_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E188">A155</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c49">m11</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E5:117_8_3_1_1_1"/><span class="lab"><font color="Green" title="E189">A156</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c49">m11</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E6:117_8_3_1_1_1"/><span class="lab"><font color="Green" title="E190">A157</font></span>: 
<font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">m11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">m11</font>,<font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c50">n11</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:117_8_3"><span class="lab"><font color="Green" title="E184">A151</font></span></a>, <a class="txt" href="#E1:117_8_3_1_1_1"><span class="lab"><font color="Green" title="E187">A154</font></span></a>, <a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></span>;<br/><a NAME="E7:117_8_3_1_1_1"/>
<font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">m11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pag</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c49">m11</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E108:117"><span class="lab"><font color="Green" title="E156">A83</font></span></a>, <a class="txt" href="#E91:117"><span class="lab"><font color="Green" title="E140">A52</font></span></a>, <a class="txt" href="#E96:117"><span class="lab"><font color="Green" title="E145">A57</font></span></a>, <a class="txt" href="#E84:117"><span class="lab"><font color="Green" title="E137">A49</font></span></a>, <a class="txt" href="#E19:117_8_3"><span class="lab"><font color="Green" title="E184">A151</font></span></a>, <a class="ref" href="glib_001.html#T33" title="GLIB_001:th.33">GLIB_001:33</a></span>;<br/><a NAME="E8:117_8_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c40">Pbg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">m11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pag</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">Pbg</font></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E109:117"><span class="lab"><font color="Green" title="E157">A84</font></span></a>, <a class="txt" href="#E5:117_8_3_1_1_1"><span class="lab"><font color="Green" title="E189">A156</font></span></a>, <a class="txt" href="#E4:117_8_3_1_1_1"><span class="lab"><font color="Green" title="E188">A155</font></span></a>, <a class="ref" href="glib_001.html#T99" title="GLIB_001:th.99">GLIB_001:99</a></span>;<br/><span class="kw">hence </span><a NAME="E9:117_8_3_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E126:117"><span class="lab"><font color="Green" title="E166">A93</font></span></a>, <a class="txt" href="#E3:117_8_3"><span class="lab"><font color="Green" title="E175">A142</font></span></a>, <a class="txt" href="#E9:117_8_3"><span class="lab"><font color="Green" title="E178">A145</font></span></a>, <a class="txt" href="#E10:117_8_3"><span class="lab"><font color="Green" title="E179">A146</font></span></a>, <a class="txt" href="#E14:117_8_3"><span class="lab"><font color="Green" title="E182">A149</font></span></a>, <a class="txt" href="#E6:117_8_3_1_1_1"><span class="lab"><font color="Green" title="E190">A157</font></span></a></span>; <a class="txt" 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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_8_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:117_8_3_1_1_2"/>
<font color="Maroon" title="c49">m11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c50">n11</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:117_8_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c49">m11</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c50">n11</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:117_8_3_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:117_8_3"><span class="lab"><font color="Green" title="E181">A148</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>; <a class="txt" 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">end;</span></div></div><span class="kw">end;</span></div><a NAME="E21:117_8_3"/>
<font color="Maroon" title="c50">n11</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="c40">Pbg</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:117_8_3"><span class="lab"><font color="Green" title="E177">A144</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E22:117_8_3"/><span class="kw">then </span>
<font color="Maroon" title="c40">Pbg</font> is  <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E100:117"><span class="lab"><font color="Green" title="E149">A61</font></span></a>, <a class="txt" href="#E125:117"><span class="lab"><font color="Green" title="E165">A92</font></span></a>, <a class="txt" href="#E3:117_8_3"><span class="lab"><font color="Green" title="E175">A142</font></span></a>, <a class="txt" href="#E9:117_8_3"><span class="lab"><font color="Green" title="E178">A145</font></span></a>, <a class="txt" href="#E10:117_8_3"><span class="lab"><font color="Green" title="E179">A146</font></span></a>, <a class="txt" href="#E14:117_8_3"><span class="lab"><font color="Green" title="E182">A149</font></span></a>, <a class="txt" href="#E20:117_8_3"><span class="lab"><font color="Green" title="E185">A152</font></span></a>, <a class="ref" href="chord.html#T85" target="_self" title="CHORD:th.85">Th85</a></span>;<br/><a NAME="E23:117_8_3"/><span class="kw">then </span>
<font color="Maroon" title="c37">Pb</font> is  <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:117"><span class="lab"><font color="Green" title="E119">A31</font></span></a>, <a class="ref" href="chord.html#T87" target="_self" title="CHORD:th.87">Th87</a></span>;<br/><span class="kw">hence </span><a NAME="E24:117_8_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:117"><span class="lab"><font color="Green" title="E134">A46</font></span></a>, <a class="ref" href="chord.html#T89" target="_self" title="CHORD:th.89">Th89</a></span>; <a class="txt" 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">end;</span></div>

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