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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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">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="E1:112"/><span class="lab"><font color="Green" title="E88">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="E2:112"/><span class="lab"><font color="Green" title="E89">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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="E3:112"/><span class="lab"><font color="Green" title="E90">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="E4:112"/><span class="lab"><font color="Green" title="E91">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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="#E1:112"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E89">A2</font></span></a>, <a class="ref" href="chord.html#T77" target="_self" onmouseover="rs('chord/T77')" onmouseout="rh()">Th76</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="E6:112"/><span class="lab"><font color="Green" title="E92">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="#E1:112"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E89">A2</font></span></a>, <a class="ref" href="chord.html#T76" target="_self" onmouseover="rs('chord/T76')" onmouseout="rh()">Th75</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="E7:112"/><span class="lab"><font color="Green" title="E93">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" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/>
<a NAME="E8:112"/><span class="lab"><font color="Green" title="E94">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" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/>
<a NAME="E9:112"/><span class="lab"><font color="Green" title="E95">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="#E1:112"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E89">A2</font></span></a>, <a class="ref" href="chord.html#T75" target="_self" onmouseover="rs('chord/T75')" onmouseout="rh()">Th74</a></span>;<br/>
<a NAME="E10:112"/><span class="lab"><font color="Green" title="E96">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="#E1:112"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E89">A2</font></span></a>, <a class="ref" href="chord.html#T75" target="_self" onmouseover="rs('chord/T75')" onmouseout="rh()">Th74</a></span>;<br/>
<a NAME="E11:112"/>
 <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" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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" onmouseover="rs('chord/D6')" onmouseout="rh()">CHORD:def 6</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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="E13:112"/><span class="lab"><font color="Green" title="E97">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="E14:112"/><span class="lab"><font color="Green" title="E98">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<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" onmouseover="rs('glib_000/T42')" onmouseout="rh()">GLIB_000:42</a></span>;<br/>
<a NAME="E16:112"/><span class="lab"><font color="Green" title="E99">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="#E4:112"><span class="lab"><font color="Green" title="E91">A4</font></span></a>, <a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/>
<a NAME="E17:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">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="tarski.html#R2" title="TARSKI: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="#E1:112"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E89">A2</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E90">A3</font></span></a>, <a class="ref" href="chord.html#T82" target="_self" onmouseover="rs('chord/T82')" onmouseout="rh()">Th81</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="E18:112"/><span class="lab"><font color="Green" title="E101">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" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="112_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">A</font> holds <br/><font color="Olive" title="b1">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></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">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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="E1:112_1"/><span class="lab"><font color="Green" title="E102">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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="E2:112_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="#E10:112"><span class="lab"><font color="Green" title="E96">A9</font></span></a>, <a class="txt" href="#E1:112_1"><span class="lab"><font color="Green" title="E102">A15</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_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="#E1:112_1"><span class="lab"><font color="Green" title="E102">A15</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E20:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">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>
 
;<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="E22:112"/><span class="lab"><font color="Green" title="E103">A17</font></span>: 
<font color="Maroon" title="c25">yag</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">A</font>
 <span class="kw">and </span><br/><a NAME="E23:112"/><span class="lab"><font color="Green" title="E104">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="#E1:112"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E89">A2</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E90">A3</font></span></a>, <a class="txt" href="#E16:112"><span class="lab"><font color="Green" title="E99">A12</font></span></a>, <a class="ref" href="chord.html#T82" target="_self" onmouseover="rs('chord/T82')" onmouseout="rh()">Th81</a></span>;<br/>
<a NAME="E24:112"/><span class="lab"><font color="Green" title="E105">A19</font></span>: 
<font color="Maroon" title="c25">yag</font> <a href="tarski.html#R2" title="TARSKI: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="#E22:112"><span class="lab"><font color="Green" title="E103">A17</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">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="E25:112"/><span class="lab"><font color="Green" title="E106">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" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/>
<a NAME="E26:112"/>
 not <font color="Maroon" title="c3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E89">A2</font></span></a>, <a class="ref" href="chord.html#D8" target="_self" onmouseover="rs('chord/D8')" onmouseout="rh()">Def8</a></span>;<br/>
<a NAME="E27:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">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" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">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="#E20:112"><span class="lab"><font color="Green" title="E102">A16</font></span></a>, <a class="ref" href="chord.html#T29" target="_self" onmouseover="rs('chord/T29')" onmouseout="rh()">Th29</a></span>;<br/>
<a NAME="E29:112"/><span class="lab"><font color="Green" title="E108">A22</font></span>: 
 not <font color="Maroon" title="c17">y</font> <a href="tarski.html#R2" title="TARSKI: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="#E13:112"><span class="lab"><font color="Green" title="E97">A10</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="112_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> holds <br/><font color="Olive" title="b1">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></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">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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="E1:112_2"/><span class="lab"><font color="Green" title="E109">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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="E2:112_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="#E9:112"><span class="lab"><font color="Green" title="E95">A8</font></span></a>, <a class="txt" href="#E1:112_2"><span class="lab"><font color="Green" title="E109">A23</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_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="#E1:112_2"><span class="lab"><font color="Green" title="E109">A23</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E31:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">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>
 
;<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="#E27:112"><span class="lab"><font color="Green" title="E107">A21</font></span></a>, <a class="ref" href="chord.html#T29" target="_self" onmouseover="rs('chord/T29')" onmouseout="rh()">Th29</a></span>;<br/>
<a NAME="E33:112"/>
<font color="Maroon" title="c17">y</font> <a href="tarski.html#R2" title="TARSKI: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" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<a NAME="E34:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A25</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="tarski.html#R2" title="TARSKI: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" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E35:112"/><span class="lab"><font color="Green" title="E111">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="tarski.html#R2" title="TARSKI: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="#E1:112"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E89">A2</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E90">A3</font></span></a>, <a class="txt" href="#E16:112"><span class="lab"><font color="Green" title="E99">A12</font></span></a>, <a class="ref" href="chord.html#T83" target="_self" onmouseover="rs('chord/T83')" onmouseout="rh()">Th82</a></span>;<br/>
<a NAME="E36:112"/>
 not <font color="Maroon" title="c16">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:112"><span class="lab"><font color="Green" title="E109">A24</font></span></a>, <a class="txt" href="#E16:112"><span class="lab"><font color="Green" title="E99">A12</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E37:112"/><span class="kw">then </span>
<font color="Maroon" title="c16">x</font> <a href="tarski.html#R2" title="TARSKI: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="#E8:112"><span class="lab"><font color="Green" title="E94">A7</font></span></a>, <a class="txt" href="#E35:112"><span class="lab"><font color="Green" title="E111">A26</font></span></a></span>;<br/>
<a NAME="E38:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">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" onmouseover="rs('chord/T57')" onmouseout="rh()">Th56</a></span>;<br/>
<a NAME="E39:112"/>
<font color="Maroon" title="c17">y</font> <a href="tarski.html#R2" title="TARSKI: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" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<a NAME="E40:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A28</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="tarski.html#R2" title="TARSKI: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" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">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="E42:112"/><span class="lab"><font color="Green" title="E114">A29</font></span>: 
<font color="Maroon" title="c30">ybg</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">B</font>
 <span class="kw">and </span><br/><a NAME="E43:112"/><span class="lab"><font color="Green" title="E115">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="#E1:112"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E89">A2</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E90">A3</font></span></a>, <a class="txt" href="#E16:112"><span class="lab"><font color="Green" title="E99">A12</font></span></a>, <a class="ref" href="chord.html#T83" target="_self" onmouseover="rs('chord/T83')" onmouseout="rh()">Th82</a></span>;<br/>
<span class="lab"><font color="Green" title="E116">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" onmouseover="rs('xboole_1/T4')" onmouseout="rh()">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" onmouseover="rs('enumset1/T1')" onmouseout="rh()">ENUMSET1:1</a></span>
