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<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="chord.html#V6" title="CHORD:attr.6">chordal</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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> holds <br/>for <font color="Olive" title="b4">H</font> being  <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Olive" title="b3">S</font><br/> for <font color="Olive" title="b5">a1</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b4">H</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">a1</font> holds <br/> ex <font color="Olive" title="b6">c</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="b6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Olive" title="b5">a1</font> &amp; ( for <font color="Olive" title="b7">x</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="b7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">S</font> holds <br/><font color="Olive" title="b6">c</font>,<font color="Olive" title="b7">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  ) )</span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp; not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">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="c2">a</font>,<font color="Maroon" title="c3">b</font>  st <font color="Olive" title="b1">S</font> is <a href="chord.html#V4" title="CHORD:attr.4">minimal</a> holds <br/>for <font color="Olive" title="b2">H</font> being  <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b3">a1</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b2">H</font>  st <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a1</font> holds <br/> ex <font color="Olive" title="b4">c</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="b4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Olive" title="b3">a1</font> &amp; ( for <font color="Olive" title="b5">x</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="b5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">S</font> holds <br/><font color="Olive" title="b4">c</font>,<font color="Olive" title="b5">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  ) ) )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:120"/><i><font color="Green" title="E92">A1</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 <b>and </b><br/><a NAME="E2:120"/><i><font color="Green" title="E93">A2</font></i>: 
not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c2">a</font>,<font color="Maroon" title="c3">b</font>  st <font color="Olive" title="b1">S</font> is <a href="chord.html#V4" title="CHORD:attr.4">minimal</a> holds <br/>for <font color="Olive" title="b2">H</font> being  <a href="glib_000.html#NM7" title="GLIB_000:NM.7">removeVertices</a> of <font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b3">a1</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b2">H</font>  st <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a1</font> holds <br/> ex <font color="Olive" title="b4">c</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="b4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Olive" title="b3">a1</font> &amp; ( for <font color="Olive" title="b5">x</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="b5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">S</font> holds <br/><font color="Olive" title="b4">c</font>,<font color="Olive" title="b5">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  ) )</span><br/>

<b>let </b><font color="Maroon" title="c4">S</font> be    <a href="chord.html#M2" title="CHORD:mode.2">VertexSeparator</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">S</font> is <a href="chord.html#V4" title="CHORD:attr.4">minimal</a> implies for <font color="Olive" title="b1">H</font> being  <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="c4">S</font><br/> for <font color="Olive" title="b2">a1</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">H</font>  st <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a1</font> holds <br/> ex <font color="Olive" title="b3">c</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="b3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Olive" title="b2">a1</font> &amp; ( for <font color="Olive" title="b4">x</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="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> holds <br/><font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  ) ) )</span><br/>

<b>assume </b><a NAME="E3:120"/><i><font color="Green" title="E94">A3</font></i>: 
<font color="Maroon" title="c4">S</font> is <a href="chord.html#V4" title="CHORD:attr.4">minimal</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">H</font> being  <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="c4">S</font><br/> for <font color="Olive" title="b2">a1</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">H</font>  st <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a1</font> holds <br/> ex <font color="Olive" title="b3">c</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="b3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Olive" title="b2">a1</font> &amp; ( for <font color="Olive" title="b4">x</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="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> holds <br/><font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  ) )</span><br/>

<b>let </b><font color="Maroon" title="c5">H</font> be   <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="c4">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a1</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c5">H</font>  st <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a1</font> holds <br/> ex <font color="Olive" title="b2">c</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="b2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Olive" title="b1">a1</font> &amp; ( for <font color="Olive" title="b3">x</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="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> holds <br/><font color="Olive" title="b2">c</font>,<font color="Olive" title="b3">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  ) )</span><br/><b>let </b><font color="Maroon" title="c6">a1</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c5">H</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a1</font> implies  ex <font color="Olive" title="b1">c</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">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c6">a1</font> &amp; ( for <font color="Olive" title="b2">x</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> holds <br/><font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  ) ) )</span><br/>



<b>assume </b><a NAME="E4:120"/><i><font color="Green" title="E95">A4</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">c</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">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c6">a1</font> &amp; ( for <font color="Olive" title="b2">x</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> holds <br/><font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  ) )</span><br/>

<b>assume </b><a NAME="E5:120"/><i><font color="Green" title="E96">A5</font></i>: 
for <font color="Olive" title="b1">c</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  holds <br/>(  not <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c6">a1</font> or  ex <font color="Olive" title="b2">x</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="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> &amp; not <font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:120_1"/>
( <font color="Maroon" title="c4">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c4">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_1"><b>suppose </b></a><a NAME="E1:120_1_1"/>
<font color="Maroon" title="c4">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:120_1_1"/><b>then </b>
for <font color="Olive" title="b1">x</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  holds <br/>(  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> or <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )
 ;<br/><a NAME="E3:120_1_1"/><b>then </b>
not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c6">a1</font>
 <b>by </b><i><a class="txt" href="chord.html#E5:120"><i><font color="Green" title="E96">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E4:120_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E4:120"><i><font color="Green" title="E95">A4</font></i></a>, <a class="ref" href="glib_002.html#T9" title="GLIB_002:th.9">GLIB_002:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2"><b>suppose </b></a><a NAME="E1:120_1_2"/>
not <font color="Maroon" title="c4">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c7">S</font> = <font color="Maroon" title="c4">S</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> ;<br/><b>set </b><font color="Maroon" title="c8">A</font> = <font color="Maroon" title="c5">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c6">a1</font>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  =  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">$1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">S</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c9">M</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) where <font color="Olive" title="b1">x</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c6">a1</font> </span>}</span> ;<br/><a NAME="E3:120_1_2"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c6">a1</font>
 <b>by </b><i><a class="txt" href="chord.html#E4:120"><i><font color="Green" title="E95">A4</font></i></a>, <a class="ref" href="glib_002.html#T9" title="GLIB_002:th.9">GLIB_002:9</a></i>;<br/><a NAME="E4:120_1_2"/><b>then </b><i><font color="Green" title="E97">A6</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">S</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) where <font color="Olive" title="b1">x</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c6">a1</font> </span>}</span> 
 ;<br/><a NAME="E5:120_1_2"/><i><font color="Green" title="E98">A7</font></i>: 
<font color="Maroon" title="c5">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c6">a1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/><a NAME="E6:120_1_2"/><i><font color="Green" title="E99">A8</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) where <font color="Olive" title="b1">x</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c6">a1</font> </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21" title="FRAENKEL:sch.21">FRAENKEL:sch 21</a>(<a class="txt" href="chord.html#E5:120_1_2"><i><font color="Green" title="E98">A7</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) where <font color="Olive" title="b1">x</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c6">a1</font> </span>}</span>  implies <font color="Maroon" title="c10">x</font> is <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> )</span><br/><b>assume </b><a NAME="E1:120_1_2_1"/><i><font color="Green" title="E100">A9</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) where <font color="Olive" title="b1">x</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c6">a1</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x</font> is <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a></span><br/><b>consider </b><font color="Maroon" title="c11">y</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:120_1_2_1"/><i><font color="Green" title="E101">A10</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">S</font></span>)</span>
 <b>and </b><br/><a NAME="E4:120_1_2_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c6">a1</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:120_1_2_1"><i><font color="Green" title="E100">A9</font></i></a></i>;<br/><b>thus </b><a NAME="E5:120_1_2_1"/>
<font color="Maroon" title="c10">x</font> is <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>
 <b>by </b><i><a class="txt" href="chord.html#E3:120_1_2_1"><i><font color="Green" title="E101">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c10">M</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) where <font color="Olive" title="b1">x</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c6">a1</font> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="membered.html#V6" title="MEMBERED:attr.6">natural-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="chord.html#E4:120_1_2"><i><font color="Green" title="E97">A6</font></i></a>, <a class="txt" href="chord.html#E6:120_1_2"><i><font color="Green" title="E99">A8</font></i></a>, <a class="ref" href="membered.html#D6" title="MEMBERED:def.6">MEMBERED:def 6</a></i>;<br/><a NAME="E9:120_1_2"/>
 <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">c</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E11:120_1_2"/><i><font color="Green" title="E100">A11</font></i>: 
 <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c10">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">S</font></span>)</span>
 <b>and </b><br/><a NAME="E12:120_1_2"/><i><font color="Green" title="E101">A12</font></i>: 
<font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c6">a1</font>
 ;<br/><b>consider </b><font color="Maroon" title="c12">y</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E14:120_1_2"/><i><font color="Green" title="E102">A13</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 <b>and </b><br/><a NAME="E15:120_1_2"/><i><font color="Green" title="E103">A14</font></i>: 
not <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c12">y</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="chord.html#E5:120"><i><font color="Green" title="E96">A5</font></i></a>, <a class="txt" href="chord.html#E12:120_1_2"><i><font color="Green" title="E101">A12</font></i></a></i>;<br/><a NAME="E16:120_1_2"/><i><font color="Green" title="E104">A15</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="chord.html#E1:120"><i><font color="Green" title="E92">A1</font></i></a>, <a class="txt" href="chord.html#E2:120"><i><font color="Green" title="E93">A2</font></i></a>, <a class="txt" href="chord.html#E4:120"><i><font color="Green" title="E95">A4</font></i></a>, <a class="ref" href="chord.html#T75" target="_self" title="CHORD:th.75">Th75</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">Ga</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c5">H</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c6">a1</font></span>)</span>;<br/><a NAME="E18:120_1_2"/><i><font color="Green" title="E105">A16</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c13">Ga</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c6">a1</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/><b>set </b><font color="Maroon" title="c14">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/><a NAME="E19:120_1_2"/>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c5">H</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/><b>then reconsider </b><font color="Maroon" title="c15">A</font> = <font color="Maroon" title="c5">H</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c6">a1</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> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E21:120_1_2"/>
not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:120"><i><font color="Green" title="E92">A1</font></i></a>, <a class="txt" href="chord.html#E2:120"><i><font color="Green" title="E93">A2</font></i></a>, <a class="ref" href="chord.html#D8" target="_self" title="CHORD:def.8">Def8</a></i>;<br/><a NAME="E22:120_1_2"/><b>then </b><i><font color="Green" title="E106">A17</font></i>: 
