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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp; not <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  holds <br/>for <font color="Olive" title="b3">S</font> being   <a href="chord.html#M2" title="CHORD:mode.2">VertexSeparator</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>  st <font color="Olive" title="b3">S</font> is  <a href="chord.html#V4" title="CHORD:attr.4">minimal</a>  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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:119"/><span class="lab"><font color="Green" title="E92">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:119"/><span class="lab"><font color="Green" title="E93">A2</font></span>: 
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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being   <a href="chord.html#M2" title="CHORD:mode.2">VertexSeparator</a> of <font color="Maroon" title="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/>

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E3:119"/><span class="lab"><font color="Green" title="E94">A3</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>



<span class="kw">assume </span><a NAME="E4:119"/><span class="lab"><font color="Green" title="E95">A4</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E5:119"/><span class="lab"><font color="Green" title="E96">A5</font></span>: 
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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:119_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="119_1_1"><span class="kw">suppose </span></a><a NAME="E1:119_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:119_1_1"/><span class="kw">then </span>
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/><span class="kw">hence </span><a NAME="E3:119_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:119"><span class="lab"><font color="Green" title="E95">A4</font></span></a>, <a class="txt" href="#E5:119"><span class="lab"><font color="Green" title="E96">A5</font></span></a>, <a class="ref" href="glib_002.html#T9" title="GLIB_002:th.9">GLIB_002:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2"><span class="kw">suppose </span></a><a NAME="E1:119_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="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/><span class="kw">set </span><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/><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <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/><span class="kw">set </span><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/><div><span class="lab"><font color="Green" title="E97">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">assume </span><a NAME="E1:119_1_2_1"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><a NAME="E2:119_1_2_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">y</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="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="Olive" title="b1">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> &amp; <font color="Olive" title="b1">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> )
 ;<br/><span class="kw">hence </span><a NAME="E3:119_1_2_1"/>
<font color="Maroon" title="c10">x</font> is  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:119_1_2"/><span class="lab"><font color="Green" title="E98">A7</font></span>: 
<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="E5:119_1_2"/><span class="lab"><font color="Green" title="E99">A8</font></span>: 
<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> 
 <span class="kw">from</span> <span class="lab"><a class="ref" href="fraenkel.html#S21" title="FRAENKEL:sch.21">FRAENKEL:sch 21</a>(<a class="txt" href="#E4:119_1_2"><span class="lab"><font color="Green" title="E98">A7</font></span></a>);<br/></span><a NAME="E6:119_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:119"><span class="lab"><font color="Green" title="E95">A4</font></span></a>, <a class="ref" href="glib_002.html#T9" title="GLIB_002:th.9">GLIB_002:9</a></span>;<br/><a NAME="E7:119_1_2"/><span class="kw">then </span>
 <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/><span class="kw">then </span><span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:119_1_2"><span class="lab"><font color="Green" title="E99">A8</font></span></a>, <a class="txt" href="#E3:119_1_2"><span class="lab"><font color="Green" title="E97">A6</font></span></a>, <a class="ref" href="membered.html#D6" title="MEMBERED:def.6">MEMBERED:def 6</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">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="E10:119_1_2"/><span class="lab"><font color="Green" title="E100">A9</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c11">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></span>;<br/><a NAME="E11:119_1_2"/>
 <a href="xxreal_2.html#K1" title="XXREAL_2: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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D8" title="XXREAL_2:def.8">XXREAL_2:def 8</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">c</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E13:119_1_2"/><span class="lab"><font color="Green" title="E101">A10</font></span>: 
 <a href="xxreal_2.html#K1" title="XXREAL_2: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="c12">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>
 <span class="kw">and </span><br/><a NAME="E14:119_1_2"/><span class="lab"><font color="Green" title="E102">A11</font></span>: 
<font color="Maroon" title="c12">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/><span class="kw">set </span><font color="Maroon" title="c13">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="c12">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/><a NAME="E15:119_1_2"/><span class="lab"><font color="Green" title="E103">A12</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119"><span class="lab"><font color="Green" title="E92">A1</font></span></a>, <a class="txt" href="#E2:119"><span class="lab"><font color="Green" title="E93">A2</font></span></a>, <a class="ref" href="chord.html#T75" target="_self" title="CHORD:th.75">Th75</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">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/><span class="kw">set </span><font color="Maroon" title="c15">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="E17:119_1_2"/><span class="lab"><font color="Green" title="E104">A13</font></span>: 
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="E18:119_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119"><span class="lab"><font color="Green" title="E92">A1</font></span></a>, <a class="txt" href="#E2:119"><span class="lab"><font color="Green" title="E93">A2</font></span></a>, <a class="ref" href="chord.html#D8" target="_self" title="CHORD:def.8">Def8</a></span>;<br/><a NAME="E19:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A14</font></span>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E20:119_1_2"/>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">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/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A</font> implies <font color="Maroon" title="c17">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/><span class="kw">assume </span><a NAME="E1:119_1_2_2"/><span class="lab"><font color="Green" title="E106">A15</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">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="E2:119_1_2_2"/>
not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:119_1_2"><span class="lab"><font color="Green" title="E103">A12</font></span></a>, <a class="txt" href="#E1:119_1_2_2"><span class="lab"><font color="Green" title="E106">A15</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:119_1_2_2"/>
<font color="Maroon" title="c17">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119_1_2_2"><span class="lab"><font color="Green" title="E106">A15</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E23:119_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">Ga</font> = <font color="Maroon" title="c11">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="c16">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:119_1_2"><span class="lab"><font color="Green" title="E105">A14</font></span></a>, <a class="ref" href="chord.html#T29" target="_self" title="CHORD:th.29">Th29</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c18">y</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E26:119_1_2"/><span class="lab"><font color="Green" title="E106">A16</font></span>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 <span class="kw">and </span><br/><a NAME="E27:119_1_2"/><span class="lab"><font color="Green" title="E107">A17</font></span>: 
not <font color="Maroon" title="c12">c</font>,<font color="Maroon" title="c18">y</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:119"><span class="lab"><font color="Green" title="E96">A5</font></span></a>, <a class="txt" href="#E14:119_1_2"><span class="lab"><font color="Green" title="E102">A11</font></span></a></span>;<br/><a NAME="E28:119_1_2"/><span class="lab"><font color="Green" title="E108">A18</font></span>: 