;<br/>
<a NAME="E45:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">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" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/>
<a NAME="E46:112"/>
<font color="Maroon" title="c16">x</font> <a href="tarski.html#R2" title="TARSKI: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" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<a NAME="E47:112"/><span class="kw">then </span>
<font color="Maroon" title="c16">x</font> <a href="tarski.html#R2" title="TARSKI: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" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">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="#E44:112"><span class="lab"><font color="Green" title="E116">A31</font></span></a>, <a class="txt" href="#E40:112"><span class="lab"><font color="Green" title="E113">A28</font></span></a>, <a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/>
<a NAME="E49:112"/><span class="lab"><font color="Green" title="E118">A33</font></span>: 
<font color="Maroon" title="c30">ybg</font> <a href="tarski.html#R2" title="TARSKI: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="#E42:112"><span class="lab"><font color="Green" title="E114">A29</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E50:112"/>
 not <font color="Maroon" title="c17">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:112"><span class="lab"><font color="Green" title="E109">A24</font></span></a>, <a class="txt" href="#E16:112"><span class="lab"><font color="Green" title="E99">A12</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E51:112"/><span class="kw">then </span>
 not <font color="Maroon" title="c19">yg</font> <a href="tarski.html#R2" title="TARSKI: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="#E18:112"><span class="lab"><font color="Green" title="E101">A14</font></span></a>, <a class="txt" href="#E29:112"><span class="lab"><font color="Green" title="E108">A22</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E52:112"/><span class="kw">then </span>
<font color="Maroon" title="c17">y</font> <a href="tarski.html#R2" title="TARSKI: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="#E18:112"><span class="lab"><font color="Green" title="E101">A14</font></span></a>, <a class="txt" href="#E43:112"><span class="lab"><font color="Green" title="E115">A30</font></span></a>, <a class="txt" href="#E49:112"><span class="lab"><font color="Green" title="E118">A33</font></span></a></span>;<br/>
<a NAME="E53:112"/><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="#E38:112"><span class="lab"><font color="Green" title="E112">A27</font></span></a>, <a class="ref" href="chord.html#T57" target="_self" onmouseover="rs('chord/T57')" onmouseout="rh()">Th56</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="E55:112"/><span class="lab"><font color="Green" title="E119">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>
 ;<br/>
<a NAME="E56:112"/>
 not <font color="Maroon" title="c16">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:112"><span class="lab"><font color="Green" title="E102">A16</font></span></a>, <a class="txt" href="#E16:112"><span class="lab"><font color="Green" title="E99">A12</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E57:112"/><span class="kw">then </span>
<font color="Maroon" title="c16">x</font> <a href="tarski.html#R2" title="TARSKI: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="#E7:112"><span class="lab"><font color="Green" title="E93">A6</font></span></a>, <a class="txt" href="#E17:112"><span class="lab"><font color="Green" title="E100">A13</font></span></a></span>;<br/>
<a NAME="E58:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">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" onmouseover="rs('chord/T57')" onmouseout="rh()">Th56</a></span>;<br/>
<span class="lab"><font color="Green" title="E121">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" onmouseover="rs('xboole_1/T4')" onmouseout="rh()">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" onmouseover="rs('enumset1/T1')" onmouseout="rh()">ENUMSET1:1</a></span>
;<br/>
<a NAME="E60:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">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" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/>
<a NAME="E61:112"/>
<font color="Maroon" title="c16">x</font> <a href="tarski.html#R2" title="TARSKI: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" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<a NAME="E62:112"/><span class="kw">then </span>
<font color="Maroon" title="c16">x</font> <a href="tarski.html#R2" title="TARSKI: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" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">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="#E59:112"><span class="lab"><font color="Green" title="E121">A36</font></span></a>, <a class="txt" href="#E34:112"><span class="lab"><font color="Green" title="E110">A25</font></span></a>, <a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/>
<a NAME="E64:112"/><span class="lab"><font color="Green" title="E123">A38</font></span>: 
 not <font color="Maroon" title="c17">y</font> <a href="tarski.html#R2" title="TARSKI: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="#E13:112"><span class="lab"><font color="Green" title="E97">A10</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E65:112"/>
 not <font color="Maroon" title="c17">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:112"><span class="lab"><font color="Green" title="E102">A16</font></span></a>, <a class="txt" href="#E16:112"><span class="lab"><font color="Green" title="E99">A12</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E66:112"/><span class="kw">then </span>
 not <font color="Maroon" title="c19">yg</font> <a href="tarski.html#R2" title="TARSKI: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="#E25:112"><span class="lab"><font color="Green" title="E106">A20</font></span></a>, <a class="txt" href="#E64:112"><span class="lab"><font color="Green" title="E123">A38</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E67:112"/><span class="kw">then </span>
<font color="Maroon" title="c17">y</font> <a href="tarski.html#R2" title="TARSKI: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="#E25:112"><span class="lab"><font color="Green" title="E106">A20</font></span></a>, <a class="txt" href="#E23:112"><span class="lab"><font color="Green" title="E104">A18</font></span></a>, <a class="txt" href="#E24:112"><span class="lab"><font color="Green" title="E105">A19</font></span></a></span>;<br/>
<a NAME="E68:112"/><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="#E58:112"><span class="lab"><font color="Green" title="E120">A35</font></span></a>, <a class="ref" href="chord.html#T57" target="_self" onmouseover="rs('chord/T57')" onmouseout="rh()">Th56</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="E70:112"/><span class="lab"><font color="Green" title="E124">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>
 ;<br/>
<span class="lab"><font color="Green" title="E125">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" onmouseover="rs('xboole_1/T24')" onmouseout="rh()">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="#E6:112"><span class="lab"><font color="Green" title="E92">A5</font></span></a></span>
;<br/>
<a NAME="E72:112"/><span class="lab"><font color="Green" title="E126">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="#E70:112"><span class="lab"><font color="Green" title="E124">A39</font></span></a></span>;<br/>
<a NAME="E73:112"/><span class="lab"><font color="Green" title="E127">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="#E70:112"><span class="lab"><font color="Green" title="E124">A39</font></span></a></span>;<br/>
<a NAME="E74:112"/><span class="lab"><font color="Green" title="E128">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="#E55:112"><span class="lab"><font color="Green" title="E119">A34</font></span></a></span>;<br/>
<a NAME="E75:112"/><span class="lab"><font color="Green" title="E129">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="#E55:112"><span class="lab"><font color="Green" title="E119">A34</font></span></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="E77:112"/><span class="lab"><font color="Green" title="E130">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="E78:112"/><span class="lab"><font color="Green" title="E131">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" onmouseover="rs('chord/T39')" onmouseout="rh()">Th38</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="E80:112"/><span class="lab"><font color="Green" title="E132">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="E81:112"/><span class="lab"><font color="Green" title="E133">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" onmouseover="rs('chord/T39')" onmouseout="rh()">Th38</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" onmouseover="rs('glib_001/T175')" onmouseout="rh()">GLIB_001:175</a></span>;<br/>
<a NAME="E83:112"/><span class="lab"><font color="Green" title="E134">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="#E75:112"><span class="lab"><font color="Green" title="E129">A44</font></span></a>, <a class="txt" href="#E77:112"><span class="lab"><font color="Green" title="E130">A45</font></span></a></span>;<br/>
<a NAME="E84:112"/>
<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" onmouseover="rs('glib_001/T98')" onmouseout="rh()">GLIB_001:98</a></span>;<br/>
<a NAME="E85:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E135">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="#E45:112"><span class="lab"><font color="Green" title="E117">A32</font></span></a></span>;<br/>
<a NAME="E86:112"/>
<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" onmouseover="rs('glib_001/T98')" onmouseout="rh()">GLIB_001:98</a></span>;<br/>
<a NAME="E87:112"/><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="#E60:112"><span class="lab"><font color="Green" title="E122">A37</font></span></a></span>;<br/>
<a NAME="E88:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E136">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="#E85:112"><span class="lab"><font color="Green" title="E135">A50</font></span></a>, <a class="txt" href="#E71:112"><span class="lab"><font color="Green" title="E125">A40</font></span></a>, <a class="ref" href="xboole_1.html#T27" onmouseover="rs('xboole_1/T27')" onmouseout="rh()">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="E89:112"/><span class="lab"><font color="Green" title="E137">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="#E72:112"><span class="lab"><font color="Green" title="E126">A41</font></span></a>, <a class="txt" href="#E80:112"><span class="lab"><font color="Green" title="E132">A47</font></span></a></span>;<br/>
<a NAME="E90:112"/><span class="lab"><font color="Green" title="E138">A53</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="#E74:112"><span class="lab"><font color="Green" title="E128">A43</font></span></a>, <a class="txt" href="#E77:112"><span class="lab"><font color="Green" title="E130">A45</font></span></a></span>;<br/>
<a NAME="E91:112"/><span class="lab"><font color="Green" title="E139">A54</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="E92:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">A55</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="tarski.html#R2" title="TARSKI: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="#E90:112"><span class="lab"><font color="Green" title="E138">A53</font></span></a>, <a class="ref" href="glib_001.html#T88" onmouseover="rs('glib_001/T88')" onmouseout="rh()">GLIB_001:88</a></span>;<br/>
<a NAME="E93:112"/><span class="lab"><font color="Green" title="E141">A56</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="E94:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E142">A57</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="tarski.html#R2" title="TARSKI: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="#E83:112"><span class="lab"><font color="Green" title="E134">A49</font></span></a>, <a class="ref" href="glib_001.html#T88" onmouseover="rs('glib_001/T88')" onmouseout="rh()">GLIB_001:88</a></span>;<br/>