<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="c7">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>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A</font> implies <font color="Maroon" title="c16">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="c7">S</font> )</span><br/><b>assume </b><a NAME="E1:120_1_2_2"/><i><font color="Green" title="E107">A18</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">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="c7">S</font></span><br/><a NAME="E2:120_1_2_2"/>
not <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 <b>by </b><i><a class="txt" href="chord.html#E16:120_1_2"><i><font color="Green" title="E104">A15</font></i></a>, <a class="txt" href="chord.html#E1:120_1_2_2"><i><font color="Green" title="E107">A18</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:120_1_2_2"/>
<font color="Maroon" title="c16">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="c7">S</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:120_1_2_2"><i><font color="Green" title="E107">A18</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E24:120_1_2"/><b>then </b>
<font color="Maroon" title="c15">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="c7">S</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c16">Ga</font> = <font color="Maroon" title="c13">Ga</font> 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="c15">A</font> <b>by </b><i><a class="txt" href="chord.html#E22:120_1_2"><i><font color="Green" title="E106">A17</font></i></a>, <a class="ref" href="chord.html#T29" target="_self" title="CHORD:th.29">Th29</a></i>;<br/><b>set </b><font color="Maroon" title="c17">Ay</font> = <font color="Maroon" title="c15">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="c12">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><b>consider </b><font color="Maroon" title="c18">Gay</font> being   <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="c15">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="c12">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/><a NAME="E27:120_1_2"/><i><font color="Green" title="E107">A19</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c18">Gay</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">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="c12">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/><a NAME="E28:120_1_2"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c12">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E29:120_1_2"/><b>then </b>
( <font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">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="c12">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">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="c12">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="chord.html#E12:120_1_2"><i><font color="Green" title="E101">A12</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c19">ca</font> = <font color="Maroon" title="c11">c</font>, <font color="Maroon" title="c20">ya</font> = <font color="Maroon" title="c12">y</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c18">Gay</font> <b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/><b>consider </b><font color="Maroon" title="c21">yaa</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E32:120_1_2"/><i><font color="Green" title="E108">A20</font></i>: 
<font color="Maroon" title="c21">yaa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A</font>
 <b>and </b><br/><a NAME="E33:120_1_2"/><i><font color="Green" title="E109">A21</font></i>: 
<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c21">yaa</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="chord.html#E1:120"><i><font color="Green" title="E92">A1</font></i></a>, <a class="txt" href="chord.html#E2:120"><i><font color="Green" title="E93">A2</font></i></a>, <a class="txt" href="chord.html#E3:120"><i><font color="Green" title="E94">A3</font></i></a>, <a class="txt" href="chord.html#E4:120"><i><font color="Green" title="E95">A4</font></i></a>, <a class="txt" href="chord.html#E14:120_1_2"><i><font color="Green" title="E102">A13</font></i></a>, <a class="ref" href="chord.html#T82" target="_self" title="CHORD:th.82">Th82</a></i>;<br/><a NAME="E34:120_1_2"/>
not <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A</font>
 <b>by </b><i><a class="txt" href="chord.html#E14:120_1_2"><i><font color="Green" title="E102">A13</font></i></a>, <a class="txt" href="chord.html#E16:120_1_2"><i><font color="Green" title="E104">A15</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E35:120_1_2"/><b>then </b>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c16">Ga</font></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E18:120_1_2"><i><font color="Green" title="E105">A16</font></i></a>, <a class="txt" href="chord.html#E32:120_1_2"><i><font color="Green" title="E108">A20</font></i></a>, <a class="txt" href="chord.html#E33:120_1_2"><i><font color="Green" title="E109">A21</font></i></a></i>;<br/><a NAME="E36:120_1_2"/><b>then </b>
<font color="Maroon" title="c18">Gay</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>
 <b>by </b><i><a class="ref" href="chord.html#T57" target="_self" title="CHORD:th.57">Th57</a></i>;<br/><b>then consider </b><font color="Maroon" title="c22">Wa</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c18">Gay</font><b> such that </b><br/><a NAME="E38:120_1_2"/><i><font color="Green" title="E110">A22</font></i>: 
<font color="Maroon" title="c22">Wa</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c19">ca</font>,<font color="Maroon" title="c20">ya</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" title="GLIB_002:def.1">GLIB_002:def 1</a></i>;<br/><a NAME="E39:120_1_2"/><i><font color="Green" title="E111">A23</font></i>: 
( <font color="Maroon" title="c22">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="c19">ca</font> &amp; <font color="Maroon" title="c22">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="c20">ya</font> )
 <b>by </b><i><a class="txt" href="chord.html#E38:120_1_2"><i><font color="Green" title="E110">A22</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>;<br/><b>consider </b><font color="Maroon" title="c23">P</font> being   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c18">Gay</font><b> such that </b><br/><a NAME="E41:120_1_2"/><i><font color="Green" title="E112">A24</font></i>: 
<font color="Maroon" title="c23">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c22">Wa</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c22">Wa</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <b>and </b><br/><a NAME="E42:120_1_2"/><i><font color="Green" title="E113">A25</font></i>: 
<font color="Maroon" title="c23">P</font> is <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>
 <b>by </b><i><a class="ref" href="chord.html#T39" target="_self" title="CHORD:th.39">Th39</a></i>;<br/><a NAME="E43:120_1_2"/><i><font color="Green" title="E114">A26</font></i>: 
<font color="Maroon" title="c23">P</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="c19">ca</font>
 <b>by </b><i><a class="txt" href="chord.html#E39:120_1_2"><i><font color="Green" title="E111">A23</font></i></a>, <a class="txt" href="chord.html#E41:120_1_2"><i><font color="Green" title="E112">A24</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>;<br/><a NAME="E44:120_1_2"/><i><font color="Green" title="E115">A27</font></i>: 
<font color="Maroon" title="c23">P</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="c12">y</font>
 <b>by </b><i><a class="txt" href="chord.html#E39:120_1_2"><i><font color="Green" title="E111">A23</font></i></a>, <a class="txt" href="chord.html#E41:120_1_2"><i><font color="Green" title="E112">A24</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>;<br/><a NAME="E45:120_1_2"/>
<font color="Maroon" title="c11">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y</font>
 <b>by </b><i><a class="txt" href="chord.html#E12:120_1_2"><i><font color="Green" title="E101">A12</font></i></a>, <a class="txt" href="chord.html#E14:120_1_2"><i><font color="Green" title="E102">A13</font></i></a>, <a class="txt" href="chord.html#E16:120_1_2"><i><font color="Green" title="E104">A15</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E46:120_1_2"/><b>then </b>
not <font color="Maroon" title="c23">P</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 <b>by </b><i><a class="txt" href="chord.html#E43:120_1_2"><i><font color="Green" title="E114">A26</font></i></a>, <a class="txt" href="chord.html#E44:120_1_2"><i><font color="Green" title="E115">A27</font></i></a>, <a class="ref" href="glib_001.html#T128" title="GLIB_001:th.128">GLIB_001:128</a></i>;<br/><a NAME="E47:120_1_2"/><b>then </b><i><font color="Green" title="E116">A28</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">P</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3
 <b>by </b><i><a class="ref" href="glib_001.html#T126" title="GLIB_001:th.126">GLIB_001:126</a></i>;<br/><div><i><font color="Green" title="E117">A29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:120_1_2_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">P</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:120_1_2_3"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">P</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 5 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <b>by </b><i><a class="ref" href="chord.html#T3" target="_self" title="CHORD:th.3">Th3</a></i>;<br/><a NAME="E3:120_1_2_3"/><b>then </b><i><font color="Green" title="E118">A30</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <b>by </b><i><a class="txt" href="chord.html#E47:120_1_2"><i><font color="Green" title="E116">A28</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:120_1_2_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">P</font>
 ;<br/><a NAME="E5:120_1_2_3"/><b>then </b>
<font color="Maroon" title="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1,<font color="Maroon" title="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c18">Gay</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></i>;<br/><a NAME="E6:120_1_2_3"/><b>then </b>
<font color="Maroon" title="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E43:120_1_2"><i><font color="Green" title="E114">A26</font></i></a>, <a class="txt" href="chord.html#E44:120_1_2"><i><font color="Green" title="E115">A27</font></i></a>, <a class="txt" href="chord.html#E3:120_1_2_3"><i><font color="Green" title="E118">A30</font></i></a>, <a class="ref" href="glib_000.html#T75" title="GLIB_000:th.75">GLIB_000:75</a></i>;<br/><b>hence </b><a NAME="E7:120_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E15:120_1_2"><i><font color="Green" title="E103">A14</font></i></a>, <a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E49:120_1_2"/><b>then </b>
5 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 2</span>)</span> <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="c23">P</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c24">j</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon" title="c25">d</font> = <font color="Maroon" title="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font>;<br/><a NAME="E51:120_1_2"/><i><font color="Green" title="E118">A31</font></i>: 
<font color="Maroon" title="c24">j</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="c23">P</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></i>;<br/><a NAME="E52:120_1_2"/><b>then </b><i><font color="Green" title="E119">A32</font></i>: 
<font color="Maroon" title="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c18">Gay</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T8" title="GLIB_001:th.8">GLIB_001:8</a></i>;<br/><a NAME="E53:120_1_2"/><b>then </b><i><font color="Green" title="E120">A33</font></i>: 
( <font color="Maroon" title="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A</font> or <font color="Maroon" title="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c12">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="chord.html#E27:120_1_2"><i><font color="Green" title="E107">A19</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div><i><font color="Green" title="E121">A34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:120_1_2_4"/>
<font color="Maroon" title="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:120_1_2_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="chord.html#E44:120_1_2"><i><font color="Green" title="E115">A27</font></i></a>, <a class="txt" href="chord.html#E51:120_1_2"><i><font color="Green" title="E118">A31</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/><a NAME="E3:120_1_2_4"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; 3 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 5 )
 ;<br/><b>hence </b><a NAME="E4:120_1_2_4"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E48:120_1_2"><i><font color="Green" title="E117">A29</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c26">d</font> = <font color="Maroon" title="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="chord.html#E52:120_1_2"><i><font color="Green" title="E119">A32</font></i></a></i>;<br/><a NAME="E56:120_1_2"/>
<font color="Maroon" title="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c26">d</font>,<font color="Maroon" title="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c18">Gay</font>
 <b>by </b><i><a class="txt" href="chord.html#E51:120_1_2"><i><font color="Green" title="E118">A31</font></i></a>, <a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></i>;<br/><a NAME="E57:120_1_2"/><b>then </b><i><font color="Green" title="E122">A35</font></i>: 
<font color="Maroon" title="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c26">d</font>,<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E44:120_1_2"><i><font color="Green" title="E115">A27</font></i></a>, <a class="ref" href="glib_000.html#T75" title="GLIB_000:th.75">GLIB_000:75</a></i>;<br/><a NAME="E58:120_1_2"/><b>then </b>
( <font color="Maroon" title="c26">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y</font> &amp; <font color="Maroon" title="c26">d</font>,<font color="Maroon" title="c12">y</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )
 <b>by </b><i><a class="txt" href="chord.html#E54:120_1_2"><i><font color="Green" title="E121">A34</font></i></a>, <a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></i>;<br/><a NAME="E59:120_1_2"/><b>then </b><i><font color="Green" title="E123">A36</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">d</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="chord.html#T52" target="_self" title="CHORD:th.52">Th52</a></i>;<br/><a NAME="E60:120_1_2"/><b>then </b><i><font color="Green" title="E124">A37</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">d</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">S</font>
 <b>by </b><i><a class="txt" href="chord.html#E14:120_1_2"><i><font color="Green" title="E102">A13</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E61:120_1_2"/><i><font color="Green" title="E125">A38</font></i>: 
not <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="chord.html#E15:120_1_2"><i><font color="Green" title="E103">A14</font></i></a>, <a class="ref" href="chord.html#T52" target="_self" title="CHORD:th.52">Th52</a></i>;<br/><b>set </b><font color="Maroon" title="c27">gds</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">d</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">S</font>;<br/><b>set </b><font color="Maroon" title="c28">gcs</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">S</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:120_1_2_5"/><i><font color="Green" title="E126">A39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">d</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:120_1_2_5"/>
not <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">S</font>