not <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:119_1_2"><span class="lab"><font color="Green" title="E106">A16</font></span></a>, <a class="txt" href="#E15:119_1_2"><span class="lab"><font color="Green" title="E103">A12</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c19">Ay</font> = <font color="Maroon" title="c16">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c20">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="c16">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/><a NAME="E30:119_1_2"/>
<font color="Maroon" title="c18">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="c18">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E31:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A19</font></span>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E32:119_1_2"/>
<font color="Maroon" title="c12">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:119_1_2"><span class="lab"><font color="Green" title="E102">A11</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">ca</font> = <font color="Maroon" title="c12">c</font>, <font color="Maroon" title="c22">ya</font> = <font color="Maroon" title="c18">y</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c20">Gay</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:119_1_2"><span class="lab"><font color="Green" title="E109">A19</font></span></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></span>;<br/><a NAME="E34:119_1_2"/>
 ex <font color="Olive" title="b1">yaa</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">yaa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A</font> &amp; <font color="Maroon" title="c18">y</font>,<font color="Olive" title="b1">yaa</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119"><span class="lab"><font color="Green" title="E92">A1</font></span></a>, <a class="txt" href="#E2:119"><span class="lab"><font color="Green" title="E93">A2</font></span></a>, <a class="txt" href="#E3:119"><span class="lab"><font color="Green" title="E94">A3</font></span></a>, <a class="txt" href="#E4:119"><span class="lab"><font color="Green" title="E95">A4</font></span></a>, <a class="txt" href="#E26:119_1_2"><span class="lab"><font color="Green" title="E106">A16</font></span></a>, <a class="ref" href="chord.html#T82" target="_self" title="CHORD:th.82">Th82</a></span>;<br/><a NAME="E35:119_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1">(<span class="default"><a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c17">Ga</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:119_1_2"><span class="lab"><font color="Green" title="E100">A9</font></span></a>, <a class="txt" href="#E28:119_1_2"><span class="lab"><font color="Green" title="E108">A18</font></span></a></span>;<br/><a NAME="E36:119_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">Gay</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T57" target="_self" title="CHORD:th.57">Th57</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">Wa</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c20">Gay</font><span class="kw"> such that </span><br/><a NAME="E38:119_1_2"/><span class="lab"><font color="Green" title="E110">A20</font></span>: 
<font color="Maroon" title="c23">Wa</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c21">ca</font>,<font color="Maroon" title="c22">ya</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D1" title="GLIB_002:def.1">GLIB_002:def 1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c24">P</font> being   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c20">Gay</font><span class="kw"> such that </span><br/><a NAME="E40:119_1_2"/><span class="lab"><font color="Green" title="E111">A21</font></span>: 
<font color="Maroon" title="c24">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c23">Wa</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c23">Wa</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <span class="kw">and </span><br/><a NAME="E41:119_1_2"/><span class="lab"><font color="Green" title="E112">A22</font></span>: 
<font color="Maroon" title="c24">P</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T39" target="_self" title="CHORD:th.39">Th39</a></span>;<br/><a NAME="E42:119_1_2"/>
<font color="Maroon" title="c23">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="c21">ca</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:119_1_2"><span class="lab"><font color="Green" title="E110">A20</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>;<br/><a NAME="E43:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A23</font></span>: 
<font color="Maroon" title="c24">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="c21">ca</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:119_1_2"><span class="lab"><font color="Green" title="E111">A21</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>;<br/><a NAME="E44:119_1_2"/>
<font color="Maroon" title="c23">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="c22">ya</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:119_1_2"><span class="lab"><font color="Green" title="E110">A20</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>;<br/><a NAME="E45:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A24</font></span>: 
<font color="Maroon" title="c24">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="c18">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:119_1_2"><span class="lab"><font color="Green" title="E111">A21</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>;<br/><a NAME="E46:119_1_2"/>
<font color="Maroon" title="c12">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:119_1_2"><span class="lab"><font color="Green" title="E102">A11</font></span></a>, <a class="txt" href="#E26:119_1_2"><span class="lab"><font color="Green" title="E106">A16</font></span></a>, <a class="txt" href="#E15:119_1_2"><span class="lab"><font color="Green" title="E103">A12</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E47:119_1_2"/><span class="kw">then </span>
not <font color="Maroon" title="c24">P</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:119_1_2"><span class="lab"><font color="Green" title="E113">A23</font></span></a>, <a class="txt" href="#E45:119_1_2"><span class="lab"><font color="Green" title="E114">A24</font></span></a>, <a class="ref" href="glib_001.html#T128" title="GLIB_001:th.128">GLIB_001:128</a></span>;<br/><a NAME="E48:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A25</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">P</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T126" title="GLIB_001:th.126">GLIB_001:126</a></span>;<br/><div><span class="lab"><font color="Green" title="E116">A26</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:119_1_2_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:119_1_2_3"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T3" target="_self" title="CHORD:th.3">Th3</a></span>;<br/><a NAME="E3:119_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A27</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:119_1_2"><span class="lab"><font color="Green" title="E115">A25</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:119_1_2_3"/><span class="kw">then </span>
<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="c24">P</font>
 ;<br/><a NAME="E5:119_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c24">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="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1,<font color="Maroon" title="c24">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="c20">Gay</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></span>;<br/><a NAME="E6:119_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c24">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="c12">c</font>,<font color="Maroon" title="c18">y</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:119_1_2"><span class="lab"><font color="Green" title="E113">A23</font></span></a>, <a class="txt" href="#E45:119_1_2"><span class="lab"><font color="Green" title="E114">A24</font></span></a>, <a class="txt" href="#E3:119_1_2_3"><span class="lab"><font color="Green" title="E117">A27</font></span></a>, <a class="ref" href="glib_000.html#T75" title="GLIB_000:th.75">GLIB_000:75</a></span>;<br/><span class="kw">hence </span><a NAME="E7:119_1_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:119_1_2"><span class="lab"><font color="Green" title="E107">A17</font></span></a>, <a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E50:119_1_2"/><span class="kw">then </span>
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="c24">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c25">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="c24">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>  <span class="kw">by</span> <span class="lab"><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></span>;<br/><span class="kw">set </span><font color="Maroon" title="c26">d</font> = <font color="Maroon" title="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font>;<br/><a NAME="E52:119_1_2"/><span class="lab"><font color="Green" title="E117">A28</font></span>: 