<a NAME="E95:112"/><span class="lab"><font color="Green" title="E143">A58</font></span>: 
<font color="Maroon" title="c40">Pbg</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:112"><span class="lab"><font color="Green" title="E97">A10</font></span></a>, <a class="txt" href="#E93:112"><span class="lab"><font color="Green" title="E141">A56</font></span></a>, <a class="txt" href="#E91:112"><span class="lab"><font color="Green" title="E139">A54</font></span></a>, <a class="txt" href="#E83:112"><span class="lab"><font color="Green" title="E134">A49</font></span></a>, <a class="txt" href="#E90:112"><span class="lab"><font color="Green" title="E138">A53</font></span></a>, <a class="ref" href="glib_001.html#T127" onmouseover="rs('glib_001/T127')" onmouseout="rh()">GLIB_001:127</a></span>;<br/>
<a NAME="E96:112"/><span class="lab"><font color="Green" title="E144">A59</font></span>: 
<font color="Maroon" title="c40">Pbg</font> is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:112"><span class="lab"><font color="Green" title="E97">A10</font></span></a>, <a class="txt" href="#E83:112"><span class="lab"><font color="Green" title="E134">A49</font></span></a>, <a class="txt" href="#E90:112"><span class="lab"><font color="Green" title="E138">A53</font></span></a></span>;<br/>
<a NAME="E97:112"/><span class="lab"><font color="Green" title="E145">A60</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="#E13:112"><span class="lab"><font color="Green" title="E97">A10</font></span></a>, <a class="txt" href="#E93:112"><span class="lab"><font color="Green" title="E141">A56</font></span></a>, <a class="txt" href="#E91:112"><span class="lab"><font color="Green" title="E139">A54</font></span></a>, <a class="txt" href="#E83:112"><span class="lab"><font color="Green" title="E134">A49</font></span></a>, <a class="txt" href="#E90:112"><span class="lab"><font color="Green" title="E138">A53</font></span></a>, <a class="ref" href="glib_001.html#D24" onmouseover="rs('glib_001/D24')" onmouseout="rh()">GLIB_001:def 24</a></span>;<br/>
<a NAME="E98:112"/><span class="lab"><font color="Green" title="E146">A61</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="#E4:112"><span class="lab"><font color="Green" title="E91">A4</font></span></a>, <a class="txt" href="#E14:112"><span class="lab"><font color="Green" title="E98">A11</font></span></a>, <a class="ref" href="chord.html#T45" target="_self" onmouseover="rs('chord/T45')" onmouseout="rh()">Th44</a></span>;<br/>
<a NAME="E99:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E147">A62</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="#E13:112"><span class="lab"><font color="Green" title="E97">A10</font></span></a>, <a class="txt" href="#E93:112"><span class="lab"><font color="Green" title="E141">A56</font></span></a>, <a class="txt" href="#E91:112"><span class="lab"><font color="Green" title="E139">A54</font></span></a>, <a class="txt" href="#E83:112"><span class="lab"><font color="Green" title="E134">A49</font></span></a>, <a class="txt" href="#E90:112"><span class="lab"><font color="Green" title="E138">A53</font></span></a>, <a class="ref" href="chord.html#T46" target="_self" onmouseover="rs('chord/T46')" onmouseout="rh()">Th45</a></span>;<br/>
<a NAME="E100:112"/><span class="lab"><font color="Green" title="E148">A63</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="#E73:112"><span class="lab"><font color="Green" title="E127">A42</font></span></a>, <a class="txt" href="#E80:112"><span class="lab"><font color="Green" title="E132">A47</font></span></a></span>;<br/>
<a NAME="E101:112"/><span class="lab"><font color="Green" title="E149">A64</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="112_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:112_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" onmouseover="rs('xboole_0/D7')" onmouseout="rh()">XBOOLE_0:def 7</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<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">object</a> <span class="kw"> such that </span><br/><a NAME="E3:112_6"/><span class="lab"><font color="Green" title="E150">A65</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" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E4:112_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="#E3:112_6"><span class="lab"><font color="Green" title="E150">A65</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E5:112_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" onmouseover="rs('glib_001/T110')" onmouseout="rh()">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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:112_6"/><span class="lab"><font color="Green" title="E151">A66</font></span>: 
<font color="Maroon" title="c45">na</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E8:112_6"/><span class="lab"><font color="Green" title="E152">A67</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="E9:112_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:112_6"/><span class="lab"><font color="Green" title="E153">A68</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#K1" title="NAT_1:func.1">+</a> 2</span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:112_6"/><span class="lab"><font color="Green" title="E154">A69</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" onmouseover="rs('glib_001/T103')" onmouseout="rh()">GLIB_001:103</a></span>;<br/>
<a NAME="E12:112_6"/><span class="lab"><font color="Green" title="E155">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>,<font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:112_6"><span class="lab"><font color="Green" title="E154">A69</font></span></a>, <a class="ref" href="glib_000.html#T72" onmouseover="rs('glib_000/T72')" onmouseout="rh()">GLIB_000:72</a></span>;<br/>
<a NAME="E13:112_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="#E3:112_6"><span class="lab"><font color="Green" title="E150">A65</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E14:112_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" onmouseover="rs('glib_001/T110')" onmouseout="rh()">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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E16:112_6"/>
<font color="Maroon" title="c48">nb</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E17:112_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="E18:112_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E19:112_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#K1" title="NAT_1:func.1">+</a> 2</span>)</span>
 <span class="kw">and </span><br/><a NAME="E20:112_6"/><span class="lab"><font color="Green" title="E156">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="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" onmouseover="rs('glib_001/T103')" onmouseout="rh()">GLIB_001:103</a></span>;<br/>
<a NAME="E21:112_6"/><span class="lab"><font color="Green" title="E157">A72</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="#E44:112"><span class="lab"><font color="Green" title="E116">A31</font></span></a>, <a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/>
<a NAME="E22:112_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="#E20:112_6"><span class="lab"><font color="Green" title="E156">A71</font></span></a>, <a class="ref" href="glib_000.html#T72" onmouseover="rs('glib_000/T72')" onmouseout="rh()">GLIB_000:72</a></span>;<br/>
<a NAME="E23:112_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E158">A73</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="#E12:112_6"><span class="lab"><font color="Green" title="E155">A70</font></span></a></span>;<br/>
<a NAME="E24:112_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E159">A74</font></span>: 
( <font color="Maroon" title="c43">a1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">B</font> or <font color="Maroon" title="c43">a1</font> <a href="tarski.html#R2" title="TARSKI: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="#E21:112_6"><span class="lab"><font color="Green" title="E157">A72</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E25:112_6"/><span class="lab"><font color="Green" title="E160">A75</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="#E59:112"><span class="lab"><font color="Green" title="E121">A36</font></span></a>, <a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/>
<a NAME="E26:112_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E161">A76</font></span>: 
( <font color="Maroon" title="c43">a1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">A</font> or <font color="Maroon" title="c43">a1</font> <a href="tarski.html#R2" title="TARSKI: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" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E27:112_6"/><span class="lab"><font color="Green" title="E162">A77</font></span>: 
( <font color="Maroon" title="c44">a2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">A</font> or <font color="Maroon" title="c44">a2</font> <a href="tarski.html#R2" title="TARSKI: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="#E25:112_6"><span class="lab"><font color="Green" title="E160">A75</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E28:112_6"/><span class="lab"><font color="Green" title="E163">A78</font></span>: 
( <font color="Maroon" title="c44">a2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">B</font> or <font color="Maroon" title="c44">a2</font> <a href="tarski.html#R2" title="TARSKI: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="#E23:112_6"><span class="lab"><font color="Green" title="E158">A73</font></span></a>, <a class="txt" href="#E21:112_6"><span class="lab"><font color="Green" title="E157">A72</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_6_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:112_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="#E6:112"><span class="lab"><font color="Green" title="E92">A5</font></span></a>, <a class="txt" href="#E26:112_6"><span class="lab"><font color="Green" title="E161">A76</font></span></a>, <a class="txt" href="#E27:112_6"><span class="lab"><font color="Green" title="E162">A77</font></span></a>, <a class="txt" href="#E24:112_6"><span class="lab"><font color="Green" title="E159">A74</font></span></a>, <a class="txt" href="#E28:112_6"><span class="lab"><font color="Green" title="E163">A78</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:112_6_1_1"/><span class="lab"><font color="Green" title="E164">A79</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:112_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#K1" title="NAT_1:func.1">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T39" onmouseover="rs('xreal_1/T39')" onmouseout="rh()">XREAL_1:39</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_6_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:112"><span class="lab"><font color="Green" title="E97">A10</font></span></a>, <a class="txt" href="#E100:112"><span class="lab"><font color="Green" title="E148">A63</font></span></a>, <a class="txt" href="#E89:112"><span class="lab"><font color="Green" title="E137">A52</font></span></a>, <a class="txt" href="#E7:112_6"><span class="lab"><font color="Green" title="E151">A66</font></span></a>, <a class="txt" href="#E8:112_6"><span class="lab"><font color="Green" title="E152">A67</font></span></a>, <a class="txt" href="#E10:112_6"><span class="lab"><font color="Green" title="E153">A68</font></span></a>, <a class="txt" href="#E1:112_6_1_1"><span class="lab"><font color="Green" title="E164">A79</font></span></a>, <a class="ref" href="glib_001.html#D28" onmouseover="rs('glib_001/D28')" onmouseout="rh()">GLIB_001:def 28</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:112_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:112_6_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E98:112"><span class="lab"><font color="Green" title="E146">A61</font></span></a>, <a class="txt" href="#E12:112_6"><span class="lab"><font color="Green" title="E155">A70</font></span></a>, <a class="ref" href="chord.html#D3" target="_self" onmouseover="rs('chord/D3')" onmouseout="rh()">Def3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E102:112"/><span class="lab"><font color="Green" title="E150">A80</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="E103:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E151">A81</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="tarski.html#R2" title="TARSKI: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="#E100:112"><span class="lab"><font color="Green" title="E148">A63</font></span></a>, <a class="ref" href="glib_001.html#T88" onmouseover="rs('glib_001/T88')" onmouseout="rh()">GLIB_001:88</a></span>;<br/>