 <b>by </b><i><a class="txt" href="chord.html#E61:120_1_2"><i><font color="Green" title="E125">A38</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:120_1_2_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">S</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">d</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">S</font>
 <b>by </b><i><a class="txt" href="chord.html#E60:120_1_2"><i><font color="Green" title="E124">A37</font></i></a>, <a class="txt" href="chord.html#E1:120_1_2_5"><i><font color="Green" title="E126">A39</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E4:120_1_2_5"/><b>then </b><i><font color="Green" title="E127">A40</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">S</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">d</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">S</font></span>)</span>
 <b>by </b><i><a class="ref" href="trees_1.html#T24" title="TREES_1:th.24">TREES_1:24</a></i>;<br/><a NAME="E5:120_1_2_5"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">d</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">S</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 <b>by </b><i><a class="txt" href="chord.html#E53:120_1_2"><i><font color="Green" title="E120">A33</font></i></a>, <a class="txt" href="chord.html#E54:120_1_2"><i><font color="Green" title="E121">A34</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:120_1_2_5"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E11:120_1_2"><i><font color="Green" title="E100">A11</font></i></a>, <a class="txt" href="chord.html#E4:120_1_2_5"><i><font color="Green" title="E127">A40</font></i></a>, <a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c29">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E64:120_1_2"/><i><font color="Green" title="E126">A41</font></i>: 
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">S</font>
 <b>and </b><br/><a NAME="E65:120_1_2"/><i><font color="Green" title="E127">A42</font></i>: 
not <font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">d</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">S</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E66:120_1_2"/><i><font color="Green" title="E128">A43</font></i>: 
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 <b>by </b><i><a class="txt" href="chord.html#E64:120_1_2"><i><font color="Green" title="E126">A41</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E67:120_1_2"/><b>then </b><i><font color="Green" title="E129">A44</font></i>: 
not <font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">d</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="chord.html#E65:120_1_2"><i><font color="Green" title="E127">A42</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c30">x</font> = <font color="Maroon" title="c29">x</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="chord.html#E64:120_1_2"><i><font color="Green" title="E126">A41</font></i></a></i>;<br/><a NAME="E69:120_1_2"/><i><font color="Green" title="E130">A45</font></i>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y</font>
 <b>by </b><i><a class="txt" href="chord.html#E14:120_1_2"><i><font color="Green" title="E102">A13</font></i></a>, <a class="txt" href="chord.html#E59:120_1_2"><i><font color="Green" title="E123">A36</font></i></a>, <a class="txt" href="chord.html#E65:120_1_2"><i><font color="Green" title="E127">A42</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c31">Gs</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="c7">S</font>;<br/><b>reconsider </b><font color="Maroon" title="c32">xs</font> = <font color="Maroon" title="c30">x</font>, <font color="Maroon" title="c33">ys</font> = <font color="Maroon" title="c12">y</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c31">Gs</font> <b>by </b><i><a class="txt" href="chord.html#E14:120_1_2"><i><font color="Green" title="E102">A13</font></i></a>, <a class="txt" href="chord.html#E66:120_1_2"><i><font color="Green" title="E128">A43</font></i></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/><a NAME="E72:120_1_2"/>
<font color="Maroon" title="c31">Gs</font> is <a href="chord.html#V2" title="CHORD:attr.2">complete</a>
 <b>by </b><i><a class="txt" href="chord.html#E1:120"><i><font color="Green" title="E92">A1</font></i></a>, <a class="txt" href="chord.html#E2:120"><i><font color="Green" title="E93">A2</font></i></a>, <a class="txt" href="chord.html#E3:120"><i><font color="Green" title="E94">A3</font></i></a>, <a class="ref" href="chord.html#T98" target="_self" title="CHORD:th.98">Th98</a></i>;<br/><a NAME="E73:120_1_2"/><b>then </b>
<font color="Maroon" title="c32">xs</font>,<font color="Maroon" title="c33">ys</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="chord.html#E69:120_1_2"><i><font color="Green" title="E130">A45</font></i></a>, <a class="ref" href="chord.html#D6" target="_self" title="CHORD:def.6">Def6</a></i>;<br/><b>then consider </b><font color="Maroon" title="c34">ej</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E75:120_1_2"/><i><font color="Green" title="E131">A46</font></i>: 
<font color="Maroon" title="c34">ej</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c32">xs</font>,<font color="Maroon" title="c33">ys</font>,<font color="Maroon" title="c31">Gs</font>
 <b>by </b><i><a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></i>;<br/><a NAME="E76:120_1_2"/>
<font color="Maroon" title="c34">ej</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c30">x</font>,<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E75:120_1_2"><i><font color="Green" title="E131">A46</font></i></a>, <a class="ref" href="glib_000.html#T75" title="GLIB_000:th.75">GLIB_000:75</a></i>;<br/><a NAME="E77:120_1_2"/><b>then </b><i><font color="Green" title="E132">A47</font></i>: 
<font color="Maroon" title="c30">x</font>,<font color="Maroon" title="c12">y</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></i>;<br/><a NAME="E78:120_1_2"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="chord.html#E64:120_1_2"><i><font color="Green" title="E126">A41</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E79:120_1_2"/><b>then </b><i><font color="Green" title="E133">A48</font></i>: 
<font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c30">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="ref" href="chord.html#T52" target="_self" title="CHORD:th.52">Th52</a></i>;<br/><a NAME="E80:120_1_2"/>
<font color="Maroon" title="c26">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c30">x</font>
 <b>by </b><i><a class="txt" href="chord.html#E16:120_1_2"><i><font color="Green" title="E104">A15</font></i></a>, <a class="txt" href="chord.html#E53:120_1_2"><i><font color="Green" title="E120">A33</font></i></a>, <a class="txt" href="chord.html#E54:120_1_2"><i><font color="Green" title="E121">A34</font></i></a>, <a class="txt" href="chord.html#E66:120_1_2"><i><font color="Green" title="E128">A43</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E81:120_1_2"/><b>then </b><i><font color="Green" title="E134">A49</font></i>: 
not <font color="Maroon" title="c26">d</font>,<font color="Maroon" title="c30">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="chord.html#E67:120_1_2"><i><font color="Green" title="E129">A44</font></i></a>, <a class="ref" href="chord.html#T52" target="_self" title="CHORD:th.52">Th52</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">P</font> &amp; not $1 is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; $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="c23">P</font> &amp;  ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c30">x</font>,<font color="Maroon" title="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1,<font color="Maroon" title="c1">G</font> );<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>() <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">P</font>;<br/><a NAME="E82:120_1_2"/><i><font color="Green" title="E135">A50</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">H<sub>2</sub></font>()
 ;<br/><a NAME="E83:120_1_2"/><i><font color="Green" title="E136">A51</font></i>: 
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="c23">P</font>
 <b>by </b><i><a class="txt" href="chord.html#E48:120_1_2"><i><font color="Green" title="E117">A29</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E84:120_1_2"/><b>then </b><i><font color="Green" title="E137">A52</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">P</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E85:120_1_2"/><i><font color="Green" title="E138">A53</font></i>: 
not <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ;<br/><a NAME="E86:120_1_2"/>
 ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c30">x</font>,<font color="Maroon" title="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E43:120_1_2"><i><font color="Green" title="E114">A26</font></i></a>, <a class="txt" href="chord.html#E79:120_1_2"><i><font color="Green" title="E133">A48</font></i></a>, <a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></i>;<br/><a NAME="E87:120_1_2"/><b>then </b><i><font color="Green" title="E139">A54</font></i>: 
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>by </b><i><a class="txt" href="chord.html#E83:120_1_2"><i><font color="Green" title="E136">A51</font></i></a>, <a class="txt" href="chord.html#E84:120_1_2"><i><font color="Green" title="E137">A52</font></i></a>, <a class="txt" href="chord.html#E85:120_1_2"><i><font color="Green" title="E138">A53</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c35">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E89:120_1_2"/><i><font color="Green" title="E140">A55</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c35">k</font>]
 <b>and </b><br/><a NAME="E90:120_1_2"/><i><font color="Green" title="E141">A56</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon" title="c35">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">i</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="chord.html#E82:120_1_2"><i><font color="Green" title="E135">A50</font></i></a>, <a class="txt" href="chord.html#E87:120_1_2"><i><font color="Green" title="E139">A54</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c36">k</font> = <font color="Maroon" title="c35">k</font> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="chord.html#E89:120_1_2"><i><font color="Green" title="E140">A55</font></i></a></i>;<br/><a NAME="E92:120_1_2"/><i><font color="Green" title="E142">A57</font></i>: 
( <font color="Maroon" title="c36">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">j</font> &amp; <font color="Maroon" title="c24">j</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="c23">P</font> )
 <b>by </b><i><a class="txt" href="chord.html#E89:120_1_2"><i><font color="Green" title="E140">A55</font></i></a>, <a class="ref" href="chord.html#T3" target="_self" title="CHORD:th.3">Th3</a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></i>;<br/><a NAME="E93:120_1_2"/><i><font color="Green" title="E143">A58</font></i>: 
<font color="Maroon" title="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font>
 <b>by </b><i><a class="txt" href="chord.html#E81:120_1_2"><i><font color="Green" title="E134">A49</font></i></a>, <a class="txt" href="chord.html#E89:120_1_2"><i><font color="Green" title="E140">A55</font></i></a>, <a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></i>;<br/><div><i><font color="Green" title="E144">A59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:120_1_2_6"/><i><font color="Green" title="E145">A60</font></i>: 
<font color="Maroon" title="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">P</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="c23">P</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:120_1_2_6"/><i><font color="Green" title="E146">A61</font></i>: 
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">k</font> &amp; <font color="Maroon" title="c36">k</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="c23">P</font> )
 <b>by </b><i><a class="txt" href="chord.html#E89:120_1_2"><i><font color="Green" title="E140">A55</font></i></a>, <a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E3:120_1_2_6"/><b>then </b><i><font color="Green" title="E147">A62</font></i>: 
<font color="Maroon" title="c23">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c36">k</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c23">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c23">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E1:120_1_2_6"><i><font color="Green" title="E145">A60</font></i></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></i>;<br/><a NAME="E4:120_1_2_6"/>
<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="c23">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c36">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="chord.html#E2:120_1_2_6"><i><font color="Green" title="E146">A61</font></i></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></i>;<br/><b>hence </b><a NAME="E5:120_1_2_6"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E42:120_1_2"><i><font color="Green" title="E113">A25</font></i></a>, <a class="txt" href="chord.html#E89:120_1_2"><i><font color="Green" title="E140">A55</font></i></a>, <a class="txt" href="chord.html#E3:120_1_2_6"><i><font color="Green" title="E147">A62</font></i></a>, <a class="ref" href="chord.html#D2" target="_self" title="CHORD:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>set </b><font color="Maroon" title="c37">Q1</font> = <font color="Maroon" title="c23">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c36">k</font>,<font color="Maroon" title="c24">j</font>;<br/><a NAME="E95:120_1_2"/><i><font color="Green" title="E145">A63</font></i>: 
<font color="Maroon" title="c23">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c36">k</font>,<font color="Maroon" title="c24">j</font> is <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>
 <b>by </b><i><a class="txt" href="chord.html#E42:120_1_2"><i><font color="Green" title="E113">A25</font></i></a>, <a class="txt" href="chord.html#E92:120_1_2"><i><font color="Green" title="E142">A57</font></i></a>, <a class="ref" href="chord.html#T42" target="_self" title="CHORD:th.42">Th42</a></i>;<br/><a NAME="E96:120_1_2"/><i><font color="Green" title="E146">A64</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c23">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c36">k</font>,<font color="Maroon" title="c24">j</font></span>)</span> <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="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c23">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c36">k</font>,<font color="Maroon" title="c24">j</font></span>)</span> <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="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font> )
 <b>by </b><i><a class="txt" href="chord.html#E92:120_1_2"><i><font color="Green" title="E142">A57</font></i></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c38">Q</font> = <font color="Maroon" title="c23">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c36">k</font>,<font color="Maroon" title="c24">j</font> as   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="glib_001.html#T176" title="GLIB_001:th.176">GLIB_001:176</a></i>;<br/><a NAME="E98:120_1_2"/><i><font color="Green" title="E147">A65</font></i>: 
<font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c36">k</font>,<font color="Maroon" title="c24">j</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#T99" title="GLIB_001:th.99">GLIB_001:99</a></i>;<br/><a NAME="E99:120_1_2"/><i><font color="Green" title="E148">A66</font></i>: 
<font color="Maroon" title="c38">Q</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="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">k</font>