<font color="Maroon" title="c25">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="c24">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></span>;<br/><div><span class="lab"><font color="Green" title="E118">A29</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_2_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:119_1_2_4"/>
<font color="Maroon" title="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:119_1_2_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:119_1_2"><span class="lab"><font color="Green" title="E114">A24</font></span></a>, <a class="txt" href="#E52:119_1_2"><span class="lab"><font color="Green" title="E117">A28</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><a NAME="E3:119_1_2_4"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">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
 ;<br/><span class="kw">hence </span><a NAME="E4:119_1_2_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:119_1_2"><span class="lab"><font color="Green" title="E116">A26</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E54:119_1_2"/><span class="lab"><font color="Green" title="E119">A30</font></span>: 
not <font color="Maroon" title="c18">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="c12">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:119_1_2"><span class="lab"><font color="Green" title="E107">A17</font></span></a>, <a class="ref" href="chord.html#T52" target="_self" title="CHORD:th.52">Th52</a></span>;<br/><a NAME="E55:119_1_2"/><span class="lab"><font color="Green" title="E120">A31</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c20">Gay</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></span>;<br/><a NAME="E56:119_1_2"/><span class="lab"><font color="Green" title="E121">A32</font></span>: 
<font color="Maroon" title="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c20">Gay</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:119_1_2"><span class="lab"><font color="Green" title="E117">A28</font></span></a>, <a class="ref" href="glib_001.html#T8" title="GLIB_001:th.8">GLIB_001:8</a></span>;<br/><a NAME="E57:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">A33</font></span>: 
( <font color="Maroon" title="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A</font> or <font color="Maroon" title="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">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="c18">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:119_1_2"><span class="lab"><font color="Green" title="E120">A31</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c27">d</font> = <font color="Maroon" title="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:119_1_2"><span class="lab"><font color="Green" title="E121">A32</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c28">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="c27">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/><a NAME="E59:119_1_2"/>
<font color="Maroon" title="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">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="c27">d</font>,<font color="Maroon" title="c24">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="c24">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="c20">Gay</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:119_1_2"><span class="lab"><font color="Green" title="E117">A28</font></span></a>, <a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></span>;<br/><a NAME="E60:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A34</font></span>: 
<font color="Maroon" title="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">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="c27">d</font>,<font color="Maroon" title="c18">y</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:119_1_2"><span class="lab"><font color="Green" title="E114">A24</font></span></a>, <a class="ref" href="glib_000.html#T75" title="GLIB_000:th.75">GLIB_000:75</a></span>;<br/><a NAME="E61:119_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c27">d</font>,<font color="Maroon" title="c18">y</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></span>;<br/><a NAME="E62:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A35</font></span>: 
<font color="Maroon" title="c18">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="c27">d</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:119_1_2"><span class="lab"><font color="Green" title="E118">A29</font></span></a>, <a class="ref" href="chord.html#T52" target="_self" title="CHORD:th.52">Th52</a></span>;<br/><a NAME="E63:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A36</font></span>: 
<font color="Maroon" title="c18">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="c27">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E26:119_1_2"><span class="lab"><font color="Green" title="E106">A16</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_2_5"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:119_1_2_5"/><span class="lab"><font color="Green" title="E126">A37</font></span>: 
<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="c12">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="c27">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:119_1_2_5"/>
not <font color="Maroon" title="c18">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="c12">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E54:119_1_2"><span class="lab"><font color="Green" title="E119">A30</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:119_1_2_5"/><span class="kw">then </span>
<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="c12">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="c27">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E63:119_1_2"><span class="lab"><font color="Green" title="E125">A36</font></span></a>, <a class="txt" href="#E1:119_1_2_5"><span class="lab"><font color="Green" title="E126">A37</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/><a NAME="E4:119_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A38</font></span>: 
 <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="c12">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="c27">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T24" title="TREES_1:th.24">TREES_1:24</a></span>;<br/><a NAME="E5:119_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="c27">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:119_1_2"><span class="lab"><font color="Green" title="E122">A33</font></span></a>, <a class="txt" href="#E53:119_1_2"><span class="lab"><font color="Green" title="E118">A29</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:119_1_2_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:119_1_2"><span class="lab"><font color="Green" title="E101">A10</font></span></a>, <a class="txt" href="#E4:119_1_2_5"><span class="lab"><font color="Green" title="E127">A38</font></span></a>, <a class="ref" href="xxreal_2.html#D8" title="XXREAL_2:def.8">XXREAL_2:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c29">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E66:119_1_2"/><span class="lab"><font color="Green" title="E126">A39</font></span>: 
<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="c12">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 class="kw">and </span><br/><a NAME="E67:119_1_2"/><span class="lab"><font color="Green" title="E127">A40</font></span>: 
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="c27">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 class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E68:119_1_2"/><span class="lab"><font color="Green" title="E128">A41</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:119_1_2"><span class="lab"><font color="Green" title="E126">A39</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E69:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A42</font></span>: 
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="c27">d</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E67:119_1_2"><span class="lab"><font color="Green" title="E127">A40</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:119_1_2"><span class="lab"><font color="Green" title="E126">A39</font></span></a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( $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="c24">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="c24">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="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1,<font color="Maroon" title="c1">G</font> );<br/><a NAME="E71:119_1_2"/><span class="lab"><font color="Green" title="E130">A43</font></span>: 
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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">P</font>
 ;<br/><a NAME="E72:119_1_2"/><span class="lab"><font color="Green" title="E131">A44</font></span>: 
not <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:119_1_2"><span class="lab"><font color="Green" title="E103">A12</font></span></a>, <a class="txt" href="#E68:119_1_2"><span class="lab"><font color="Green" title="E128">A41</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E73:119_1_2"/><span class="lab"><font color="Green" title="E132">A45</font></span>: 