<a NAME="E104:112"/><span class="lab"><font color="Green" title="E152">A82</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="E105:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E153">A83</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_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="#E89:112"><span class="lab"><font color="Green" title="E137">A52</font></span></a>, <a class="txt" href="#E93:112"><span class="lab"><font color="Green" title="E141">A56</font></span></a>, <a class="txt" href="#E83:112"><span class="lab"><font color="Green" title="E134">A49</font></span></a>, <a class="ref" href="glib_001.html#T28" onmouseover="rs('glib_001/T28')" onmouseout="rh()">GLIB_001:28</a></span>;<br/>
<a NAME="E106:112"/><span class="lab"><font color="Green" title="E154">A84</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="#E13:112"><span class="lab"><font color="Green" title="E97">A10</font></span></a>, <a class="txt" href="#E98:112"><span class="lab"><font color="Green" title="E146">A61</font></span></a>, <a class="txt" href="#E102:112"><span class="lab"><font color="Green" title="E150">A80</font></span></a>, <a class="txt" href="#E104:112"><span class="lab"><font color="Green" title="E152">A82</font></span></a>, <a class="txt" href="#E100:112"><span class="lab"><font color="Green" title="E148">A63</font></span></a>, <a class="txt" href="#E89:112"><span class="lab"><font color="Green" title="E137">A52</font></span></a>, <a class="ref" href="chord.html#T46" target="_self" onmouseover="rs('chord/T46')" onmouseout="rh()">Th45</a></span>;<br/>
<a NAME="E107:112"/>
<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#K1" title="NAT_1:func.1">+</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="#E104:112"><span class="lab"><font color="Green" title="E152">A82</font></span></a>, <a class="txt" href="#E89:112"><span class="lab"><font color="Green" title="E137">A52</font></span></a>, <a class="txt" href="#E93:112"><span class="lab"><font color="Green" title="E141">A56</font></span></a>, <a class="txt" href="#E83:112"><span class="lab"><font color="Green" title="E134">A49</font></span></a>, <a class="ref" href="chord.html#T28" target="_self" onmouseover="rs('chord/T28')" onmouseout="rh()">Th28</a></span>;<br/>
<a NAME="E108:112"/><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#K1" title="NAT_1:func.1">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E106:112"><span class="lab"><font color="Green" title="E154">A84</font></span></a>, <a class="txt" href="#E99:112"><span class="lab"><font color="Green" title="E147">A62</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E109:112"/><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#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<a NAME="E110:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E155">A85</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" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E111:112"/><span class="lab"><font color="Green" title="E156">A86</font></span>: 
<font color="Maroon" title="c39">Pag</font> is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:112"><span class="lab"><font color="Green" title="E97">A10</font></span></a>, <a class="txt" href="#E100:112"><span class="lab"><font color="Green" title="E148">A63</font></span></a>, <a class="txt" href="#E89:112"><span class="lab"><font color="Green" title="E137">A52</font></span></a></span>;<br/>
<a NAME="E112:112"/><span class="lab"><font color="Green" title="E157">A87</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="tarski.html#R2" title="TARSKI: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="#E104:112"><span class="lab"><font color="Green" title="E152">A82</font></span></a>, <a class="txt" href="#E89:112"><span class="lab"><font color="Green" title="E137">A52</font></span></a>, <a class="ref" href="glib_001.html#T88" onmouseover="rs('glib_001/T88')" onmouseout="rh()">GLIB_001:88</a></span>;<br/>
<a NAME="E113:112"/>
<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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="112_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">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="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="E1:112_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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="E2:112_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" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:112_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="#E103:112"><span class="lab"><font color="Green" title="E151">A81</font></span></a>, <a class="txt" href="#E112:112"><span class="lab"><font color="Green" title="E157">A87</font></span></a>, <a class="txt" href="#E92:112"><span class="lab"><font color="Green" title="E140">A55</font></span></a>, <a class="txt" href="#E94:112"><span class="lab"><font color="Green" title="E142">A57</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E114:112"/><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="#E88:112"><span class="lab"><font color="Green" title="E136">A51</font></span></a></span>;<br/>
<a NAME="E115:112"/><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="#E100:112"><span class="lab"><font color="Green" title="E148">A63</font></span></a>, <a class="txt" href="#E89:112"><span class="lab"><font color="Green" title="E137">A52</font></span></a>, <a class="txt" href="#E111:112"><span class="lab"><font color="Green" title="E156">A86</font></span></a>, <a class="txt" href="#E83:112"><span class="lab"><font color="Green" title="E134">A49</font></span></a>, <a class="txt" href="#E90:112"><span class="lab"><font color="Green" title="E138">A53</font></span></a>, <a class="txt" href="#E96:112"><span class="lab"><font color="Green" title="E144">A59</font></span></a>, <a class="txt" href="#E95:112"><span class="lab"><font color="Green" title="E143">A58</font></span></a>, <a class="txt" href="#E101:112"><span class="lab"><font color="Green" title="E149">A64</font></span></a>, <a class="ref" href="chord.html#T27" target="_self" onmouseover="rs('chord/T27')" onmouseout="rh()">Th27</a></span>;<br/>
<a NAME="E116:112"/><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="#E110:112"><span class="lab"><font color="Green" title="E155">A85</font></span></a>, <a class="ref" href="chord.html#D11" target="_self" onmouseover="rs('chord/D11')" onmouseout="rh()">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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E118:112"/><span class="lab"><font color="Green" title="E158">A88</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="E119:112"/><span class="lab"><font color="Green" title="E159">A89</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="E120:112"/><span class="lab"><font color="Green" title="E160">A90</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="E121:112"/><span class="lab"><font color="Green" title="E161">A91</font></span>: 
 ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e</font> <a href="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="E122:112"/><span class="lab"><font color="Green" title="E162">A92</font></span>: 
 for <font color="Olive" title="b1">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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>
 ;<br/>
<a NAME="E123:112"/><span class="lab"><font color="Green" title="E163">A93</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="#E118:112"><span class="lab"><font color="Green" title="E158">A88</font></span></a>, <a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">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">object</a> <span class="kw"> such that </span><br/><a NAME="E125:112"/><span class="lab"><font color="Green" title="E164">A94</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="#E121:112"><span class="lab"><font color="Green" title="E161">A91</font></span></a></span>;<br/>
<a NAME="E126:112"/><span class="lab"><font color="Green" title="E165">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="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="#E125:112"><span class="lab"><font color="Green" title="E164">A94</font></span></a></span>;<br/>
<a NAME="E127:112"/><span class="lab"><font color="Green" title="E166">A96</font></span>: 
<font color="Maroon" title="c42">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E128:112"/><span class="lab"><font color="Green" title="E167">A97</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="#E13:112"><span class="lab"><font color="Green" title="E97">A10</font></span></a>, <a class="txt" href="#E102:112"><span class="lab"><font color="Green" title="E150">A80</font></span></a>, <a class="txt" href="#E104:112"><span class="lab"><font color="Green" title="E152">A82</font></span></a>, <a class="txt" href="#E100:112"><span class="lab"><font color="Green" title="E148">A63</font></span></a>, <a class="txt" href="#E89:112"><span class="lab"><font color="Green" title="E137">A52</font></span></a>, <a class="ref" href="glib_001.html#D24" onmouseover="rs('glib_001/D24')" onmouseout="rh()">GLIB_001:def 24</a></span>;<br/>
<a NAME="E129:112"/><span class="lab"><font color="Green" title="E168">A98</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" onmouseover="rs('abian/T12')" onmouseout="rh()">ABIAN:12</a></span>;<br/>
<a NAME="E130:112"/><span class="lab"><font color="Green" title="E169">A99</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" onmouseover="rs('abian/T12')" onmouseout="rh()">ABIAN:12</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_8"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:112_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="112_8_1"><span class="kw">suppose </span></a><a NAME="E1:112_8_1"/><span class="lab"><font color="Green" title="E170">A100</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:112_8_1"/><span class="kw">then </span>
<font color="Maroon" title="c43">n</font> <a href="tarski.html#R2" title="TARSKI: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="#E129:112"><span class="lab"><font color="Green" title="E168">A98</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E3:112_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E171">A101</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" onmouseover="rs('glib_001/T32')" onmouseout="rh()">GLIB_001:32</a></span>;<br/><a NAME="E4:112_8_1"/>