 <b>by </b><i><a class="txt" href="chord.html#E96:120_1_2"><i><font color="Green" title="E146">A64</font></i></a></i>;<br/><a NAME="E100:120_1_2"/><i><font color="Green" title="E149">A67</font></i>: 
<font color="Maroon" title="c38">Q</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="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font>
 <b>by </b><i><a class="txt" href="chord.html#E96:120_1_2"><i><font color="Green" title="E146">A64</font></i></a></i>;<br/><a NAME="E101:120_1_2"/><i><font color="Green" title="E150">A68</font></i>: 
<font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E81:120_1_2"><i><font color="Green" title="E134">A49</font></i></a>, <a class="txt" href="chord.html#E89:120_1_2"><i><font color="Green" title="E140">A55</font></i></a>, <a class="txt" href="chord.html#E96:120_1_2"><i><font color="Green" title="E146">A64</font></i></a>, <a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></i>;<br/><a NAME="E102:120_1_2"/><i><font color="Green" title="E151">A69</font></i>: 
not <font color="Maroon" title="c38">Q</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 <b>by </b><i><a class="txt" href="chord.html#E93:120_1_2"><i><font color="Green" title="E143">A58</font></i></a>, <a class="txt" href="chord.html#E99:120_1_2"><i><font color="Green" title="E148">A66</font></i></a>, <a class="txt" href="chord.html#E100:120_1_2"><i><font color="Green" title="E149">A67</font></i></a>, <a class="ref" href="glib_001.html#T128" title="GLIB_001:th.128">GLIB_001:128</a></i>;<br/><a NAME="E103:120_1_2"/><b>then </b>
( <font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c38">Q</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="glib_001.html#D26" title="GLIB_001:def.26">GLIB_001:def 26</a></i>;<br/><a NAME="E104:120_1_2"/><b>then </b><i><font color="Green" title="E152">A70</font></i>: 
<font color="Maroon" title="c38">Q</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E105:120_1_2"/><i><font color="Green" title="E153">A71</font></i>: 
not <font color="Maroon" title="c38">Q</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 <b>by </b><i><a class="txt" href="chord.html#E101:120_1_2"><i><font color="Green" title="E150">A68</font></i></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></i>;<br/><a NAME="E106:120_1_2"/><i><font color="Green" title="E154">A72</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c36">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="chord.html#E92:120_1_2"><i><font color="Green" title="E142">A57</font></i></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></i>;<br/><b>set </b><font color="Maroon" title="c39">cc</font> = <font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ;<br/><b>set </b><font color="Maroon" title="c40">dd</font> = <font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ;<br/><a NAME="E107:120_1_2"/><i><font color="Green" title="E155">A73</font></i>: 
( not <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A</font> &amp; not <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c12">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="chord.html#E16:120_1_2"><i><font color="Green" title="E104">A15</font></i></a>, <a class="txt" href="chord.html#E66:120_1_2"><i><font color="Green" title="E128">A43</font></i></a>, <a class="txt" href="chord.html#E69:120_1_2"><i><font color="Green" title="E130">A45</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E108:120_1_2"/>
<font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c36">k</font>,<font color="Maroon" title="c24">j</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E98:120_1_2"><i><font color="Green" title="E147">A65</font></i></a>, <a class="ref" href="glib_001.html#T89" title="GLIB_001:th.89">GLIB_001:89</a></i>;<br/><a NAME="E109:120_1_2"/><b>then </b><i><font color="Green" title="E156">A74</font></i>: 
<font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c30">x</font>
 <b>by </b><i><a class="txt" href="chord.html#E27:120_1_2"><i><font color="Green" title="E107">A19</font></i></a>, <a class="txt" href="chord.html#E107:120_1_2"><i><font color="Green" title="E155">A73</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E110:120_1_2"/>
<font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c36">k</font>,<font color="Maroon" title="c24">j</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E98:120_1_2"><i><font color="Green" title="E147">A65</font></i></a>, <a class="ref" href="glib_001.html#T89" title="GLIB_001:th.89">GLIB_001:89</a></i>;<br/><a NAME="E111:120_1_2"/><b>then </b><i><font color="Green" title="E157">A75</font></i>: 
<font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c30">x</font>
 <b>by </b><i><a class="txt" href="chord.html#E27:120_1_2"><i><font color="Green" title="E107">A19</font></i></a>, <a class="txt" href="chord.html#E107:120_1_2"><i><font color="Green" title="E155">A73</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E112:120_1_2"/><i><font color="Green" title="E158">A76</font></i>: 
<font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c12">y</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="chord.html#E57:120_1_2"><i><font color="Green" title="E122">A35</font></i></a>, <a class="txt" href="chord.html#E96:120_1_2"><i><font color="Green" title="E146">A64</font></i></a>, <a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></i>;<br/><a NAME="E113:120_1_2"/><i><font color="Green" title="E159">A77</font></i>: 
<font color="Maroon" title="c30">x</font>,<font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="chord.html#E89:120_1_2"><i><font color="Green" title="E140">A55</font></i></a>, <a class="txt" href="chord.html#E96:120_1_2"><i><font color="Green" title="E146">A64</font></i></a>, <a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></i>;<br/><div><i><font color="Green" title="E160">A78</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c41">i</font> be  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c41">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">Q</font> &amp; <font color="Maroon" title="c41">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">Q</font> implies for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c38">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">i</font>,<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c1">G</font> )</span><br/><b>assume </b><a NAME="E1:120_1_2_7"/><i><font color="Green" title="E161">A79</font></i>: 
( <font color="Maroon" title="c41">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">Q</font> &amp; <font color="Maroon" title="c41">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">Q</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c38">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">i</font>,<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c1">G</font></span><br/><b>assume </b><a NAME="E2:120_1_2_7"/>
 ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c38">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">i</font>,<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c42">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:120_1_2_7"/><i><font color="Green" title="E162">A80</font></i>: 
<font color="Maroon" title="c42">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c38">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">i</font>,<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c1">G</font>
 ;<br/><a NAME="E5:120_1_2_7"/><i><font color="Green" title="E163">A81</font></i>: 
<font color="Maroon" title="c41">i</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="c38">Q</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:120_1_2_7"><i><font color="Green" title="E161">A79</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:120_1_2_7"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c41">i</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:120_1_2_7"><i><font color="Green" title="E161">A79</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:120_1_2_7"/><b>then </b>
1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c41">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c43">i1</font> = <font color="Maroon" title="c41">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c44">ki1</font> = <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">i1</font> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E10:120_1_2_7"/><i><font color="Green" title="E164">A82</font></i>: 
<font color="Maroon" title="c41">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c41">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E11:120_1_2_7"/><b>then </b>
<font color="Maroon" title="c43">i1</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="c38">Q</font>
 <b>by </b><i><a class="txt" href="chord.html#E5:120_1_2_7"><i><font color="Green" title="E163">A81</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E12:120_1_2_7"/><b>then </b><i><font color="Green" title="E165">A83</font></i>: 
( <font color="Maroon" title="c38">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">ki1</font> &amp; <font color="Maroon" title="c44">ki1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">P</font> )
 <b>by </b><i><a class="txt" href="chord.html#E92:120_1_2"><i><font color="Green" title="E142">A57</font></i></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></i>;<br/><a NAME="E13:120_1_2_7"/><b>then </b><i><font color="Green" title="E166">A84</font></i>: 
<font color="Maroon" title="c44">ki1</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="c23">P</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:120_1_2_7_1"/><i><font color="Green" title="E167">A85</font></i>: 
<font color="Maroon" title="c44">ki1</font> <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="c23">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:120_1_2_7_1"/>
<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="c23">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c36">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <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="c23">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c36">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T32" title="XREAL_1:th.32">XREAL_1:32</a></i>;<br/><b>hence </b><a NAME="E3:120_1_2_7_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E106:120_1_2"><i><font color="Green" title="E154">A72</font></i></a>, <a class="txt" href="chord.html#E5:120_1_2_7"><i><font color="Green" title="E163">A81</font></i></a>, <a class="txt" href="chord.html#E10:120_1_2_7"><i><font color="Green" title="E164">A82</font></i></a>, <a class="txt" href="chord.html#E1:120_1_2_7_1"><i><font color="Green" title="E167">A85</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E15:120_1_2_7"/><b>then </b>
<font color="Maroon" title="c44">ki1</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="c23">P</font>
 <b>by </b><i><a class="txt" href="chord.html#E13:120_1_2_7"><i><font color="Green" title="E166">A84</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E16:120_1_2_7"/><b>then </b><i><font color="Green" title="E167">A86</font></i>: 
<font color="Maroon" title="c44">ki1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">P</font>
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></i>;<br/><a NAME="E17:120_1_2_7"/>
<font color="Maroon" title="c44">ki1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">P</font>
 <b>by </b><i><a class="txt" href="chord.html#E106:120_1_2"><i><font color="Green" title="E154">A72</font></i></a>, <a class="txt" href="chord.html#E1:120_1_2_7"><i><font color="Green" title="E161">A79</font></i></a></i>;<br/><a NAME="E18:120_1_2_7"/><b>then </b><i><font color="Green" title="E168">A87</font></i>: 
<font color="Maroon" title="c44">ki1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <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="c23">P</font>
 <b>by </b><i><a class="txt" href="chord.html#E16:120_1_2_7"><i><font color="Green" title="E167">A86</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E19:120_1_2_7"/>
( <font color="Maroon" title="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">ki1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">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="c12">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c23">P</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="c23">P</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">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="c12">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="chord.html#E27:120_1_2"><i><font color="Green" title="E107">A19</font></i></a>, <a class="txt" href="chord.html#E13:120_1_2_7"><i><font color="Green" title="E166">A84</font></i></a>, <a class="ref" href="glib_001.html#T8" title="GLIB_001:th.8">GLIB_001:8</a></i>;<br/><b>hence </b><a NAME="E20:120_1_2_7"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E42:120_1_2"><i><font color="Green" title="E113">A25</font></i></a>, <a class="txt" href="chord.html#E44:120_1_2"><i><font color="Green" title="E115">A27</font></i></a>, <a class="txt" href="chord.html#E4:120_1_2_7"><i><font color="Green" title="E162">A80</font></i></a>, <a class="txt" href="chord.html#E12:120_1_2_7"><i><font color="Green" title="E165">A83</font></i></a>, <a class="txt" href="chord.html#E18:120_1_2_7"><i><font color="Green" title="E168">A87</font></i></a>, <a class="ref" href="chord.html#T19" target="_self" title="CHORD:th.19">Th19</a>, <a class="ref" href="chord.html#T40" target="_self" title="CHORD:th.40">Th40</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><i><font color="Green" title="E161">A88</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c41">i</font> be  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c41">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">Q</font> &amp; <font color="Maroon" title="c41">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 implies for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c38">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">i</font>,<font color="Maroon" title="c30">x</font>,<font color="Maroon" title="c1">G</font> )</span><br/><b>assume </b><a NAME="E1:120_1_2_8"/><i><font color="Green" title="E162">A89</font></i>: 
( <font color="Maroon" title="c41">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">Q</font> &amp; <font color="Maroon" title="c41">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c38">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">i</font>,<font color="Maroon" title="c30">x</font>,<font color="Maroon" title="c1">G</font></span><br/><b>assume </b><a NAME="E2:120_1_2_8"/>
 ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c38">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">i</font>,<font color="Maroon" title="c30">x</font>,<font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c42">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:120_1_2_8"/><i><font color="Green" title="E163">A90</font></i>: 
<font color="Maroon" title="c42">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c38">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">i</font>,<font color="Maroon" title="c30">x</font>,<font color="Maroon" title="c1">G</font>
 ;<br/><a NAME="E5:120_1_2_8"/><i><font color="Green" title="E164">A91</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c41">i</font> &amp; <font color="Maroon" title="c41">i</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="c38">Q</font> )
 <b>by </b><i><a class="txt" href="chord.html#E1:120_1_2_8"><i><font color="Green" title="E162">A89</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:120_1_2_8"/><b>then </b>