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="c24">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:119_1_2"><span class="lab"><font color="Green" title="E116">A26</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E74:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A46</font></span>: 
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="c24">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E75:119_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="c12">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:119_1_2"><span class="lab"><font color="Green" title="E126">A39</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E76:119_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">c</font>,<font color="Maroon" title="c30">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T52" target="_self" title="CHORD:th.52">Th52</a></span>;<br/><a NAME="E77:119_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c30">x</font>,<font color="Maroon" title="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:119_1_2"><span class="lab"><font color="Green" title="E113">A23</font></span></a>, <a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></span>;<br/><a NAME="E78:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">A47</font></span>: 
 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>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E73:119_1_2"><span class="lab"><font color="Green" title="E132">A45</font></span></a>, <a class="txt" href="#E74:119_1_2"><span class="lab"><font color="Green" title="E133">A46</font></span></a>, <a class="txt" href="#E17:119_1_2"><span class="lab"><font color="Green" title="E104">A13</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c31">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E80:119_1_2"/><span class="lab"><font color="Green" title="E135">A48</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c31">k</font>]
 <span class="kw">and </span><br/><a NAME="E81:119_1_2"/><span class="lab"><font color="Green" title="E136">A49</font></span>: 
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="c31">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">i</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="#E71:119_1_2"><span class="lab"><font color="Green" title="E130">A43</font></span></a>, <a class="txt" href="#E78:119_1_2"><span class="lab"><font color="Green" title="E134">A47</font></span></a>);<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c32">k</font> = <font color="Maroon" title="c31">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:119_1_2"><span class="lab"><font color="Green" title="E135">A48</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c33">Q1</font> = <font color="Maroon" title="c24">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c32">k</font>,<font color="Maroon" title="c25">j</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c34">Q</font> = <font color="Maroon" title="c24">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c32">k</font>,<font color="Maroon" title="c25">j</font> as   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T176" title="GLIB_001:th.176">GLIB_001:176</a></span>;<br/><a NAME="E84:119_1_2"/><span class="lab"><font color="Green" title="E137">A50</font></span>: 
<font color="Maroon" title="c32">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:119_1_2"><span class="lab"><font color="Green" title="E135">A48</font></span></a>, <a class="ref" href="chord.html#T3" target="_self" title="CHORD:th.3">Th3</a></span>;<br/><a NAME="E85:119_1_2"/><span class="lab"><font color="Green" title="E138">A51</font></span>: 
<font color="Maroon" title="c25">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="c24">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></span>;<br/><a NAME="E86:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E139">A52</font></span>: 
<font color="Maroon" title="c24">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c32">k</font>,<font color="Maroon" title="c25">j</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:119_1_2"><span class="lab"><font color="Green" title="E112">A22</font></span></a>, <a class="txt" href="#E84:119_1_2"><span class="lab"><font color="Green" title="E137">A50</font></span></a>, <a class="ref" href="chord.html#T42" target="_self" title="CHORD:th.42">Th42</a></span>;<br/><a NAME="E87:119_1_2"/><span class="lab"><font color="Green" title="E140">A53</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c32">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E84:119_1_2"><span class="lab"><font color="Green" title="E137">A50</font></span></a>, <a class="txt" href="#E85:119_1_2"><span class="lab"><font color="Green" title="E138">A51</font></span></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></span>;<br/><div><span class="lab"><font color="Green" title="E141">A54</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_2_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c35">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c35">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="c34">Q</font> &amp; <font color="Maroon" title="c35">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="c34">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</font>,<font color="Maroon" title="c30">x</font>,<font color="Maroon" title="c1">G</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:119_1_2_6"/><span class="lab"><font color="Green" title="E142">A55</font></span>: 
<font color="Maroon" title="c35">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="c34">Q</font>
 <span class="kw">and </span><br/><a NAME="E2:119_1_2_6"/><span class="lab"><font color="Green" title="E143">A56</font></span>: 
<font color="Maroon" title="c35">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">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="c34">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</font>,<font color="Maroon" title="c30">x</font>,<font color="Maroon" title="c1">G</font></span><br/><a NAME="E3:119_1_2_6"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119_1_2_6"><span class="lab"><font color="Green" title="E142">A55</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E4:119_1_2_6"/><span class="kw">then </span>
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="c35">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c36">i1</font> = <font color="Maroon" title="c35">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c37">ki1</font> = <font color="Maroon" title="c32">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c36">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="E7:119_1_2_6"/><span class="lab"><font color="Green" title="E144">A57</font></span>: 
<font color="Maroon" title="c35">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="c35">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E8:119_1_2_6"/>
<font color="Maroon" title="c35">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="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119_1_2_6"><span class="lab"><font color="Green" title="E142">A55</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E9:119_1_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E145">A58</font></span>: 
<font color="Maroon" title="c36">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="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:119_1_2_6"><span class="lab"><font color="Green" title="E144">A57</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E10:119_1_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E146">A59</font></span>: 
<font color="Maroon" title="c37">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="c24">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E84:119_1_2"><span class="lab"><font color="Green" title="E137">A50</font></span></a>, <a class="txt" href="#E85:119_1_2"><span class="lab"><font color="Green" title="E138">A51</font></span></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></span>;<br/><a NAME="E11:119_1_2_6"/><span class="lab"><font color="Green" title="E147">A60</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">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="c24">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><span class="kw">assume </span><a NAME="E12:119_1_2_6"/><span class="lab"><font color="Green" title="E148">A61</font></span>: 
 ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c34">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</font>,<font color="Maroon" title="c30">x</font>,<font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E13:119_1_2_6"/>
<font color="Maroon" title="c36">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c32">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="c24">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="c32">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c32">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E87:119_1_2"><span class="lab"><font color="Green" title="E140">A53</font></span></a>, <a class="txt" href="#E9:119_1_2_6"><span class="lab"><font color="Green" title="E145">A58</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E14:119_1_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E149">A62</font></span>: 