<font color="Maroon" title="c42">m</font> <a href="tarski.html#R2" title="TARSKI: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="#E130:112"><span class="lab"><font color="Green" title="E169">A99</font></span></a>, <a class="txt" href="#E1:112_8_1"><span class="lab"><font color="Green" title="E170">A100</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E5:112_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" onmouseover="rs('glib_001/T32')" onmouseout="rh()">GLIB_001:32</a></span>;<br/><a NAME="E6:112_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="#E128:112"><span class="lab"><font color="Green" title="E167">A97</font></span></a>, <a class="txt" href="#E118:112"><span class="lab"><font color="Green" title="E158">A88</font></span></a>, <a class="txt" href="#E121:112"><span class="lab"><font color="Green" title="E161">A91</font></span></a>, <a class="txt" href="#E1:112_8_1"><span class="lab"><font color="Green" title="E170">A100</font></span></a>, <a class="txt" href="#E3:112_8_1"><span class="lab"><font color="Green" title="E171">A101</font></span></a>, <a class="ref" href="chord.html#T85" target="_self" onmouseover="rs('chord/T85')" onmouseout="rh()">Th84</a></span>;<br/><a NAME="E7:112_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="#E59:112"><span class="lab"><font color="Green" title="E121">A36</font></span></a>, <a class="ref" href="chord.html#T87" target="_self" onmouseover="rs('chord/T87')" onmouseout="rh()">Th86</a></span>;<br/><span class="kw">hence </span><a NAME="E8:112_8_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E81:112"><span class="lab"><font color="Green" title="E133">A48</font></span></a>, <a class="ref" href="chord.html#T89" target="_self" onmouseover="rs('chord/T89')" onmouseout="rh()">Th88</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_8_2"><span class="kw">suppose </span></a><a NAME="E1:112_8_2"/><span class="lab"><font color="Green" title="E170">A102</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:112_8_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E171">A103</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="tarski.html#R2" title="TARSKI: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="#E127:112"><span class="lab"><font color="Green" title="E166">A96</font></span></a>, <a class="ref" href="glib_001.html#T7" onmouseover="rs('glib_001/T7')" onmouseout="rh()">GLIB_001:7</a></span>;<br/><a NAME="E3:112_8_2"/>
<font color="Maroon" title="c42">m</font> <a href="tarski.html#R2" title="TARSKI: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="#E130:112"><span class="lab"><font color="Green" title="E169">A99</font></span></a>, <a class="txt" href="#E1:112_8_2"><span class="lab"><font color="Green" title="E170">A102</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E4:112_8_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E172">A104</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" onmouseover="rs('glib_001/T32')" onmouseout="rh()">GLIB_001:32</a></span>;<br/><a NAME="E5:112_8_2"/><span class="lab"><font color="Green" title="E173">A105</font></span>: 
 not <font color="Maroon" title="c43">n</font> <a href="tarski.html#R2" title="TARSKI: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="#E1:112_8_2"><span class="lab"><font color="Green" title="E170">A102</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E6:112_8_2"/>
<font color="Maroon" title="c43">n</font> <a href="tarski.html#R2" title="TARSKI: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="#E119:112"><span class="lab"><font color="Green" title="E159">A89</font></span></a>, <a class="txt" href="#E129:112"><span class="lab"><font color="Green" title="E168">A98</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E8:112_8_2"/><span class="lab"><font color="Green" title="E174">A106</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="E9:112_8_2"/><span class="lab"><font color="Green" title="E175">A107</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#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c45">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:112_8_2"><span class="lab"><font color="Green" title="E173">A105</font></span></a>, <a class="ref" href="glib_001.html#T34" onmouseover="rs('glib_001/T34')" onmouseout="rh()">GLIB_001:34</a></span>;<br/><a NAME="E10:112_8_2"/><span class="lab"><font color="Green" title="E176">A108</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E104:112"><span class="lab"><font color="Green" title="E152">A82</font></span></a>, <a class="txt" href="#E89:112"><span class="lab"><font color="Green" title="E137">A52</font></span></a>, <a class="txt" href="#E93:112"><span class="lab"><font color="Green" title="E141">A56</font></span></a>, <a class="txt" href="#E83:112"><span class="lab"><font color="Green" title="E134">A49</font></span></a>, <a class="txt" href="#E8:112_8_2"><span class="lab"><font color="Green" title="E174">A106</font></span></a>, <a class="ref" href="glib_001.html#T33" onmouseover="rs('glib_001/T33')" onmouseout="rh()">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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:112_8_2"><span class="lab"><font color="Green" title="E175">A107</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#K1" title="NAT_1:func.1">+</a> 1 as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/><a NAME="E13:112_8_2"/><span class="lab"><font color="Green" title="E177">A109</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="#E8:112_8_2"><span class="lab"><font color="Green" title="E174">A106</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E14:112_8_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E178">A110</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="tarski.html#R2" title="TARSKI: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" onmouseover="rs('glib_001/T7')" onmouseout="rh()">GLIB_001:7</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_8_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:112_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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="#E60:112"><span class="lab"><font color="Green" title="E122">A37</font></span></a>, <a class="txt" href="#E45:112"><span class="lab"><font color="Green" title="E117">A32</font></span></a>, <a class="txt" href="#E2:112_8_2"><span class="lab"><font color="Green" title="E171">A103</font></span></a>, <a class="txt" href="#E14:112_8_2"><span class="lab"><font color="Green" title="E178">A110</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_8_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:112_8_2_1_1"/><span class="lab"><font color="Green" title="E179">A111</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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_8_2_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:112_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="#E100:112"><span class="lab"><font color="Green" title="E148">A63</font></span></a>, <a class="txt" href="#E89:112"><span class="lab"><font color="Green" title="E137">A52</font></span></a>, <a class="txt" href="#E1:112_8_2_1_1"><span class="lab"><font color="Green" title="E179">A111</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_8_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:112_8_2_1_1_1_1"/><span class="lab"><font color="Green" title="E180">A112</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="112_8_2_1_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not 1 <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> <font color="Maroon" title="c42">m</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:112_8_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E181">A113</font></span>: 
1 <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> <font color="Maroon" title="c42">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_8_2_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:112_8_2_1_1_1_1_1_1"/>
( 1 <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> <font color="Maroon" title="c42">m</font> or 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="#E1:112_8_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E181">A113</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_8_2_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:112_8_2_1_1_1_1_1_1_1"/>
1 <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> <font color="Maroon" title="c42">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:112_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" onmouseover="rs('chord/T4')" onmouseout="rh()">Th4</a>, <a class="ref" href="jordan12.html#T2" onmouseover="rs('jordan12/T2')" onmouseout="rh()">JORDAN12:2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_8_2_1_1_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E120:112"><span class="lab"><font color="Green" title="E160">A90</font></span></a>, <a class="txt" href="#E130:112"><span class="lab"><font color="Green" title="E169">A99</font></span></a>, <a class="txt" href="#E4:112_8_2"><span class="lab"><font color="Green" title="E172">A104</font></span></a>, <a class="txt" href="#E9:112_8_2"><span class="lab"><font color="Green" title="E175">A107</font></span></a>, <a class="txt" href="#E10:112_8_2"><span class="lab"><font color="Green" title="E176">A108</font></span></a>, <a class="txt" href="#E1:112_8_2_1_1_1_1"><span class="lab"><font color="Green" title="E180">A112</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_8_2_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:112_8_2_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E182">A114</font></span>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:112_8_2_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E183">A115</font></span>: 
1 <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="txt" href="#E1:112_8_2"><span class="lab"><font color="Green" title="E170">A102</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:112_8_2_1_1_1_1_1_1_2"/><span class="kw">then </span>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI: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" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E4:112_8_2_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E184">A116</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#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">1 <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" onmouseover="rs('glib_001/T32')" onmouseout="rh()">GLIB_001:32</a></span>;<br/><a NAME="E5:112_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="#E2:112_8_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E183">A115</font></span></a>, <a class="ref" href="xreal_1.html#T20" onmouseover="rs('xreal_1/T20')" onmouseout="rh()">XREAL_1:20</a></span>;<br/><a NAME="E6:112_8_2_1_1_1_1_1_1_2"/><span class="kw">then </span>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E7:112_8_2_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E185">A117</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI: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="#E105:112"><span class="lab"><font color="Green" title="E153">A83</font></span></a>, <a class="ref" href="glib_001.html#T100" onmouseover="rs('glib_001/T100')" onmouseout="rh()">GLIB_001:100</a>, <a class="ref" href="jordan12.html#T2" onmouseover="rs('jordan12/T2')" onmouseout="rh()">JORDAN12:2</a></span>;<br/><a NAME="E8:112_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="#E130:112"><span class="lab"><font color="Green" title="E169">A99</font></span></a>, <a class="txt" href="#E1:112_8_2"><span class="lab"><font color="Green" title="E170">A102</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E9:112_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#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> 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="#E1:112_8_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E182">A114</font></span></a>, <a class="ref" href="glib_001.html#D3" onmouseover="rs('glib_001/D3')" onmouseout="rh()">GLIB_001:def 3</a>, <a class="ref" href="jordan12.html#T2" onmouseover="rs('jordan12/T2')" onmouseout="rh()">JORDAN12:2</a></span>;<br/><span class="kw">hence </span><a NAME="E10:112_8_2_1_1_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E122:112"><span class="lab"><font color="Green" title="E162">A92</font></span></a>, <a class="txt" href="#E4:112_8_2"><span class="lab"><font color="Green" title="E172">A104</font></span></a>, <a class="txt" href="#E9:112_8_2"><span class="lab"><font color="Green" title="E175">A107</font></span></a>, <a class="txt" href="#E10:112_8_2"><span class="lab"><font color="Green" title="E176">A108</font></span></a>, <a class="txt" href="#E1:112_8_2_1_1_1_1"><span class="lab"><font color="Green" title="E180">A112</font></span></a>, <a class="txt" href="#E4:112_8_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E184">A116</font></span></a>, <a class="txt" href="#E7:112_8_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E185">A117</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E3:112_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="#E128:112"><span class="lab"><font color="Green" title="E167">A97</font></span></a>, <a class="txt" href="#E126:112"><span class="lab"><font