1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c41">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c43">i1</font> = <font color="Maroon" title="c41">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c44">ki1</font> = <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">i1</font> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E9:120_1_2_8"/>
<font color="Maroon" title="c41">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c41">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E10:120_1_2_8"/><b>then </b><i><font color="Green" title="E165">A92</font></i>: 
<font color="Maroon" title="c43">i1</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="c38">Q</font>
 <b>by </b><i><a class="txt" href="chord.html#E5:120_1_2_8"><i><font color="Green" title="E164">A91</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E11:120_1_2_8"/><b>then </b><i><font color="Green" title="E166">A93</font></i>: 
( <font color="Maroon" title="c38">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">ki1</font> &amp; <font color="Maroon" title="c44">ki1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">P</font> )
 <b>by </b><i><a class="txt" href="chord.html#E92:120_1_2"><i><font color="Green" title="E142">A57</font></i></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></i>;<br/><a NAME="E12:120_1_2_8"/><i><font color="Green" title="E167">A94</font></i>: 
<font color="Maroon" title="c43">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c36">k</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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c36">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c36">k</font>
 <b>by </b><i><a class="txt" href="chord.html#E106:120_1_2"><i><font color="Green" title="E154">A72</font></i></a>, <a class="txt" href="chord.html#E10:120_1_2_8"><i><font color="Green" title="E165">A92</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E13:120_1_2_8"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">P</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <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="c23">P</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E14:120_1_2_8"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">ki1</font> &amp; <font color="Maroon" title="c44">ki1</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="c23">P</font> )
 <b>by </b><i><a class="txt" href="chord.html#E12:120_1_2_8"><i><font color="Green" title="E167">A94</font></i></a>, <a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E15:120_1_2_8"/><b>then </b><i><font color="Green" title="E168">A95</font></i>: 
( <font color="Maroon" title="c44">ki1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">P</font> &amp; <font color="Maroon" title="c44">ki1</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="c23">P</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:120_1_2_8_1"/>
<font color="Maroon" title="c43">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:120_1_2_8_1"/><b>then </b>
( <font color="Maroon" title="c43">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c43">i1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><a NAME="E3:120_1_2_8_1"/><b>then </b>
<font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">i1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E4:120_1_2_8_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E90:120_1_2"><i><font color="Green" title="E141">A56</font></i></a>, <a class="txt" href="chord.html#E4:120_1_2_8"><i><font color="Green" title="E163">A90</font></i></a>, <a class="txt" href="chord.html#E11:120_1_2_8"><i><font color="Green" title="E166">A93</font></i></a>, <a class="txt" href="chord.html#E15:120_1_2_8"><i><font color="Green" title="E168">A95</font></i></a>, <a class="ref" href="glib_000.html#T17" title="GLIB_000:th.17">GLIB_000:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E17:120_1_2_8"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E1:120_1_2_8"><i><font color="Green" title="E162">A89</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><i><font color="Green" title="E162">A96</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c41">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c41">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  implies <font color="Maroon" title="c41">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A</font> )</span><br/><b>assume </b><a NAME="E1:120_1_2_9"/><i><font color="Green" title="E163">A97</font></i>: 
<font color="Maroon" title="c41">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c41">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A</font></span><br/><a NAME="E2:120_1_2_9"/>
<font color="Maroon" title="c38">Q</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="c23">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E92:120_1_2"><i><font color="Green" title="E142">A57</font></i></a>, <a class="txt" href="chord.html#E98:120_1_2"><i><font color="Green" title="E147">A65</font></i></a>, <a class="ref" href="glib_001.html#T95" title="GLIB_001:th.95">GLIB_001:95</a></i>;<br/><a NAME="E3:120_1_2_9"/><b>then </b><i><font color="Green" title="E164">A98</font></i>: 
<font color="Maroon" title="c41">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E1:120_1_2_9"><i><font color="Green" title="E163">A97</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c42">n</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:120_1_2_9"/><i><font color="Green" title="E165">A99</font></i>: 
( <font color="Maroon" title="c42">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="c38">Q</font> &amp; <font color="Maroon" title="c38">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">v</font> )
 <b>by </b><i><a class="txt" href="chord.html#E1:120_1_2_9"><i><font color="Green" title="E163">A97</font></i></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/><a NAME="E6:120_1_2_9"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c42">n</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E7:120_1_2_9"/><b>then </b>
1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c42">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c43">n1</font> = <font color="Maroon" title="c42">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c44">kn1</font> = <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">n1</font> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E10:120_1_2_9"/>
<font color="Maroon" title="c42">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c42">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E11:120_1_2_9"/><b>then </b><i><font color="Green" title="E166">A100</font></i>: 
<font color="Maroon" title="c43">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="c38">Q</font>
 <b>by </b><i><a class="txt" href="chord.html#E5:120_1_2_9"><i><font color="Green" title="E165">A99</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E12:120_1_2_9"/><b>then </b><i><font color="Green" title="E167">A101</font></i>: 
( <font color="Maroon" title="c38">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">kn1</font> &amp; <font color="Maroon" title="c44">kn1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">P</font> )
 <b>by </b><i><a class="txt" href="chord.html#E92:120_1_2"><i><font color="Green" title="E142">A57</font></i></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></i>;<br/><a NAME="E13:120_1_2_9"/><i><font color="Green" title="E168">A102</font></i>: 
<font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">n1</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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c36">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c36">k</font>
 <b>by </b><i><a class="txt" href="chord.html#E106:120_1_2"><i><font color="Green" title="E154">A72</font></i></a>, <a class="txt" href="chord.html#E11:120_1_2_9"><i><font color="Green" title="E166">A100</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E14:120_1_2_9"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">P</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <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="c23">P</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E15:120_1_2_9"/><b>then </b><i><font color="Green" title="E169">A103</font></i>: 
<font color="Maroon" title="c44">kn1</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="c23">P</font>
 <b>by </b><i><a class="txt" href="chord.html#E13:120_1_2_9"><i><font color="Green" title="E168">A102</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_9_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:120_1_2_9_1"/><i><font color="Green" title="E170">A104</font></i>: 
<font color="Maroon" title="c41">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:120_1_2_9_1"/><b>then </b><i><font color="Green" title="E171">A105</font></i>: 
<font color="Maroon" title="c36">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="chord.html#E44:120_1_2"><i><font color="Green" title="E115">A27</font></i></a>, <a class="txt" href="chord.html#E5:120_1_2_9"><i><font color="Green" title="E165">A99</font></i></a>, <a class="txt" href="chord.html#E12:120_1_2_9"><i><font color="Green" title="E167">A101</font></i></a>, <a class="txt" href="chord.html#E15:120_1_2_9"><i><font color="Green" title="E169">A103</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/><a NAME="E3:120_1_2_9_1"/><i><font color="Green" title="E172">A106</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">k</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c42">n</font> )
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_9_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:120_1_2_9_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c42">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:120_1_2_9_1_1"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c42">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:120_1_2_9_1_1"/><b>then </b>
2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c42">n</font>
 <b>by </b><i><a class="txt" href="chord.html#E3:120_1_2_9_1"><i><font color="Green" title="E172">A106</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E4:120_1_2_9_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E2:120_1_2_9_1"><i><font color="Green" title="E171">A105</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E5:120_1_2_9_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E44:120_1_2"><i><font color="Green" title="E115">A27</font></i></a>, <a class="txt" href="chord.html#E94:120_1_2"><i><font color="Green" title="E144">A59</font></i></a>, <a class="txt" href="chord.html#E96:120_1_2"><i><font color="Green" title="E146">A64</font></i></a>, <a class="txt" href="chord.html#E5:120_1_2_9"><i><font color="Green" title="E165">A99</font></i></a>, <a class="txt" href="chord.html#E1:120_1_2_9_1"><i><font color="Green" title="E170">A104</font></i></a>, <a class="txt" href="chord.html#E3:120_1_2_9_1"><i><font color="Green" title="E172">A106</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E17:120_1_2_9"/><b>then </b>
not <font color="Maroon" title="c41">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c12">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E18:120_1_2_9"/>
<font color="Maroon" title="c41">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A</font>
 <b>by </b><i><a class="txt" href="chord.html#E27:120_1_2"><i><font color="Green" title="E107">A19</font></i></a>, <a class="txt" href="chord.html#E3:120_1_2_9"><i><font color="Green" title="E164">A98</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c41">R</font> being   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E118:120_1_2"/><i><font color="Green" title="E163">A107</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 7 &amp; <font color="Maroon" title="c41">R</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 <b>and </b><br/><a NAME="E119:120_1_2"/><i><font color="Green" title="E164">A108</font></i>: 
<font color="Maroon" title="c41">R</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span>,<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c30">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> </span>)</span></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
 <b>and </b><br/><a NAME="E120:120_1_2"/><i><font color="Green" title="E165">A109</font></i>: 
( <font color="Maroon" title="c41">R</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="c38">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  &amp; <font color="Maroon" title="c41">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y</font> &amp; <font color="Maroon" title="c41">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">x</font> &amp; <font color="Maroon" title="c41">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  )
 <b>by </b><i><a class="txt" href="chord.html#E44:120_1_2"><i><font color="Green" title="E115">A27</font></i></a>, <a class="txt" href="chord.html#E54:120_1_2"><i><font color="Green" title="E121">A34</font></i></a>, <a class="txt" href="chord.html#E69:120_1_2"><i><font color="Green" title="E130">A45</font></i></a>, <a class="txt" href="chord.html#E77:120_1_2"><i><font color="Green" title="E132">A47</font></i></a>, <a class="txt" href="chord.html#E94:120_1_2"><i><font color="Green" title="E144">A59</font></i></a>, <a class="txt" href="chord.html#E96:120_1_2"><i><font color="Green" title="E146">A64</font></i></a>, <a class="txt" href="chord.html#E109:120_1_2"><i><font color="Green" title="E156">A74</font></i></a>, <a class="txt" href="chord.html#E111:120_1_2"><i><font color="Green" title="E157">A75</font></i></a>, <a class="txt" href="chord.html#E112:120_1_2"><i><font color="Green" title="E158">A76</font></i></a>, <a class="txt" href="chord.html#E113:120_1_2"><i><font color="Green" title="E159">A77</font></i></a>, <a class="ref" href="chord.html#T48" target="_self" title="CHORD:th.48">Th48</a></i>;<br/><a NAME="E121:120_1_2"/><i><font color="Green" title="E166">A110</font></i>: 
<font color="Maroon" title="c41">R</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E120:120_1_2"><i><font color="Green" title="E165">A109</font></i></a></i>;<br/><a NAME="E122:120_1_2"/><i><font color="Green" title="E167">A111</font></i>: 
<font color="Maroon" title="c41">R</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E118:120_1_2"><i><font color="Green" title="E163">A107</font></i></a>, <a class="txt" href="chord.html#E120:120_1_2"><i><font color="Green" title="E165">A109</font></i></a></i>;<br/><a NAME="E123:120_1_2"/><i><font color="Green" title="E168">A112</font></i>: 
not <font color="Maroon" title="c41">R</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 <b>by </b><i><a class="txt" href="chord.html#E118:120_1_2"><i><font color="Green" title="E163">A107</font></i></a>, <a class="ref" href="glib_001.html#T127" title="GLIB_001:th.127">GLIB_001:127</a></i>;<br/><a NAME="E124:120_1_2"/><i><font color="Green" title="E169">A113</font></i>: 
not <font color="Maroon" title="c41">R</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 <b>by </b><i><a class="txt" href="chord.html#E101:120_1_2"><i><font color="Green" title="E150">A68</font></i></a>, <a class="txt" href="chord.html#E121:120_1_2"><i><font color="Green" title="E166">A110</font></i></a>, <a class="txt" href="chord.html#E122:120_1_2"><i><font color="Green" title="E167">A111</font></i></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></i>;<br/><a NAME="E125:120_1_2"/><i><font color="Green" title="E170">A114</font></i>: 
<font color="Maroon" title="c38">Q</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="c41">R</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_10"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:120_1_2_10"/>