<font color="Maroon" title="c37">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="c24">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:119_1_2_6"><span class="lab"><font color="Green" title="E147">A60</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E15:119_1_2_6"/><span class="lab"><font color="Green" title="E150">A63</font></span>: 
<font color="Maroon" title="c34">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">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="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">ki1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E84:119_1_2"><span class="lab"><font color="Green" title="E137">A50</font></span></a>, <a class="txt" href="#E85:119_1_2"><span class="lab"><font color="Green" title="E138">A51</font></span></a>, <a class="txt" href="#E9:119_1_2_6"><span class="lab"><font color="Green" title="E145">A58</font></span></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_2_6_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:119_1_2_6_1"/>
<font color="Maroon" title="c36">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:119_1_2_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c32">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="c32">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c36">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><span class="kw">hence </span><a NAME="E3:119_1_2_6_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E81:119_1_2"><span class="lab"><font color="Green" title="E136">A49</font></span></a>, <a class="txt" href="#E12:119_1_2_6"><span class="lab"><font color="Green" title="E148">A61</font></span></a>, <a class="txt" href="#E15:119_1_2_6"><span class="lab"><font color="Green" title="E150">A63</font></span></a>, <a class="txt" href="#E10:119_1_2_6"><span class="lab"><font color="Green" title="E146">A59</font></span></a>, <a class="txt" href="#E14:119_1_2_6"><span class="lab"><font color="Green" title="E149">A62</font></span></a>, <a class="ref" href="glib_000.html#T17" title="GLIB_000:th.17">GLIB_000:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E17:119_1_2_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:119_1_2_6"><span class="lab"><font color="Green" title="E143">A56</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">set </span><font color="Maroon" title="c35">cc</font> = <font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c36">dd</font> = <font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ;<br/><a NAME="E89:119_1_2"/><span class="lab"><font color="Green" title="E142">A64</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c32">k</font>,<font color="Maroon" title="c25">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="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E84:119_1_2"><span class="lab"><font color="Green" title="E137">A50</font></span></a>, <a class="txt" href="#E85:119_1_2"><span class="lab"><font color="Green" title="E138">A51</font></span></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></span>;<br/><a NAME="E90:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E143">A65</font></span>: 
<font color="Maroon" title="c30">x</font>,<font color="Maroon" title="c34">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:119_1_2"><span class="lab"><font color="Green" title="E135">A48</font></span></a>, <a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></span>;<br/><a NAME="E91:119_1_2"/><span class="lab"><font color="Green" title="E144">A66</font></span>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:119_1_2"><span class="lab"><font color="Green" title="E106">A16</font></span></a>, <a class="txt" href="#E62:119_1_2"><span class="lab"><font color="Green" title="E124">A35</font></span></a>, <a class="txt" href="#E67:119_1_2"><span class="lab"><font color="Green" title="E127">A40</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E92:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E145">A67</font></span>: 
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="c18">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c37">xs</font> = <font color="Maroon" title="c30">x</font>, <font color="Maroon" title="c38">ys</font> = <font color="Maroon" title="c18">y</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c14">Gs</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:119_1_2"><span class="lab"><font color="Green" title="E106">A16</font></span></a>, <a class="txt" href="#E68:119_1_2"><span class="lab"><font color="Green" title="E128">A41</font></span></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></span>;<br/><a NAME="E94:119_1_2"/>
<font color="Maroon" title="c14">Gs</font> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119"><span class="lab"><font color="Green" title="E92">A1</font></span></a>, <a class="txt" href="#E2:119"><span class="lab"><font color="Green" title="E93">A2</font></span></a>, <a class="txt" href="#E3:119"><span class="lab"><font color="Green" title="E94">A3</font></span></a>, <a class="ref" href="chord.html#T98" target="_self" title="CHORD:th.98">Th98</a></span>;<br/><a NAME="E95:119_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c37">xs</font>,<font color="Maroon" title="c38">ys</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E91:119_1_2"><span class="lab"><font color="Green" title="E144">A66</font></span></a>, <a class="ref" href="chord.html#D6" target="_self" title="CHORD:def.6">Def6</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c39">ej</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E97:119_1_2"/><span class="lab"><font color="Green" title="E146">A68</font></span>: 
<font color="Maroon" title="c39">ej</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c37">xs</font>,<font color="Maroon" title="c38">ys</font>,<font color="Maroon" title="c14">Gs</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></span>;<br/><a NAME="E98:119_1_2"/>
<font color="Maroon" title="c39">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="c18">y</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E97:119_1_2"><span class="lab"><font color="Green" title="E146">A68</font></span></a>, <a class="ref" href="glib_000.html#T75" title="GLIB_000:th.75">GLIB_000:75</a></span>;<br/><a NAME="E99:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E147">A69</font></span>: 
<font color="Maroon" title="c30">x</font>,<font color="Maroon" title="c18">y</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></span>;<br/><a NAME="E100:119_1_2"/><span class="lab"><font color="Green" title="E148">A70</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c32">k</font>,<font color="Maroon" title="c25">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="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E84:119_1_2"><span class="lab"><font color="Green" title="E137">A50</font></span></a>, <a class="txt" href="#E85:119_1_2"><span class="lab"><font color="Green" title="E138">A51</font></span></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></span>;<br/><a NAME="E101:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E149">A71</font></span>: 
<font color="Maroon" title="c34">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="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font>
 ;<br/><a NAME="E102:119_1_2"/>
<font color="Maroon" title="c27">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c30">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:119_1_2"><span class="lab"><font color="Green" title="E103">A12</font></span></a>, <a class="txt" href="#E57:119_1_2"><span class="lab"><font color="Green" title="E122">A33</font></span></a>, <a class="txt" href="#E53:119_1_2"><span class="lab"><font color="Green" title="E118">A29</font></span></a>, <a class="txt" href="#E68:119_1_2"><span class="lab"><font color="Green" title="E128">A41</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E103:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E150">A72</font></span>: 
not <font color="Maroon" title="c27">d</font>,<font color="Maroon" title="c30">x</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:119_1_2"><span class="lab"><font color="Green" title="E129">A42</font></span></a>, <a class="ref" href="chord.html#T52" target="_self" title="CHORD:th.52">Th52</a></span>;<br/><a NAME="E104:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E151">A73</font></span>: 
<font color="Maroon" title="c34">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="c34">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:119_1_2"><span class="lab"><font color="Green" title="E135">A48</font></span></a>, <a class="txt" href="#E89:119_1_2"><span class="lab"><font color="Green" title="E142">A64</font></span></a>, <a class="txt" href="#E100:119_1_2"><span class="lab"><font color="Green" title="E148">A70</font></span></a>, <a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></span>;<br/><a NAME="E105:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E152">A74</font></span>: 
not <font color="Maroon" title="c34">Q</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></span>;<br/><a NAME="E106:119_1_2"/><span class="lab"><font color="Green" title="E153">A75</font></span>: 