color="Green" title="E165">A95</font></span></a>, <a class="txt" href="#E1:112_8_2"><span class="lab"><font color="Green" title="E170">A102</font></span></a>, <a class="txt" href="#E4:112_8_2"><span class="lab"><font color="Green" title="E172">A104</font></span></a>, <a class="txt" href="#E9:112_8_2"><span class="lab"><font color="Green" title="E175">A107</font></span></a>, <a class="txt" href="#E10:112_8_2"><span class="lab"><font color="Green" title="E176">A108</font></span></a>, <a class="txt" href="#E1:112_8_2_1_1_1_1"><span class="lab"><font color="Green" title="E180">A112</font></span></a>, <a class="ref" href="chord.html#T85" target="_self" onmouseover="rs('chord/T85')" onmouseout="rh()">Th84</a>, <a class="ref" href="jordan12.html#T2" onmouseover="rs('jordan12/T2')" onmouseout="rh()">JORDAN12:2</a></span>;<br/><a NAME="E4:112_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="#E59:112"><span class="lab"><font color="Green" title="E121">A36</font></span></a>, <a class="ref" href="chord.html#T87" target="_self" onmouseover="rs('chord/T87')" onmouseout="rh()">Th86</a></span>;<br/><span class="kw">hence </span><a NAME="E5:112_8_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E81:112"><span class="lab"><font color="Green" title="E133">A48</font></span></a>, <a class="ref" href="chord.html#T89" target="_self" onmouseover="rs('chord/T89')" onmouseout="rh()">Th88</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_8_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:112_8_2_1_1_1_2"/><span class="lab"><font color="Green" title="E180">A118</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="112_8_2_1_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <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></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="E1:112_8_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E181">A119</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_8_2_1_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:112_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="#E1:112_8_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E181">A119</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_8_2_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:112_8_2_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E182">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="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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:112_8_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E183">A121</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#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E3:112_8_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E184">A122</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" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:112_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" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a></span>;<br/><a NAME="E5:112_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="tarski.html#R2" title="TARSKI: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="#E3:112_8_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E184">A122</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E6:112_8_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E185">A123</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" onmouseover="rs('glib_001/T32')" onmouseout="rh()">GLIB_001:32</a></span>;<br/><a NAME="E7:112_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="#E2:112_8_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E183">A121</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E8:112_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#K5" title="NUMBERS:func.5">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" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E9:112_8_2_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E186">A124</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="tarski.html#R2" title="TARSKI: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="#E105:112"><span class="lab"><font color="Green" title="E153">A83</font></span></a>, <a class="ref" href="glib_001.html#T100" onmouseover="rs('glib_001/T100')" onmouseout="rh()">GLIB_001:100</a></span>;<br/><a NAME="E10:112_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="#E3:112_8_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E184">A122</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E11:112_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="#E127:112"><span class="lab"><font color="Green" title="E166">A96</font></span></a>, <a class="txt" href="#E1:112_8_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E182">A120</font></span></a>, <a class="ref" href="glib_001.html#D3" onmouseover="rs('glib_001/D3')" onmouseout="rh()">GLIB_001:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E12:112_8_2_1_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E122:112"><span class="lab"><font color="Green" title="E162">A92</font></span></a>, <a class="txt" href="#E4:112_8_2"><span class="lab"><font color="Green" title="E172">A104</font></span></a>, <a class="txt" href="#E9:112_8_2"><span class="lab"><font color="Green" title="E175">A107</font></span></a>, <a class="txt" href="#E10:112_8_2"><span class="lab"><font color="Green" title="E176">A108</font></span></a>, <a class="txt" href="#E1:112_8_2_1_1_1_2"><span class="lab"><font color="Green" title="E180">A118</font></span></a>, <a class="txt" href="#E6:112_8_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E185">A123</font></span></a>, <a class="txt" href="#E9:112_8_2_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E186">A124</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_8_2_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:112_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:112_8_2_1_1_1_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_8_2"><span class="lab"><font color="Green" title="E170">A102</font></span></a>, <a class="ref" href="chord.html#T4" target="_self" onmouseover="rs('chord/T4')" onmouseout="rh()">Th4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E3:112_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="#E128:112"><span class="lab"><font color="Green" title="E167">A97</font></span></a>, <a class="txt" href="#E121:112"><span class="lab"><font color="Green" title="E161">A91</font></span></a>, <a class="txt" href="#E4:112_8_2"><span class="lab"><font color="Green" title="E172">A104</font></span></a>, <a class="txt" href="#E9:112_8_2"><span class="lab"><font color="Green" title="E175">A107</font></span></a>, <a class="txt" href="#E10:112_8_2"><span class="lab"><font color="Green" title="E176">A108</font></span></a>, <a class="txt" href="#E1:112_8_2_1_1_1_2"><span class="lab"><font color="Green" title="E180">A118</font></span></a>, <a class="ref" href="chord.html#T85" target="_self" onmouseover="rs('chord/T85')" onmouseout="rh()">Th84</a></span>;<br/><a NAME="E4:112_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="#E59:112"><span class="lab"><font color="Green" title="E121">A36</font></span></a>, <a class="ref" href="chord.html#T87" target="_self" onmouseover="rs('chord/T87')" onmouseout="rh()">Th86</a></span>;<br/><span class="kw">hence </span><a NAME="E5:112_8_2_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E81:112"><span class="lab"><font color="Green" title="E133">A48</font></span></a>, <a class="ref" href="chord.html#T89" target="_self" onmouseover="rs('chord/T89')" onmouseout="rh()">Th88</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_8_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:112_8_2_1_2"/><span class="lab"><font color="Green" title="E179">A125</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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">B</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><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" onmouseover="rs('glib_000/D37')" onmouseout="rh()">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="#E1:112_8_2_1_2"><span class="lab"><font color="Green" title="E179">A125</font></span></a>, <a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">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="E4:112_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="#E20:112"><span class="lab"><font color="Green" title="E102">A16</font></span></a>, <a class="txt" href="#E31:112"><span class="lab"><font color="Green" title="E109">A24</font></span></a>, <a class="txt" href="#E125:112"><span class="lab"><font color="Green" title="E164">A94</font></span></a>, <a class="txt" href="#E4:112_8_2"><span class="lab"><font color="Green" title="E172">A104</font></span></a>, <a class="txt" href="#E9:112_8_2"><span class="lab"><font color="Green" title="E175">A107</font></span></a>, <a class="txt" href="#E10:112_8_2"><span class="lab"><font color="Green" title="E176">A108</font></span></a>, <a class="txt" href="#E1:112_8_2_1_2"><span class="lab"><font color="Green" title="E179">A125</font></span></a>, <a class="ref" href="chord.html#T19" target="_self" onmouseover="rs('chord/T19')" onmouseout="rh()">Th19</a></span>;<br/><a NAME="E5:112_8_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E180">A126</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" onmouseover="rs('glib_001/T15')" onmouseout="rh()">GLIB_001:15</a></span>;<br/><a NAME="E6:112_8_2_1_2"/>
<font color="Maroon" title="c4">b</font> <a href="tarski.html#R2" title="TARSKI: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" onmouseover="rs('glib_002/T9')" onmouseout="rh()">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" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/><a NAME="E8:112_8_2_1_2"/>
<font color="Maroon" title="c3">a</font> <a href="tarski.html#R2" title="TARSKI: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" onmouseover="rs('glib_002/T9')" onmouseout="rh()">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" onmouseover="rs('glib_000/D37')" onmouseout="rh()">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="E11:112_8_2_1_2"/><span class="lab"><font color="Green" title="E181">A127</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" onmouseover="rs('glib_002/D1')" onmouseout="rh()">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="E13:112_8_2_1_2"/><span class="lab"><font color="Green" title="E182">A128</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" onmouseover="rs('glib_002/D1')" onmouseout="rh()">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" onmouseover="rs('glib_001/T167')" onmouseout="rh()">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="E16:112_8_2_1_2"/><span class="lab"><font color="Green" title="E183">A129</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="#E13:112_8_2_1_2"><span class="lab"><font color="Green" title="E182">A128</font></span></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="E17:112_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="#E11:112_8_2_1_2"><span class="lab"><font color="Green" title="E181">A127</font></span></a></span>;<br/><a NAME="E18:112_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="#E5:112_8_2_1_2"><span class="lab"><font color="Green" title="E180">A126</font></span></a>, <a class="ref" href="glib_001.html#T31" onmouseover="rs('glib_001/T31')" onmouseout="rh()">GLIB_001:31</a></span>;<br/><a NAME="E19:112_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="#E16:112_8_2_1_2"><span class="lab"><font color="Green" title="E183">A129</font></span></a>, <a class="ref" href="glib_001.html#T31" onmouseover="rs('glib_001/T31')" onmouseout="rh()">GLIB_001:31</a></span>;<br/><span class="kw">hence </span><a NAME="E20:112_8_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E89">A2</font></span></a>, <a class="ref" href="chord.html#D8" target="_self" onmouseover="rs('chord/D8')" onmouseout="rh()">Def8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_8_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:112_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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">B</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:112_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" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E179">A130</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="#E100:112"><span class="lab"><font color="Green" title="E148">A63</font></span></a>, <a class="txt" href="#E89:112"><span class="lab"><font color="Green" title="E137">A52</font></span></a>, <a class="txt" href="#E127:112"><span class="lab"><font color="Green" title="E166">A96</font></span></a>, <a class="txt" href="#E1:112_8_2"><span class="lab"><font color="Green" title="E170">A102</font></span></a>, <a class="ref" href="glib_001.html#D28" onmouseover="rs('glib_001/D28')" onmouseout="rh()">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="#E74:112"><span class="lab"><font color="Green" title="E128">A43</font></span></a>, <a class="txt" href="#E77:112"><span class="lab"><font color="Green" title="E130">A45</font></span></a></span>