not <font color="Maroon" title="c38">Q</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="c41">R</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:120_1_2_10"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</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="c41">R</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> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c42">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:120_1_2_10"/><i><font color="Green" title="E171">A115</font></i>: 
<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="c38">Q</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="c41">R</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E5:120_1_2_10"/><i><font color="Green" title="E172">A116</font></i>: 
<font color="Maroon" title="c42">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 
<b>by </b><i><a class="txt" href="chord.html#E4:120_1_2_10"><i><font color="Green" title="E171">A115</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:120_1_2_10"/>
<font color="Maroon" title="c42">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">R</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 
<b>by </b><i><a class="txt" href="chord.html#E4:120_1_2_10"><i><font color="Green" title="E171">A115</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c43">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E8:120_1_2_10"/><i><font color="Green" title="E173">A117</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">n</font> &amp; <font color="Maroon" title="c43">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 7 &amp; <font color="Maroon" title="c41">R</font> <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="c42">e</font> )
 <b>by </b><i><a class="txt" href="chord.html#E118:120_1_2"><i><font color="Green" title="E163">A107</font></i></a>, <a class="ref" href="glib_001.html#T100" title="GLIB_001:th.100">GLIB_001:100</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:120_1_2_10_1"/>
( <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6 )
 </span><b>by </b><i><a class="txt" href="chord.html#E8:120_1_2_10"><i><font color="Green" title="E173">A117</font></i></a>, <a class="ref" href="chord.html#T13" target="_self" title="CHORD:th.13">Th13</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_10_1_1"><b>suppose </b></a><a NAME="E1:120_1_2_10_1_1"/><i><font color="Green" title="E174">A118</font></i>: 
<font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:120_1_2_10_1_1"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">R</font>
 <b>by </b><i><a class="txt" href="chord.html#E118:120_1_2"><i><font color="Green" title="E163">A107</font></i></a></i>;<br/><a NAME="E3:120_1_2_10_1_1"/><b>then </b>
<font color="Maroon" title="c41">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c41">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1,<font color="Maroon" title="c41">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></i>;<br/><a NAME="E4:120_1_2_10_1_1"/><b>then </b>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E120:120_1_2"><i><font color="Green" title="E165">A109</font></i></a>, <a class="txt" href="chord.html#E5:120_1_2_10"><i><font color="Green" title="E172">A116</font></i></a>, <a class="txt" href="chord.html#E8:120_1_2_10"><i><font color="Green" title="E173">A117</font></i></a>, <a class="txt" href="chord.html#E1:120_1_2_10_1_1"><i><font color="Green" title="E174">A118</font></i></a>, <a class="ref" href="glib_001.html#T106" title="GLIB_001:th.106">GLIB_001:106</a></i>;<br/><a NAME="E5:120_1_2_10_1_1"/><b>then </b>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A</font>
 <b>by </b><i><a class="txt" href="chord.html#E116:120_1_2"><i><font color="Green" title="E162">A96</font></i></a></i>;<br/><b>hence </b><a NAME="E6:120_1_2_10_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E14:120_1_2"><i><font color="Green" title="E102">A13</font></i></a>, <a class="txt" href="chord.html#E16:120_1_2"><i><font color="Green" title="E104">A15</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_10_1_2"><b>suppose </b></a><a NAME="E1:120_1_2_10_1_2"/><i><font color="Green" title="E174">A119</font></i>: 
<font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:120_1_2_10_1_2"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">R</font>
 <b>by </b><i><a class="txt" href="chord.html#E118:120_1_2"><i><font color="Green" title="E163">A107</font></i></a></i>;<br/><a NAME="E3:120_1_2_10_1_2"/><b>then </b>
<font color="Maroon" title="c41">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c41">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3,<font color="Maroon" title="c41">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></i>;<br/><a NAME="E4:120_1_2_10_1_2"/><b>then </b>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E120:120_1_2"><i><font color="Green" title="E165">A109</font></i></a>, <a class="txt" href="chord.html#E5:120_1_2_10"><i><font color="Green" title="E172">A116</font></i></a>, <a class="txt" href="chord.html#E8:120_1_2_10"><i><font color="Green" title="E173">A117</font></i></a>, <a class="txt" href="chord.html#E1:120_1_2_10_1_2"><i><font color="Green" title="E174">A119</font></i></a>, <a class="ref" href="glib_001.html#T106" title="GLIB_001:th.106">GLIB_001:106</a></i>;<br/><a NAME="E5:120_1_2_10_1_2"/><b>then </b>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A</font>
 <b>by </b><i><a class="txt" href="chord.html#E116:120_1_2"><i><font color="Green" title="E162">A96</font></i></a></i>;<br/><b>hence </b><a NAME="E6:120_1_2_10_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E14:120_1_2"><i><font color="Green" title="E102">A13</font></i></a>, <a class="txt" href="chord.html#E16:120_1_2"><i><font color="Green" title="E104">A15</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_10_1_3"><b>suppose </b></a><a NAME="E1:120_1_2_10_1_3"/><i><font color="Green" title="E174">A120</font></i>: 
<font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:120_1_2_10_1_3"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">R</font>
 <b>by </b><i><a class="txt" href="chord.html#E118:120_1_2"><i><font color="Green" title="E163">A107</font></i></a></i>;<br/><a NAME="E3:120_1_2_10_1_3"/><b>then </b>
<font color="Maroon" title="c41">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c41">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 5,<font color="Maroon" title="c41">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></i>;<br/><a NAME="E4:120_1_2_10_1_3"/><b>then </b>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E120:120_1_2"><i><font color="Green" title="E165">A109</font></i></a>, <a class="txt" href="chord.html#E5:120_1_2_10"><i><font color="Green" title="E172">A116</font></i></a>, <a class="txt" href="chord.html#E8:120_1_2_10"><i><font color="Green" title="E173">A117</font></i></a>, <a class="txt" href="chord.html#E1:120_1_2_10_1_3"><i><font color="Green" title="E174">A120</font></i></a>, <a class="ref" href="glib_001.html#T106" title="GLIB_001:th.106">GLIB_001:106</a></i>;<br/><a NAME="E5:120_1_2_10_1_3"/><b>then </b>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A</font>
 <b>by </b><i><a class="txt" href="chord.html#E116:120_1_2"><i><font color="Green" title="E162">A96</font></i></a></i>;<br/><b>hence </b><a NAME="E6:120_1_2_10_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E16:120_1_2"><i><font color="Green" title="E104">A15</font></i></a>, <a class="txt" href="chord.html#E66:120_1_2"><i><font color="Green" title="E128">A43</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c42">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c42">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</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="c41">R</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> implies <font color="Maroon" title="c42">v</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"><span class="p2">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )</span><br/><b>assume </b><a NAME="E1:120_1_2_11"/><i><font color="Green" title="E171">A121</font></i>: 
<font color="Maroon" title="c42">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</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="c41">R</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c42">v</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"><span class="p2">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/><a NAME="E2:120_1_2_11"/>
<font color="Maroon" title="c42">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E1:120_1_2_11"><i><font color="Green" title="E171">A121</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:120_1_2_11"/><b>then </b><i><font color="Green" title="E172">A122</font></i>: 
<font color="Maroon" title="c42">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A</font>
 <b>by </b><i><a class="txt" href="chord.html#E116:120_1_2"><i><font color="Green" title="E162">A96</font></i></a></i>;<br/><a NAME="E4:120_1_2_11"/>
<font color="Maroon" title="c42">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span>,<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c30">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> </span>)</span></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="chord.html#E119:120_1_2"><i><font color="Green" title="E164">A108</font></i></a>, <a class="txt" href="chord.html#E1:120_1_2_11"><i><font color="Green" title="E171">A121</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:120_1_2_11"/><b>then </b>
( <font color="Maroon" title="c42">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  or <font color="Maroon" title="c42">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y</font> or <font color="Maroon" title="c42">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">x</font> or <font color="Maroon" title="c42">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  )
 <b>by </b><i><a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/><b>hence </b><a NAME="E6:120_1_2_11"/>
<font color="Maroon" title="c42">v</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"><span class="p2">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>by </b><i><a class="txt" href="chord.html#E14:120_1_2"><i><font color="Green" title="E102">A13</font></i></a>, <a class="txt" href="chord.html#E16:120_1_2"><i><font color="Green" title="E104">A15</font></i></a>, <a class="txt" href="chord.html#E66:120_1_2"><i><font color="Green" title="E128">A43</font></i></a>, <a class="txt" href="chord.html#E3:120_1_2_11"><i><font color="Green" title="E172">A122</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E127:120_1_2"/><b>then </b><i><font color="Green" title="E171">A123</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</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="c41">R</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"><span class="p2">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_12"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c42">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c42">v</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"><span class="p2">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</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="c41">R</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> )</span><br/><b>assume </b><a NAME="E1:120_1_2_12"/><i><font color="Green" title="E172">A124</font></i>: 
<font color="Maroon" title="c42">v</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"><span class="p2">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</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="c41">R</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_12_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:120_1_2_12_1"/>
( <font color="Maroon" title="c42">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Maroon" title="c42">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  )
 </span><b>by </b><i><a class="txt" href="chord.html#E1:120_1_2_12"><i><font color="Green" title="E172">A124</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_12_1_1"><b>suppose </b></a><a NAME="E1:120_1_2_12_1_1"/><i><font color="Green" title="E173">A125</font></i>: 
<font color="Maroon" title="c42">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</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="c41">R</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span></span><br/><div class="add"><a NAME="E2:120_1_2_12_1_1"/><b>then </b><i><font color="Green" title="E174">A126</font></i>: 
<font color="Maroon" title="c42">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#T89" title="GLIB_001:th.89">GLIB_001:89</a></i>;<br/><a NAME="E3:120_1_2_12_1_1"/>
<font color="Maroon" title="c42">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">R</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E118:120_1_2"><i><font color="Green" title="E163">A107</font></i></a>, <a class="txt" href="chord.html#E120:120_1_2"><i><font color="Green" title="E165">A109</font></i></a>, <a class="txt" href="chord.html#E1:120_1_2_12_1_1"><i><font color="Green" title="E173">A125</font></i></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/><b>hence </b><a NAME="E4:120_1_2_12_1_1"/>
<font color="Maroon" title="c42">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</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="c41">R</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E2:120_1_2_12_1_1"><i><font color="Green" title="E174">A126</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_12_1_2"><b>suppose </b></a><a NAME="E1:120_1_2_12_1_2"/><i><font color="Green" title="E173">A127</font></i>: 
<font color="Maroon" title="c42">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</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="c41">R</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span></span><br/><div class="add"><a NAME="E2:120_1_2_12_1_2"/><b>then </b><i><font color="Green" title="E174">A128</font></i>: 
<font color="Maroon" title="c42">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#T89" title="GLIB_001:th.89">GLIB_001:89</a></i>;<br/><a NAME="E3:120_1_2_12_1_2"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c41">R</font>
 <b>by </b><i><a class="txt" href="chord.html#E118:120_1_2"><i><font color="Green" title="E163">A107</font></i></a></i>;<br/><a NAME="E4:120_1_2_12_1_2"/><b>then </b>
<font color="Maroon" title="c42">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">R</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E120:120_1_2"><i><font color="Green" title="E165">A109</font></i></a>, <a class="txt" href="chord.html#E1:120_1_2_12_1_2"><i><font color="Green" title="E173">A127</font></i></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/><b>hence </b><a NAME="E5:120_1_2_12_1_2"/>
<font color="Maroon" title="c42">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</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="c41">R</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E2:120_1_2_12_1_2"><i><font color="Green" title="E174">A128</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E129:120_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span></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="c38">Q</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="c41">R</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E130:120_1_2"/><b>then </b><i><font color="Green" title="E172">A129</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</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="c41">R</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"><span class="p2">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>by </b><i><a class="txt" href="chord.html#E127:120_1_2"><i><font color="Green" title="E171">A123</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>set </b><font color="Maroon" title="c42">C</font> = <font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c41">R</font>;<br/><a NAME="E131:120_1_2"/><i><font color="Green" title="E173">A130</font></i>: 