<font color="Maroon" title="c34">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="c24">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c32">k</font>,<font color="Maroon" title="c25">j</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T99" title="GLIB_001:th.99">GLIB_001:99</a></span>;<br/><a NAME="E107:119_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c34">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="c24">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c32">k</font>,<font color="Maroon" title="c25">j</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T89" title="GLIB_001:th.89">GLIB_001:89</a></span>;<br/><a NAME="E108:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E154">A76</font></span>: 
<font color="Maroon" title="c34">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:119_1_2"><span class="lab"><font color="Green" title="E120">A31</font></span></a>, <a class="txt" href="#E72:119_1_2"><span class="lab"><font color="Green" title="E131">A44</font></span></a>, <a class="txt" href="#E92:119_1_2"><span class="lab"><font color="Green" title="E145">A67</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div><span class="lab"><font color="Green" title="E155">A77</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_2_7"><span class="kw">now </span></a><div class="add"><a NAME="E1:119_1_2_7"/><span class="lab"><font color="Green" title="E156">A78</font></span>: 
<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="c32">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><a NAME="E2:119_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E157">A79</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c32">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="c32">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:119_1_2"><span class="lab"><font color="Green" title="E135">A48</font></span></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></span>;<br/><span class="kw">assume </span><a NAME="E3:119_1_2_7"/>
<font color="Maroon" title="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">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="c24">P</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E4:119_1_2_7"/><span class="kw">then </span>
<font color="Maroon" title="c24">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c32">k</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c24">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c24">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:119_1_2"><span class="lab"><font color="Green" title="E135">A48</font></span></a>, <a class="txt" href="#E1:119_1_2_7"><span class="lab"><font color="Green" title="E156">A78</font></span></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></span>;<br/><span class="kw">hence </span><a NAME="E5:119_1_2_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:119_1_2"><span class="lab"><font color="Green" title="E112">A22</font></span></a>, <a class="txt" href="#E80:119_1_2"><span class="lab"><font color="Green" title="E135">A48</font></span></a>, <a class="txt" href="#E2:119_1_2_7"><span class="lab"><font color="Green" title="E157">A79</font></span></a>, <a class="ref" href="chord.html#D2" target="_self" title="CHORD:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E156">A80</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_2_8"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c40">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c40">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  implies <font color="Maroon" title="c40">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A</font> )</span><br/><span class="kw">assume </span><a NAME="E1:119_1_2_8"/><span class="lab"><font color="Green" title="E157">A81</font></span>: 
<font color="Maroon" title="c40">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c40">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c41">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> <span class="kw"> such that </span><br/><a NAME="E3:119_1_2_8"/><span class="lab"><font color="Green" title="E158">A82</font></span>: 
<font color="Maroon" title="c41">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="c34">Q</font>
 <span class="kw">and </span><br/><a NAME="E4:119_1_2_8"/><span class="lab"><font color="Green" title="E159">A83</font></span>: 
<font color="Maroon" title="c34">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119_1_2_8"><span class="lab"><font color="Green" title="E157">A81</font></span></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></span>;<br/><a NAME="E5:119_1_2_8"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c41">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><a NAME="E6:119_1_2_8"/><span class="kw">then </span>
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">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c42">n1</font> = <font color="Maroon" title="c41">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c43">kn1</font> = <font color="Maroon" title="c32">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c42">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="E9:119_1_2_8"/><span class="lab"><font color="Green" title="E160">A84</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">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="c24">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E10:119_1_2_8"/>
<font color="Maroon" title="c41">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="c41">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E11:119_1_2_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E161">A85</font></span>: 
<font color="Maroon" title="c42">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="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:119_1_2_8"><span class="lab"><font color="Green" title="E158">A82</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:119_1_2_8"/><span class="kw">then </span>
<font color="Maroon" title="c32">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c42">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="c24">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="c32">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c32">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E87:119_1_2"><span class="lab"><font color="Green" title="E140">A53</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E13:119_1_2_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E162">A86</font></span>: 
<font color="Maroon" title="c43">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="c24">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:119_1_2_8"><span class="lab"><font color="Green" title="E160">A84</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E14:119_1_2_8"/><span class="lab"><font color="Green" title="E163">A87</font></span>: 
<font color="Maroon" title="c34">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">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="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">kn1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E84:119_1_2"><span class="lab"><font color="Green" title="E137">A50</font></span></a>, <a class="txt" href="#E85:119_1_2"><span class="lab"><font color="Green" title="E138">A51</font></span></a>, <a class="txt" href="#E11:119_1_2_8"><span class="lab"><font color="Green" title="E161">A85</font></span></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_2_8_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:119_1_2_8_1"/><span class="lab"><font color="Green" title="E164">A88</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><span class="kw">assume </span><a NAME="E2:119_1_2_8_1"/><span class="lab"><font color="Green" title="E165">A89</font></span>: 
<font color="Maroon" title="c40">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:119_1_2_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E166">A90</font></span>: 
<font color="Maroon" title="c32">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:119_1_2"><span class="lab"><font color="Green" title="E114">A24</font></span></a>, <a class="txt" href="#E4:119_1_2_8"><span class="lab"><font color="Green" title="E159">A83</font></span></a>, <a class="txt" href="#E14:119_1_2_8"><span class="lab"><font color="Green" title="E163">A87</font></span></a>, <a class="txt" href="#E13:119_1_2_8"><span class="lab"><font color="Green" title="E162">A86</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><div><span class="lab"><font color="Green" title="E167">A91</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_2_8_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:119_1_2_8_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c41">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:119_1_2_8_1_1"/><span class="kw">then </span>
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="c41">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:119_1_2_8_1_1"/><span class="kw">then </span>