;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="112_8_2_1_3_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c47">n11</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="c40">Pbg</font></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="E1:112_8_2_1_3_2"/><span class="lab"><font color="Green" title="E180">A131</font></span>: 
<font color="Maroon" title="c47">n11</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="c40">Pbg</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_8_2_1_3_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:112_8_2_1_3_2_1"/>
( <font color="Maroon" title="c47">n11</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="c40">Pbg</font> or <font color="Maroon" title="c47">n11</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="c40">Pbg</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_8_2_1_3_2"><span class="lab"><font color="Green" title="E180">A131</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_8_2_1_3_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:112_8_2_1_3_2_1_1"/><span class="lab"><font color="Green" title="E181">A132</font></span>: 
<font color="Maroon" title="c47">n11</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="c40">Pbg</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:112_8_2_1_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E182">A133</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E3:112_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" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:112_8_2_1_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E183">A134</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#K1" title="NAT_1:func.1">+</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" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E5:112_8_2_1_3_2_1_1"/>
<font color="Maroon" title="c47">n11</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="c40">Pbg</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:112_8_2_1_3_2_1_1"><span class="lab"><font color="Green" title="E182">A133</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E6:112_8_2_1_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E184">A135</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" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E7:112_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#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="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="#E1:112_8_2_1_3_2_1_1"><span class="lab"><font color="Green" title="E181">A132</font></span></a>, <a class="ref" href="glib_001.html#D3" onmouseover="rs('glib_001/D3')" onmouseout="rh()">GLIB_001:def 3</a></span>;<br/><a NAME="E8:112_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#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="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>
 ;<br/><a NAME="E9:112_8_2_1_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E185">A136</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#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="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" onmouseover="rs('glib_000/T72')" onmouseout="rh()">GLIB_000:72</a></span>;<br/><a NAME="E10:112_8_2_1_3_2_1_1"/><span class="lab"><font color="Green" title="E186">A137</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#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">n11</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" onmouseover="rs('abian/T12')" onmouseout="rh()">ABIAN:12</a>, <a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a></span>;<br/><a NAME="E11:112_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#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"><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#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">n11</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E104:112"><span class="lab"><font color="Green" title="E152">A82</font></span></a>, <a class="txt" href="#E89:112"><span class="lab"><font color="Green" title="E137">A52</font></span></a>, <a class="txt" href="#E93:112"><span class="lab"><font color="Green" title="E141">A56</font></span></a>, <a class="txt" href="#E83:112"><span class="lab"><font color="Green" title="E134">A49</font></span></a>, <a class="txt" href="#E6:112_8_2_1_3_2_1_1"><span class="lab"><font color="Green" title="E184">A135</font></span></a>, <a class="ref" href="glib_001.html#T33" onmouseover="rs('glib_001/T33')" onmouseout="rh()">GLIB_001:33</a></span>;<br/><a NAME="E12:112_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI: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="#E105:112"><span class="lab"><font color="Green" title="E153">A83</font></span></a>, <a class="txt" href="#E10:112_8_2_1_3_2_1_1"><span class="lab"><font color="Green" title="E186">A137</font></span></a>, <a class="txt" href="#E4:112_8_2_1_3_2_1_1"><span class="lab"><font color="Green" title="E183">A134</font></span></a>, <a class="ref" href="glib_001.html#T99" onmouseover="rs('glib_001/T99')" onmouseout="rh()">GLIB_001:99</a></span>;<br/><span class="kw">hence </span><a NAME="E13:112_8_2_1_3_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E122:112"><span class="lab"><font color="Green" title="E162">A92</font></span></a>, <a class="txt" href="#E4:112_8_2"><span class="lab"><font color="Green" title="E172">A104</font></span></a>, <a class="txt" href="#E9:112_8_2"><span class="lab"><font color="Green" title="E175">A107</font></span></a>, <a class="txt" href="#E10:112_8_2"><span class="lab"><font color="Green" title="E176">A108</font></span></a>, <a class="txt" href="#E3:112_8_2_1_3"><span class="lab"><font color="Green" title="E179">A130</font></span></a>, <a class="txt" href="#E9:112_8_2_1_3_2_1_1"><span class="lab"><font color="Green" title="E185">A136</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_8_2_1_3_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:112_8_2_1_3_2_1_2"/>
<font color="Maroon" title="c47">n11</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="c40">Pbg</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:112_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" onmouseover="rs('chord/T4')" onmouseout="rh()">Th4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_8_2_1_3_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E120:112"><span class="lab"><font color="Green" title="E160">A90</font></span></a>, <a class="txt" href="#E4:112_8_2"><span class="lab"><font color="Green" title="E172">A104</font></span></a>, <a class="txt" href="#E9:112_8_2"><span class="lab"><font color="Green" title="E175">A107</font></span></a>, <a class="txt" href="#E10:112_8_2"><span class="lab"><font color="Green" title="E176">A108</font></span></a>, <a class="txt" href="#E13:112_8_2"><span class="lab"><font color="Green" title="E177">A109</font></span></a>, <a class="txt" href="#E3:112_8_2_1_3"><span class="lab"><font color="Green" title="E179">A130</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E5:112_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="#E97:112"><span class="lab"><font color="Green" title="E145">A60</font></span></a>, <a class="txt" href="#E126:112"><span class="lab"><font color="Green" title="E165">A95</font></span></a>, <a class="txt" href="#E4:112_8_2"><span class="lab"><font color="Green" title="E172">A104</font></span></a>, <a class="txt" href="#E9:112_8_2"><span class="lab"><font color="Green" title="E175">A107</font></span></a>, <a class="txt" href="#E10:112_8_2"><span class="lab"><font color="Green" title="E176">A108</font></span></a>, <a class="txt" href="#E3:112_8_2_1_3"><span class="lab"><font color="Green" title="E179">A130</font></span></a>, <a class="ref" href="chord.html#T85" target="_self" onmouseover="rs('chord/T85')" onmouseout="rh()">Th84</a></span>;<br/><a NAME="E6:112_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="#E44:112"><span class="lab"><font color="Green" title="E116">A31</font></span></a>, <a class="ref" href="chord.html#T87" target="_self" onmouseover="rs('chord/T87')" onmouseout="rh()">Th86</a></span>;<br/><span class="kw">hence </span><a NAME="E7:112_8_2_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:112"><span class="lab"><font color="Green" title="E131">A46</font></span></a>, <a class="ref" href="chord.html#T89" target="_self" onmouseover="rs('chord/T89')" onmouseout="rh()">Th88</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_8_2_1_4"><span class="kw">suppose </span></a><a NAME="E1:112_8_2_1_4"/><span class="lab"><font color="Green" title="E179">A138</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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:112_8_2_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E180">A139</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" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><a NAME="E3:112_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="#E1:112_8_2_1_4"><span class="lab"><font color="Green" title="E179">A138</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><a NAME="E4:112_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="#E120:112"><span class="lab"><font color="Green" title="E160">A90</font></span></a>, <a class="txt" href="#E121:112"><span class="lab"><font color="Green" title="E161">A91</font></span></a>, <a class="txt" href="#E4:112_8_2"><span class="lab"><font color="Green" title="E172">A104</font></span></a>, <a class="txt" href="#E9:112_8_2"><span class="lab"><font color="Green" title="E175">A107</font></span></a>, <a class="txt" href="#E10:112_8_2"><span class="lab"><font color="Green" title="E176">A108</font></span></a>, <a class="txt" href="#E2:112_8_2_1_4"><span class="lab"><font color="Green" title="E180">A139</font></span></a>, <a class="ref" href="chord.html#D3" target="_self" onmouseover="rs('chord/D3')" onmouseout="rh()">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E5:112_8_2_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E91">A4</font></span></a>, <a class="txt" href="#E14:112"><span class="lab"><font color="Green" title="E98">A11</font></span></a>, <a class="ref" href="chord.html#T45" target="_self" onmouseover="rs('chord/T45')" onmouseout="rh()">Th44</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_8_3"><span class="kw">suppose </span></a><a NAME="E1:112_8_3"/><span class="lab"><font color="Green" title="E170">A140</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c45">m1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:112_8_3"/><span class="lab"><font color="Green" title="E171">A141</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="xcmplx_0.html#K2" title="XCMPLX_0: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" onmouseover="rs('nat_1/T10')" onmouseout="rh()">NAT_1:10</a></span>;<br/><a NAME="E4:112_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="#E123:112"><span class="lab"><font color="Green" title="E163">A93</font></span></a>, <a class="txt" href="#E1:112_8_3"><span class="lab"><font color="Green" title="E170">A140</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E5:112_8_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E172">A142</font></span>: 
 not <font color="Maroon" title="c43">n</font> <a href="tarski.html#R2" title="TARSKI: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" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E6:112_8_3"/>