<font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c41">R</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>
 <b>by </b><i><a class="txt" href="chord.html#E102:120_1_2"><i><font color="Green" title="E151">A69</font></i></a>, <a class="txt" href="chord.html#E105:120_1_2"><i><font color="Green" title="E153">A71</font></i></a>, <a class="txt" href="chord.html#E121:120_1_2"><i><font color="Green" title="E166">A110</font></i></a>, <a class="txt" href="chord.html#E122:120_1_2"><i><font color="Green" title="E167">A111</font></i></a>, <a class="txt" href="chord.html#E123:120_1_2"><i><font color="Green" title="E168">A112</font></i></a>, <a class="txt" href="chord.html#E124:120_1_2"><i><font color="Green" title="E169">A113</font></i></a>, <a class="txt" href="chord.html#E125:120_1_2"><i><font color="Green" title="E170">A114</font></i></a>, <a class="txt" href="chord.html#E130:120_1_2"><i><font color="Green" title="E172">A129</font></i></a>, <a class="ref" href="chord.html#T27" target="_self" title="CHORD:th.27">Th27</a></i>;<br/><a NAME="E132:120_1_2"/><i><font color="Green" title="E174">A131</font></i>: 
<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> <span class="p3">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c41">R</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">R</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E121:120_1_2"><i><font color="Green" title="E166">A110</font></i></a>, <a class="ref" href="glib_001.html#T29" title="GLIB_001:th.29">GLIB_001:29</a></i>;<br/><a NAME="E133:120_1_2"/><i><font color="Green" title="E175">A132</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c41">R</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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">6 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E118:120_1_2"><i><font color="Green" title="E163">A107</font></i></a>, <a class="txt" href="chord.html#E121:120_1_2"><i><font color="Green" title="E166">A110</font></i></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">GLIB_001:34</a></i>;<br/><a NAME="E134:120_1_2"/><i><font color="Green" title="E176">A133</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c41">R</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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E118:120_1_2"><i><font color="Green" title="E163">A107</font></i></a>, <a class="txt" href="chord.html#E121:120_1_2"><i><font color="Green" title="E166">A110</font></i></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">GLIB_001:34</a></i>;<br/><a NAME="E135:120_1_2"/><i><font color="Green" title="E177">A134</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c41">R</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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E118:120_1_2"><i><font color="Green" title="E163">A107</font></i></a>, <a class="txt" href="chord.html#E121:120_1_2"><i><font color="Green" title="E166">A110</font></i></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">GLIB_001:34</a></i>;<br/><a NAME="E136:120_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c41">R</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="c38">Q</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3
 <b>by </b><i><a class="txt" href="chord.html#E118:120_1_2"><i><font color="Green" title="E163">A107</font></i></a>, <a class="txt" href="chord.html#E121:120_1_2"><i><font color="Green" title="E166">A110</font></i></a>, <a class="ref" href="chord.html#T28" target="_self" title="CHORD:th.28">Th28</a></i>;<br/><a NAME="E137:120_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c41">R</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> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3
 <b>by </b><i><a class="txt" href="chord.html#E104:120_1_2"><i><font color="Green" title="E152">A70</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E138:120_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c41">R</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
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E139:120_1_2"/><b>then </b>
<font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c41">R</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a>
 <b>by </b><i><a class="txt" href="chord.html#E131:120_1_2"><i><font color="Green" title="E173">A130</font></i></a>, <a class="ref" href="chord.html#D11" target="_self" title="CHORD:def.11">Def11</a></i>;<br/><b>then consider </b><font color="Maroon" title="c43">m</font>, <font color="Maroon" title="c44">n</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E141:120_1_2"/><i><font color="Green" title="E178">A135</font></i>: 
<font color="Maroon" title="c43">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="c44">n</font>
 <b>and </b><br/><a NAME="E142:120_1_2"/><i><font color="Green" title="E179">A136</font></i>: 
<font color="Maroon" title="c44">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="c38">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c41">R</font></span>)</span>
 <b>and </b><br/><a NAME="E143:120_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c41">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c41">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">n</font>
 <b>and </b><br/><a NAME="E144:120_1_2"/><i><font color="Green" title="E180">A137</font></i>: 
 ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c41">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c41">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">n</font>,<font color="Maroon" title="c1">G</font>
 <b>and </b><br/><a NAME="E145:120_1_2"/><i><font color="Green" title="E181">A138</font></i>: 
( <font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c41">R</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> implies ( (  not <font color="Maroon" title="c43">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Maroon" title="c44">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c41">R</font></span>)</span> ) &amp; (  not <font color="Maroon" title="c43">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Maroon" title="c44">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> <span class="p2">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c41">R</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 ) &amp; (  not <font color="Maroon" title="c43">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or  not <font color="Maroon" title="c44">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c41">R</font></span>)</span> ) ) )
 <b>by </b><i><a class="ref" href="chord.html#T84" target="_self" title="CHORD:th.84">Th84</a></i>;<br/><a NAME="E146:120_1_2"/><i><font color="Green" title="E182">A139</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">m</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E147:120_1_2"/><i><font color="Green" title="E183">A140</font></i>: 
( <font color="Maroon" title="c44">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="c38">Q</font> or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_13"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_13_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:120_1_2_13_1"/>
( <font color="Maroon" title="c44">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="c38">Q</font> or <font color="Maroon" title="c44">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="c38">Q</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_13_1_1"><b>suppose </b></a><a NAME="E1:120_1_2_13_1_1"/>
<font color="Maroon" title="c44">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="c38">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c44">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="c38">Q</font> or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6 )</span><br/><div class="add"><b>hence </b><a NAME="E2:120_1_2_13_1_1"/>
( <font color="Maroon" title="c44">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="c38">Q</font> or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_13_1_2"><b>suppose </b></a><a NAME="E1:120_1_2_13_1_2"/>
<font color="Maroon" title="c44">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="c38">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c44">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="c38">Q</font> or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6 )</span><br/><div class="add"><a NAME="E2:120_1_2_13_1_2"/><b>then </b><i><font color="Green" title="E184">A141</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">n</font>
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_13_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:120_1_2_13_1_2_1"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">n</font> or <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c44">n</font> )
 </span><b>by </b><i><a class="txt" href="chord.html#E2:120_1_2_13_1_2"><i><font color="Green" title="E184">A141</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_13_1_2_1_1"><b>suppose </b></a><a NAME="E1:120_1_2_13_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c44">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="c38">Q</font> or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6 )</span><br/><div class="add"><b>hence </b><a NAME="E2:120_1_2_13_1_2_1_1"/>
( <font color="Maroon" title="c44">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="c38">Q</font> or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_13_1_2_1_2"><b>suppose </b></a><a NAME="E1:120_1_2_13_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c44">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c44">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="c38">Q</font> or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6 )</span><br/><div class="add"><a NAME="E2:120_1_2_13_1_2_1_2"/><b>then </b><i><font color="Green" title="E185">A142</font></i>: 
<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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">n</font>
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_13_1_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:120_1_2_13_1_2_1_2_1"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">n</font> or <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c44">n</font> )
 </span><b>by </b><i><a class="txt" href="chord.html#E2:120_1_2_13_1_2_1_2"><i><font color="Green" title="E185">A142</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_13_1_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:120_1_2_13_1_2_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c44">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="c38">Q</font> or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6 )</span><br/><div class="add"><b>hence </b><a NAME="E2:120_1_2_13_1_2_1_2_1_1"/>
( <font color="Maroon" title="c44">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="c38">Q</font> or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_13_1_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:120_1_2_13_1_2_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c44">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c44">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="c38">Q</font> or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6 )</span><br/><div class="add"><a NAME="E2:120_1_2_13_1_2_1_2_1_2"/><b>then </b>
<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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">n</font>
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></i>;<br/><b>hence </b><a NAME="E3:120_1_2_13_1_2_1_2_1_2"/>
( <font color="Maroon" title="c44">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="c38">Q</font> or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6 )
 <b>by </b><i><a class="txt" href="chord.html#E118:120_1_2"><i><font color="Green" title="E163">A107</font></i></a>, <a class="txt" href="chord.html#E132:120_1_2"><i><font color="Green" title="E174">A131</font></i></a>, <a class="txt" href="chord.html#E142:120_1_2"><i><font color="Green" title="E179">A136</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c45">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E149:120_1_2"/><i><font color="Green" title="E184">A143</font></i>: 
<font color="Maroon" title="c45">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="c38">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c41">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c41">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">n</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E144:120_1_2"><i><font color="Green" title="E180">A137</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_14"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:120_1_2_14"/>
( <font color="Maroon" title="c44">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="c38">Q</font> or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 or <font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6 )
 </span><b>by </b><i><a class="txt" href="chord.html#E147:120_1_2"><i><font color="Green" title="E183">A140</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_14_1"><b>suppose </b></a><a NAME="E1:120_1_2_14_1"/><i><font color="Green" title="E185">A144</font></i>: 
<font color="Maroon" title="c44">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="c38">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:120_1_2_14_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">n</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E3:120_1_2_14_1"/><b>then </b><i><font color="Green" title="E186">A145</font></i>: 
<font color="Maroon" title="c44">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">Q</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:120_1_2_14_1"><i><font color="Green" title="E185">A144</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:120_1_2_14_1"/>
( <font color="Maroon" title="c43">m</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="c38">Q</font> &amp; <font color="Maroon" title="c43">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 )
 <b>by </b><i><a class="txt" href="chord.html#E141:120_1_2"><i><font color="Green" title="E178">A135</font></i></a>, <a class="txt" href="chord.html#E1:120_1_2_14_1"><i><font color="Green" title="E185">A144</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:120_1_2_14_1"/><b>then </b>
<font color="Maroon" title="c43">m</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="c38">Q</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:120_1_2_14_1"/><b>then </b>
<font color="Maroon" title="c43">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">Q</font>
 <b>by </b><i><a class="txt" href="chord.html#E146:120_1_2"><i><font color="Green" title="E182">A139</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:120_1_2_14_1"/><b>then </b>
( <font color="Maroon" title="c38">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c41">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">m</font> &amp; <font color="Maroon" title="c38">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c41">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">n</font> )
 <b>by </b><i><a class="txt" href="chord.html#E3:120_1_2_14_1"><i><font color="Green" title="E186">A145</font></i></a>, <a class="ref" href="glib_001.html#T33" title="GLIB_001:th.33">GLIB_001:33</a></i>;<br/><b>hence </b><a NAME="E8:120_1_2_14_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E95:120_1_2"><i><font color="Green" title="E145">A63</font></i></a>, <a class="txt" href="chord.html#E141:120_1_2"><i><font color="Green" title="E178">A135</font></i></a>, <a class="txt" href="chord.html#E149:120_1_2"><i><font color="Green" title="E184">A143</font></i></a>, <a class="txt" href="chord.html#E1:120_1_2_14_1"><i><font color="Green" title="E185">A144</font></i></a>, <a class="ref" href="chord.html#T41" target="_self" title="CHORD:th.41">Th41</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_14_2"><b>suppose </b></a><a NAME="E1:120_1_2_14_2"/><i><font color="Green" title="E185">A146</font></i>: 