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="c32">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c41">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119_1_2_8_1"><span class="lab"><font color="Green" title="E164">A88</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><span class="kw">hence </span><a NAME="E4:119_1_2_8_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:119_1_2_8_1"><span class="lab"><font color="Green" title="E166">A90</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E5:119_1_2_8_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c41">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><span class="kw">hence </span><a NAME="E6:119_1_2_8_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:119_1_2"><span class="lab"><font color="Green" title="E114">A24</font></span></a>, <a class="txt" href="#E109:119_1_2"><span class="lab"><font color="Green" title="E155">A77</font></span></a>, <a class="txt" href="#E89:119_1_2"><span class="lab"><font color="Green" title="E142">A64</font></span></a>, <a class="txt" href="#E4:119_1_2_8"><span class="lab"><font color="Green" title="E159">A83</font></span></a>, <a class="txt" href="#E2:119_1_2_8_1"><span class="lab"><font color="Green" title="E165">A89</font></span></a>, <a class="txt" href="#E4:119_1_2_8_1"><span class="lab"><font color="Green" title="E167">A91</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E16:119_1_2_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E164">A92</font></span>: 
not <font color="Maroon" title="c40">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="c18">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E17:119_1_2_8"/>
<font color="Maroon" title="c34">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="c24">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E84:119_1_2"><span class="lab"><font color="Green" title="E137">A50</font></span></a>, <a class="txt" href="#E85:119_1_2"><span class="lab"><font color="Green" title="E138">A51</font></span></a>, <a class="txt" href="#E106:119_1_2"><span class="lab"><font color="Green" title="E153">A75</font></span></a>, <a class="ref" href="glib_001.html#T95" title="GLIB_001:th.95">GLIB_001:95</a></span>;<br/><a NAME="E18:119_1_2_8"/><span class="kw">then </span>
<font color="Maroon" title="c40">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119_1_2_8"><span class="lab"><font color="Green" title="E157">A81</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E19:119_1_2_8"/>
<font color="Maroon" title="c40">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:119_1_2"><span class="lab"><font color="Green" title="E120">A31</font></span></a>, <a class="txt" href="#E16:119_1_2_8"><span class="lab"><font color="Green" title="E164">A92</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E111:119_1_2"/>
<font color="Maroon" title="c34">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="c24">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c32">k</font>,<font color="Maroon" title="c25">j</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E106:119_1_2"><span class="lab"><font color="Green" title="E153">A75</font></span></a>, <a class="ref" href="glib_001.html#T89" title="GLIB_001:th.89">GLIB_001:89</a></span>;<br/><a NAME="E112:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E157">A93</font></span>: 
<font color="Maroon" title="c34">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:119_1_2"><span class="lab"><font color="Green" title="E120">A31</font></span></a>, <a class="txt" href="#E72:119_1_2"><span class="lab"><font color="Green" title="E131">A44</font></span></a>, <a class="txt" href="#E92:119_1_2"><span class="lab"><font color="Green" title="E145">A67</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E113:119_1_2"/>
<font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c18">y</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:119_1_2"><span class="lab"><font color="Green" title="E123">A34</font></span></a>, <a class="txt" href="#E100:119_1_2"><span class="lab"><font color="Green" title="E148">A70</font></span></a>, <a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c40">R</font> being   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E115:119_1_2"/><span class="lab"><font color="Green" title="E158">A94</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 7
 <span class="kw">and </span><br/><a NAME="E116:119_1_2"/><span class="lab"><font color="Green" title="E159">A95</font></span>: 
<font color="Maroon" title="c40">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
 <span class="kw">and </span><br/><a NAME="E117:119_1_2"/><span class="lab"><font color="Green" title="E160">A96</font></span>: 
<font color="Maroon" title="c40">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="c34">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span>,<font color="Maroon" title="c18">y</font>,<font color="Maroon" title="c30">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c34">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>
 <span class="kw">and </span><br/><a NAME="E118:119_1_2"/><span class="lab"><font color="Green" title="E161">A97</font></span>: 
<font color="Maroon" title="c40">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="c34">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <span class="kw">and </span><br/><a NAME="E119:119_1_2"/><span class="lab"><font color="Green" title="E162">A98</font></span>: 
<font color="Maroon" title="c40">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="c18">y</font>
 <span class="kw">and </span><br/><a NAME="E120:119_1_2"/><span class="lab"><font color="Green" title="E163">A99</font></span>: 
<font color="Maroon" title="c40">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>
 <span class="kw">and </span><br/><a NAME="E121:119_1_2"/><span class="lab"><font color="Green" title="E164">A100</font></span>: 
<font color="Maroon" title="c40">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="c34">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:119_1_2"><span class="lab"><font color="Green" title="E114">A24</font></span></a>, <a class="txt" href="#E53:119_1_2"><span class="lab"><font color="Green" title="E118">A29</font></span></a>, <a class="txt" href="#E91:119_1_2"><span class="lab"><font color="Green" title="E144">A66</font></span></a>, <a class="txt" href="#E99:119_1_2"><span class="lab"><font color="Green" title="E147">A69</font></span></a>, <a class="txt" href="#E109:119_1_2"><span class="lab"><font color="Green" title="E155">A77</font></span></a>, <a class="txt" href="#E89:119_1_2"><span class="lab"><font color="Green" title="E142">A64</font></span></a>, <a class="txt" href="#E100:119_1_2"><span class="lab"><font color="Green" title="E148">A70</font></span></a>, <a class="txt" href="#E108:119_1_2"><span class="lab"><font color="Green" title="E154">A76</font></span></a>, <a class="txt" href="#E112:119_1_2"><span class="lab"><font color="Green" title="E157">A93</font></span></a>, <a class="txt" href="#E90:119_1_2"><span class="lab"><font color="Green" title="E143">A65</font></span></a>, <a class="ref" href="chord.html#T48" target="_self" title="CHORD:th.48">Th48</a></span>;<br/><a NAME="E122:119_1_2"/><span class="lab"><font color="Green" title="E165">A101</font></span>: 
not <font color="Maroon" title="c40">R</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E115:119_1_2"><span class="lab"><font color="Green" title="E158">A94</font></span></a>, <a class="ref" href="glib_001.html#T127" title="GLIB_001:th.127">GLIB_001:127</a></span>;<br/><a NAME="E123:119_1_2"/><span class="lab"><font color="Green" title="E166">A102</font></span>: 
<font color="Maroon" title="c34">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="c40">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="119_1_2_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:119_1_2_9"/>
not <font color="Maroon" title="c34">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="c40">R</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:119_1_2_9"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c34">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="c40">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c41">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:119_1_2_9"/><span class="lab"><font color="Green" title="E167">A103</font></span>: 
<font color="Maroon" title="c41">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">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="c40">R</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E5:119_1_2_9"/><span class="lab"><font color="Green" title="E168">A104</font></span>: 
<font color="Maroon" title="c41">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:119_1_2_9"><span class="lab"><font color="Green" title="E167">A103</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E6:119_1_2_9"/>
<font color="Maroon" title="c41">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c40">R</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:119_1_2_9"><span class="lab"><font color="Green" title="E167">A103</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c42">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> <span class="kw"> such that </span><br/><a NAME="E8:119_1_2_9"/><span class="lab"><font color="Green" title="E169">A105</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c42">n</font>
 <span class="kw">and </span><br/><a NAME="E9:119_1_2_9"/><span class="lab"><font color="Green" title="E170">A106</font></span>: 
<font color="Maroon" title="c42">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 7