<font color="Maroon" title="c43">n</font> <a href="tarski.html#R2" title="TARSKI: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="#E119:112"><span class="lab"><font color="Green" title="E159">A89</font></span></a>, <a class="txt" href="#E129:112"><span class="lab"><font color="Green" title="E168">A98</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E8:112_8_3"/><span class="lab"><font color="Green" title="E173">A143</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="E9:112_8_3"/><span class="lab"><font color="Green" title="E174">A144</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#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c46">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:112_8_3"><span class="lab"><font color="Green" title="E172">A142</font></span></a>, <a class="ref" href="glib_001.html#T34" onmouseover="rs('glib_001/T34')" onmouseout="rh()">GLIB_001:34</a></span>;<br/><a NAME="E10:112_8_3"/><span class="lab"><font color="Green" title="E175">A145</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E104:112"><span class="lab"><font color="Green" title="E152">A82</font></span></a>, <a class="txt" href="#E89:112"><span class="lab"><font color="Green" title="E137">A52</font></span></a>, <a class="txt" href="#E93:112"><span class="lab"><font color="Green" title="E141">A56</font></span></a>, <a class="txt" href="#E83:112"><span class="lab"><font color="Green" title="E134">A49</font></span></a>, <a class="txt" href="#E8:112_8_3"><span class="lab"><font color="Green" title="E173">A143</font></span></a>, <a class="ref" href="glib_001.html#T33" onmouseover="rs('glib_001/T33')" onmouseout="rh()">GLIB_001:33</a></span>;<br/><a NAME="E11:112_8_3"/><span class="lab"><font color="Green" title="E176">A146</font></span>: 
<font color="Maroon" title="c45">m1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><a NAME="E12:112_8_3"/><span class="lab"><font color="Green" title="E177">A147</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="#E123:112"><span class="lab"><font color="Green" title="E163">A93</font></span></a>, <a class="txt" href="#E3:112_8_3"><span class="lab"><font color="Green" title="E171">A141</font></span></a>, <a class="txt" href="#E9:112_8_3"><span class="lab"><font color="Green" title="E174">A144</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E13:112_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="#E8:112_8_3"><span class="lab"><font color="Green" title="E173">A143</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E14:112_8_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E178">A148</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="xcmplx_0.html#K2" title="XCMPLX_0: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="#E104:112"><span class="lab"><font color="Green" title="E152">A82</font></span></a>, <a class="txt" href="#E89:112"><span class="lab"><font color="Green" title="E137">A52</font></span></a>, <a class="txt" href="#E93:112"><span class="lab"><font color="Green" title="E141">A56</font></span></a>, <a class="txt" href="#E83:112"><span class="lab"><font color="Green" title="E134">A49</font></span></a>, <a class="txt" href="#E11:112_8_3"><span class="lab"><font color="Green" title="E176">A146</font></span></a>, <a class="ref" href="glib_001.html#T33" onmouseover="rs('glib_001/T33')" onmouseout="rh()">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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:112_8_3"><span class="lab"><font color="Green" title="E171">A141</font></span></a>, <a class="txt" href="#E9:112_8_3"><span class="lab"><font color="Green" title="E174">A144</font></span></a>, <a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">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#K1" title="NAT_1:func.1">+</a> 1, <font color="Maroon" title="c50">n11</font> = <font color="Maroon" title="c47">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/><a NAME="E17:112_8_3"/><span class="lab"><font color="Green" title="E179">A149</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="#E12:112_8_3"><span class="lab"><font color="Green" title="E177">A147</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E18:112_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="#E8:112_8_3"><span class="lab"><font color="Green" title="E173">A143</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E19:112_8_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E180">A150</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="#E17:112_8_3"><span class="lab"><font color="Green" title="E179">A149</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><div><span class="lab"><font color="Green" title="E181">A151</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="112_8_3_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c49">m11</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> <font color="Maroon" title="c50">n11</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:112_8_3_1"/><span class="lab"><font color="Green" title="E182">A152</font></span>: 
<font color="Maroon" title="c49">m11</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> <font color="Maroon" title="c50">n11</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_8_3_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:112_8_3_1_1"/>
( <font color="Maroon" title="c49">m11</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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="#E1:112_8_3_1"><span class="lab"><font color="Green" title="E182">A152</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_8_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:112_8_3_1_1_1"/><span class="lab"><font color="Green" title="E183">A153</font></span>: 
<font color="Maroon" title="c49">m11</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c50">n11</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:112_8_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c49">m11</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="c40">Pbg</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:112_8_3"><span class="lab"><font color="Green" title="E173">A143</font></span></a></span>;<br/><a NAME="E3:112_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" onmouseover="rs('xreal_1/T20')" onmouseout="rh()">XREAL_1:20</a></span>;<br/><a NAME="E4:112_8_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E184">A154</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#K1" title="NAT_1:func.1">+</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" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E5:112_8_3_1_1_1"/><span class="lab"><font color="Green" title="E185">A155</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#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c49">m11</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" onmouseover="rs('abian/T12')" onmouseout="rh()">ABIAN:12</a>, <a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a></span>;<br/><a NAME="E6:112_8_3_1_1_1"/><span class="lab"><font color="Green" title="E186">A156</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#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="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="#E19:112_8_3"><span class="lab"><font color="Green" title="E180">A150</font></span></a>, <a class="txt" href="#E1:112_8_3_1_1_1"><span class="lab"><font color="Green" title="E183">A153</font></span></a>, <a class="ref" href="glib_001.html#D3" onmouseover="rs('glib_001/D3')" onmouseout="rh()">GLIB_001:def 3</a></span>;<br/><a NAME="E7:112_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#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"><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#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c49">m11</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E104:112"><span class="lab"><font color="Green" title="E152">A82</font></span></a>, <a class="txt" href="#E89:112"><span class="lab"><font color="Green" title="E137">A52</font></span></a>, <a class="txt" href="#E93:112"><span class="lab"><font color="Green" title="E141">A56</font></span></a>, <a class="txt" href="#E83:112"><span class="lab"><font color="Green" title="E134">A49</font></span></a>, <a class="txt" href="#E19:112_8_3"><span class="lab"><font color="Green" title="E180">A150</font></span></a>, <a class="ref" href="glib_001.html#T33" onmouseover="rs('glib_001/T33')" onmouseout="rh()">GLIB_001:33</a></span>;<br/><a NAME="E8:112_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI: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="#E105:112"><span class="lab"><font color="Green" title="E153">A83</font></span></a>, <a class="txt" href="#E5:112_8_3_1_1_1"><span class="lab"><font color="Green" title="E185">A155</font></span></a>, <a class="txt" href="#E4:112_8_3_1_1_1"><span class="lab"><font color="Green" title="E184">A154</font></span></a>, <a class="ref" href="glib_001.html#T99" onmouseover="rs('glib_001/T99')" onmouseout="rh()">GLIB_001:99</a></span>;<br/><span class="kw">hence </span><a NAME="E9:112_8_3_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E122:112"><span class="lab"><font color="Green" title="E162">A92</font></span></a>, <a class="txt" href="#E3:112_8_3"><span class="lab"><font color="Green" title="E171">A141</font></span></a>, <a class="txt" href="#E9:112_8_3"><span class="lab"><font color="Green" title="E174">A144</font></span></a>, <a class="txt" href="#E10:112_8_3"><span class="lab"><font color="Green" title="E175">A145</font></span></a>, <a class="txt" href="#E14:112_8_3"><span class="lab"><font color="Green" title="E178">A148</font></span></a>, <a class="txt" href="#E6:112_8_3_1_1_1"><span class="lab"><font color="Green" title="E186">A156</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_8_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:112_8_3_1_1_2"/>
<font color="Maroon" title="c49">m11</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> <font color="Maroon" title="c50">n11</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:112_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" onmouseover="rs('chord/T4')" onmouseout="rh()">Th4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_8_3_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:112_8_3"><span class="lab"><font color="Green" title="E177">A147</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E21:112_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="#E8:112_8_3"><span class="lab"><font color="Green" title="E173">A143</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E22:112_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="#E97:112"><span class="lab"><font color="Green" title="E145">A60</font></span></a>, <a class="txt" href="#E121:112"><span class="lab"><font color="Green" title="E161">A91</font></span></a>, <a class="txt" href="#E3:112_8_3"><span class="lab"><font color="Green" title="E171">A141</font></span></a>, <a class="txt" href="#E9:112_8_3"><span class="lab"><font color="Green" title="E174">A144</font></span></a>, <a class="txt" href="#E10:112_8_3"><span class="lab"><font color="Green" title="E175">A145</font></span></a>, <a class="txt" href="#E14:112_8_3"><span class="lab"><font color="Green" title="E178">A148</font></span></a>, <a class="txt" href="#E20:112_8_3"><span class="lab"><font color="Green" title="E181">A151</font></span></a>, <a class="ref" href="chord.html#T85" target="_self" onmouseover="rs('chord/T85')" onmouseout="rh()">Th84</a></span>;<br/><a NAME="E23:112_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="#E44:112"><span class="lab"><font color="Green" title="E116">A31</font></span></a>, <a class="ref" href="chord.html#T87" target="_self" onmouseover="rs('chord/T87')" onmouseout="rh()">Th86</a></span>;<br/><span class="kw">hence </span><a NAME="E24:112_8_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:112"><span class="lab"><font color="Green" title="E131">A46</font></span></a>, <a class="ref" href="chord.html#T89" target="_self" onmouseover="rs('chord/T89')" onmouseout="rh()">Th88</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