<font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:120_1_2_14_2"/><b>then </b>
<font color="Maroon" title="c43">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="c38">Q</font>
 <b>by </b><i><a class="txt" href="chord.html#E141:120_1_2"><i><font color="Green" title="E178">A135</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:120_1_2_14_2"/><b>then </b><i><font color="Green" title="E186">A147</font></i>: 
<font color="Maroon" title="c43">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">Q</font>
 <b>by </b><i><a class="txt" href="chord.html#E146:120_1_2"><i><font color="Green" title="E182">A139</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:120_1_2_14_2"/><b>then </b>
<font color="Maroon" title="c45">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c38">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">m</font>,<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E120:120_1_2"><i><font color="Green" title="E165">A109</font></i></a>, <a class="txt" href="chord.html#E135:120_1_2"><i><font color="Green" title="E177">A134</font></i></a>, <a class="txt" href="chord.html#E149:120_1_2"><i><font color="Green" title="E184">A143</font></i></a>, <a class="txt" href="chord.html#E1:120_1_2_14_2"><i><font color="Green" title="E185">A146</font></i></a>, <a class="ref" href="glib_001.html#T33" title="GLIB_001:th.33">GLIB_001:33</a></i>;<br/><b>hence </b><a NAME="E5:120_1_2_14_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E114:120_1_2"><i><font color="Green" title="E160">A78</font></i></a>, <a class="txt" href="chord.html#E141:120_1_2"><i><font color="Green" title="E178">A135</font></i></a>, <a class="txt" href="chord.html#E1:120_1_2_14_2"><i><font color="Green" title="E185">A146</font></i></a>, <a class="txt" href="chord.html#E3:120_1_2_14_2"><i><font color="Green" title="E186">A147</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_14_3"><b>suppose </b></a><a NAME="E1:120_1_2_14_3"/><i><font color="Green" title="E185">A148</font></i>: 
<font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:120_1_2_14_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c43">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4
 <b>by </b><i><a class="txt" href="chord.html#E141:120_1_2"><i><font color="Green" title="E178">A135</font></i></a>, <a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></i>;<br/><a NAME="E3:120_1_2_14_3"/><b>then </b>
<font color="Maroon" title="c43">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4 <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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4
 ;<br/><a NAME="E4:120_1_2_14_3"/><b>then </b>
<font color="Maroon" title="c43">m</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="c38">Q</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E5:120_1_2_14_3"/><b>then </b><i><font color="Green" title="E186">A149</font></i>: 
<font color="Maroon" title="c43">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">Q</font>
 <b>by </b><i><a class="txt" href="chord.html#E146:120_1_2"><i><font color="Green" title="E182">A139</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:120_1_2_14_3"/><b>then </b><i><font color="Green" title="E187">A150</font></i>: 
<font color="Maroon" title="c45">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c38">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">m</font>,<font color="Maroon" title="c30">x</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E120:120_1_2"><i><font color="Green" title="E165">A109</font></i></a>, <a class="txt" href="chord.html#E134:120_1_2"><i><font color="Green" title="E176">A133</font></i></a>, <a class="txt" href="chord.html#E149:120_1_2"><i><font color="Green" title="E184">A143</font></i></a>, <a class="txt" href="chord.html#E1:120_1_2_14_3"><i><font color="Green" title="E185">A148</font></i></a>, <a class="ref" href="glib_001.html#T33" title="GLIB_001:th.33">GLIB_001:33</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_14_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:120_1_2_14_3_1"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">m</font> or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c43">m</font> )
 </span><b>by </b><i><a class="txt" href="chord.html#E146:120_1_2"><i><font color="Green" title="E182">A139</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_14_3_1_1"><b>suppose </b></a><a NAME="E1:120_1_2_14_3_1_1"/>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:120_1_2_14_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E102:120_1_2"><i><font color="Green" title="E151">A69</font></i></a>, <a class="txt" href="chord.html#E105:120_1_2"><i><font color="Green" title="E153">A71</font></i></a>, <a class="txt" href="chord.html#E118:120_1_2"><i><font color="Green" title="E163">A107</font></i></a>, <a class="txt" href="chord.html#E121:120_1_2"><i><font color="Green" title="E166">A110</font></i></a>, <a class="txt" href="chord.html#E122:120_1_2"><i><font color="Green" title="E167">A111</font></i></a>, <a class="txt" href="chord.html#E123:120_1_2"><i><font color="Green" title="E168">A112</font></i></a>, <a class="txt" href="chord.html#E124:120_1_2"><i><font color="Green" title="E169">A113</font></i></a>, <a class="txt" href="chord.html#E125:120_1_2"><i><font color="Green" title="E170">A114</font></i></a>, <a class="txt" href="chord.html#E130:120_1_2"><i><font color="Green" title="E172">A129</font></i></a>, <a class="txt" href="chord.html#E132:120_1_2"><i><font color="Green" title="E174">A131</font></i></a>, <a class="txt" href="chord.html#E145:120_1_2"><i><font color="Green" title="E181">A138</font></i></a>, <a class="txt" href="chord.html#E1:120_1_2_14_3"><i><font color="Green" title="E185">A148</font></i></a>, <a class="ref" href="chord.html#T27" target="_self" title="CHORD:th.27">Th27</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_14_3_1_2"><b>suppose </b></a><a NAME="E1:120_1_2_14_3_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c43">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:120_1_2_14_3_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E115:120_1_2"><i><font color="Green" title="E161">A88</font></i></a>, <a class="txt" href="chord.html#E5:120_1_2_14_3"><i><font color="Green" title="E186">A149</font></i></a>, <a class="txt" href="chord.html#E6:120_1_2_14_3"><i><font color="Green" title="E187">A150</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_14_4"><b>suppose </b></a><a NAME="E1:120_1_2_14_4"/><i><font color="Green" title="E185">A151</font></i>: 
<font color="Maroon" title="c44">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:120_1_2_14_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c43">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6
 <b>by </b><i><a class="txt" href="chord.html#E141:120_1_2"><i><font color="Green" title="E178">A135</font></i></a>, <a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></i>;<br/><a NAME="E3:120_1_2_14_4"/><b>then </b>
<font color="Maroon" title="c43">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4
 ;<br/><a NAME="E4:120_1_2_14_4"/><b>then </b><i><font color="Green" title="E186">A152</font></i>: 
<font color="Maroon" title="c43">m</font> <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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_14_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:120_1_2_14_4_1"/>
( <font color="Maroon" title="c43">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c43">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 )
 </span><b>by </b><i><a class="txt" href="chord.html#E4:120_1_2_14_4"><i><font color="Green" title="E186">A152</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_14_4_1_1"><b>suppose </b></a><a NAME="E1:120_1_2_14_4_1_1"/>
<font color="Maroon" title="c43">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:120_1_2_14_4_1_1"/><b>then </b><i><font color="Green" title="E187">A153</font></i>: 
<font color="Maroon" title="c43">m</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="c38">Q</font>
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></i>;<br/><a NAME="E3:120_1_2_14_4_1_1"/><b>then </b>
<font color="Maroon" title="c43">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">Q</font>
 <b>by </b><i><a class="txt" href="chord.html#E146:120_1_2"><i><font color="Green" title="E182">A139</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:120_1_2_14_4_1_1"/><b>then </b><i><font color="Green" title="E188">A154</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c38">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c41">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">m</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T33" title="GLIB_001:th.33">GLIB_001:33</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2_14_4_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:120_1_2_14_4_1_1_1"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c43">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:120_1_2_14_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c43">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="chord.html#E102:120_1_2"><i><font color="Green" title="E151">A69</font></i></a>, <a class="txt" href="chord.html#E105:120_1_2"><i><font color="Green" title="E153">A71</font></i></a>, <a class="txt" href="chord.html#E118:120_1_2"><i><font color="Green" title="E163">A107</font></i></a>, <a class="txt" href="chord.html#E121:120_1_2"><i><font color="Green" title="E166">A110</font></i></a>, <a class="txt" href="chord.html#E122:120_1_2"><i><font color="Green" title="E167">A111</font></i></a>, <a class="txt" href="chord.html#E123:120_1_2"><i><font color="Green" title="E168">A112</font></i></a>, <a class="txt" href="chord.html#E124:120_1_2"><i><font color="Green" title="E169">A113</font></i></a>, <a class="txt" href="chord.html#E125:120_1_2"><i><font color="Green" title="E170">A114</font></i></a>, <a class="txt" href="chord.html#E130:120_1_2"><i><font color="Green" title="E172">A129</font></i></a>, <a class="txt" href="chord.html#E132:120_1_2"><i><font color="Green" title="E174">A131</font></i></a>, <a class="txt" href="chord.html#E145:120_1_2"><i><font color="Green" title="E181">A138</font></i></a>, <a class="txt" href="chord.html#E1:120_1_2_14_4"><i><font color="Green" title="E185">A151</font></i></a>, <a class="ref" href="chord.html#T27" target="_self" title="CHORD:th.27">Th27</a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:120_1_2_14_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c43">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <b>by </b><i><a class="ref" href="chord.html#T3" target="_self" title="CHORD:th.3">Th3</a></i>;<br/><a NAME="E4:120_1_2_14_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c43">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <b>by </b><i><a class="txt" href="chord.html#E102:120_1_2"><i><font color="Green" title="E151">A69</font></i></a>, <a class="txt" href="chord.html#E105:120_1_2"><i><font color="Green" title="E153">A71</font></i></a>, <a class="txt" href="chord.html#E118:120_1_2"><i><font color="Green" title="E163">A107</font></i></a>, <a class="txt" href="chord.html#E121:120_1_2"><i><font color="Green" title="E166">A110</font></i></a>, <a class="txt" href="chord.html#E122:120_1_2"><i><font color="Green" title="E167">A111</font></i></a>, <a class="txt" href="chord.html#E123:120_1_2"><i><font color="Green" title="E168">A112</font></i></a>, <a class="txt" href="chord.html#E124:120_1_2"><i><font color="Green" title="E169">A113</font></i></a>, <a class="txt" href="chord.html#E125:120_1_2"><i><font color="Green" title="E170">A114</font></i></a>, <a class="txt" href="chord.html#E130:120_1_2"><i><font color="Green" title="E172">A129</font></i></a>, <a class="txt" href="chord.html#E132:120_1_2"><i><font color="Green" title="E174">A131</font></i></a>, <a class="txt" href="chord.html#E145:120_1_2"><i><font color="Green" title="E181">A138</font></i></a>, <a class="txt" href="chord.html#E1:120_1_2_14_4"><i><font color="Green" title="E185">A151</font></i></a>, <a class="ref" href="chord.html#T27" target="_self" title="CHORD:th.27">Th27</a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E5:120_1_2_14_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E6:120_1_2_14_4_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c43">m</font>
 ;<br/><b>hence </b><a NAME="E7:120_1_2_14_4_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E95:120_1_2"><i><font color="Green" title="E145">A63</font></i></a>, <a class="txt" href="chord.html#E120:120_1_2"><i><font color="Green" title="E165">A109</font></i></a>, <a class="txt" href="chord.html#E133:120_1_2"><i><font color="Green" title="E175">A132</font></i></a>, <a class="txt" href="chord.html#E149:120_1_2"><i><font color="Green" title="E184">A143</font></i></a>, <a class="txt" href="chord.html#E1:120_1_2_14_4"><i><font color="Green" title="E185">A151</font></i></a>, <a class="txt" href="chord.html#E2:120_1_2_14_4_1_1"><i><font color="Green" title="E187">A153</font></i></a>, <a class="txt" href="chord.html#E4:120_1_2_14_4_1_1"><i><font color="Green" title="E188">A154</font></i></a>, <a class="ref" href="chord.html#T41" target="_self" title="CHORD:th.41">Th41</a>, <a class="ref" href="glib_000.html#T17" title="GLIB_000:th.17">GLIB_000:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="120_1_2_14_4_1_2"><b>suppose </b></a><a NAME="E1:120_1_2_14_4_1_2"/><i><font color="Green" title="E187">A155</font></i>: 
<font color="Maroon" title="c43">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="c38">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:120_1_2_14_4_1_2"/>
( 3 <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="c38">Q</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 )
 <b>by </b><i><a class="txt" href="chord.html#E102:120_1_2"><i><font color="Green" title="E151">A69</font></i></a>, <a class="ref" href="glib_001.html#T126" title="GLIB_001:th.126">GLIB_001:126</a></i>;<br/><a NAME="E3:120_1_2_14_4_1_2"/><b>then </b>
1 <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="c38">Q</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:120_1_2_14_4_1_2"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">Q</font> &amp; 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">Q</font> )
 <b>by </b><i><a class="txt" href="chord.html#E81:120_1_2"><i><font color="Green" title="E134">A49</font></i></a>, <a class="txt" href="chord.html#E89:120_1_2"><i><font color="Green" title="E140">A55</font></i></a>, <a class="txt" href="chord.html#E96:120_1_2"><i><font color="Green" title="E146">A64</font></i></a>, <a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E5:120_1_2_14_4_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E114:120_1_2"><i><font color="Green" title="E160">A78</font></i></a>, <a class="txt" href="chord.html#E120:120_1_2"><i><font color="Green" title="E165">A109</font></i></a>, <a class="txt" href="chord.html#E133:120_1_2"><i><font color="Green" title="E175">A132</font></i></a>, <a class="txt" href="chord.html#E135:120_1_2"><i><font color="Green" title="E177">A134</font></i></a>, <a class="txt" href="chord.html#E149:120_1_2"><i><font color="Green" title="E184">A143</font></i></a>, <a class="txt" href="chord.html#E1:120_1_2_14_4"><i><font color="Green" title="E185">A151</font></i></a>, <a class="txt" href="chord.html#E1:120_1_2_14_4_1_2"><i><font color="Green" title="E187">A155</font></i></a>, <a class="ref" href="glib_000.html#T17" title="GLIB_000:th.17">GLIB_000:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