 <span class="kw">and </span><br/><a NAME="E10:119_1_2_9"/><span class="lab"><font color="Green" title="E171">A107</font></span>: 
<font color="Maroon" title="c40">R</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">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E115:119_1_2"><span class="lab"><font color="Green" title="E158">A94</font></span></a>, <a class="ref" href="glib_001.html#T100" title="GLIB_001:th.100">GLIB_001:100</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_9_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:119_1_2_9_1"/>
( <font color="Maroon" title="c42">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c42">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or <font color="Maroon" title="c42">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:119_1_2_9"><span class="lab"><font color="Green" title="E169">A105</font></span></a>, <a class="txt" href="#E9:119_1_2_9"><span class="lab"><font color="Green" title="E170">A106</font></span></a>, <a class="ref" href="chord.html#T13" target="_self" title="CHORD:th.13">Th13</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_9_1_1"><span class="kw">suppose </span></a><a NAME="E1:119_1_2_9_1_1"/><span class="lab"><font color="Green" title="E172">A108</font></span>: 
<font color="Maroon" title="c42">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:119_1_2_9_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="c40">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E115:119_1_2"><span class="lab"><font color="Green" title="E158">A94</font></span></a></span>;<br/><a NAME="E3:119_1_2_9_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c40">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="c40">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1,<font color="Maroon" title="c40">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></span>;<br/><a NAME="E4:119_1_2_9_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E119:119_1_2"><span class="lab"><font color="Green" title="E162">A98</font></span></a>, <a class="txt" href="#E5:119_1_2_9"><span class="lab"><font color="Green" title="E168">A104</font></span></a>, <a class="txt" href="#E10:119_1_2_9"><span class="lab"><font color="Green" title="E171">A107</font></span></a>, <a class="txt" href="#E1:119_1_2_9_1_1"><span class="lab"><font color="Green" title="E172">A108</font></span></a>, <a class="ref" href="glib_001.html#T106" title="GLIB_001:th.106">GLIB_001:106</a></span>;<br/><a NAME="E5:119_1_2_9_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E110:119_1_2"><span class="lab"><font color="Green" title="E156">A80</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:119_1_2_9_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:119_1_2"><span class="lab"><font color="Green" title="E106">A16</font></span></a>, <a class="txt" href="#E15:119_1_2"><span class="lab"><font color="Green" title="E103">A12</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_9_1_2"><span class="kw">suppose </span></a><a NAME="E1:119_1_2_9_1_2"/><span class="lab"><font color="Green" title="E172">A109</font></span>: 
<font color="Maroon" title="c42">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:119_1_2_9_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="c40">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E115:119_1_2"><span class="lab"><font color="Green" title="E158">A94</font></span></a></span>;<br/><a NAME="E3:119_1_2_9_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c40">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="c40">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3,<font color="Maroon" title="c40">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></span>;<br/><a NAME="E4:119_1_2_9_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E119:119_1_2"><span class="lab"><font color="Green" title="E162">A98</font></span></a>, <a class="txt" href="#E5:119_1_2_9"><span class="lab"><font color="Green" title="E168">A104</font></span></a>, <a class="txt" href="#E10:119_1_2_9"><span class="lab"><font color="Green" title="E171">A107</font></span></a>, <a class="txt" href="#E1:119_1_2_9_1_2"><span class="lab"><font color="Green" title="E172">A109</font></span></a>, <a class="ref" href="glib_001.html#T106" title="GLIB_001:th.106">GLIB_001:106</a></span>;<br/><a NAME="E5:119_1_2_9_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E110:119_1_2"><span class="lab"><font color="Green" title="E156">A80</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:119_1_2_9_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:119_1_2"><span class="lab"><font color="Green" title="E106">A16</font></span></a>, <a class="txt" href="#E15:119_1_2"><span class="lab"><font color="Green" title="E103">A12</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_9_1_3"><span class="kw">suppose </span></a><a NAME="E1:119_1_2_9_1_3"/><span class="lab"><font color="Green" title="E172">A110</font></span>: 
<font color="Maroon" title="c42">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:119_1_2_9_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="c40">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E115:119_1_2"><span class="lab"><font color="Green" title="E158">A94</font></span></a></span>;<br/><a NAME="E3:119_1_2_9_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c40">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="c40">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 5,<font color="Maroon" title="c40">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></span>;<br/><a NAME="E4:119_1_2_9_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E120:119_1_2"><span class="lab"><font color="Green" title="E163">A99</font></span></a>, <a class="txt" href="#E5:119_1_2_9"><span class="lab"><font color="Green" title="E168">A104</font></span></a>, <a class="txt" href="#E10:119_1_2_9"><span class="lab"><font color="Green" title="E171">A107</font></span></a>, <a class="txt" href="#E1:119_1_2_9_1_3"><span class="lab"><font color="Green" title="E172">A110</font></span></a>, <a class="ref" href="glib_001.html#T106" title="GLIB_001:th.106">GLIB_001:106</a></span>;<br/><a NAME="E5:119_1_2_9_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E110:119_1_2"><span class="lab"><font color="Green" title="E156">A80</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:119_1_2_9_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:119_1_2"><span class="lab"><font color="Green" title="E103">A12</font></span></a>, <a class="txt" href="#E68:119_1_2"><span class="lab"><font color="Green" title="E128">A41</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_2_10"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c34">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="c34">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="c34">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="c40">R</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:119_1_2_10"/><span class="lab"><font color="Green" title="E167">A111</font></span>: 
<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#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c34">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="c34">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c34">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="c40">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="119_1_2_10_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:119_1_2_10_1"/>
( <font color="Maroon" title="c41">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Maroon" title="c41">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119_1_2_10"><span class="lab"><font color="Green" title="E167">A111</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_10_1_1"><span class="kw">suppose </span></a><a NAME="E1:119_1_2_10_1_1"/><span class="lab"><font color="Green" title="E168">A112</font></span>: 
<font color="Maroon" title="c41">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c34">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="c40">R</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span></span><br/><div class="add"><a NAME="E2:119_1_2_10_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E169">A113</font></span>: 
<font color="Maroon" title="c41">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T89" title="GLIB_001:th.89">GLIB_001:89</a></span>;<br/><a NAME="E3:119_1_2_10_1_1"/>
<font color="Maroon" title="c41">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c40">R</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E115:119_1_2"><span class="lab"><font color="Green" title="E158">A94</font></span></a>, <a class="txt" href="#E121:119_1_2"><span class="lab"><font color="Green" title="E164">A100</font></span></a>, <a class="txt" href="#E1:119_1_2_10_1_1"><span class="lab"><font color="Green" title="E168">A112</font></span></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></span>;<br/><span class="kw">hence </span><a NAME="E4:119_1_2_10_1_1"/>
<font color="Maroon" title="c41">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">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="c40">R</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:119_1_2_10_1_1"><span class="lab"><font color="Green" title="E169">A113</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_10_1_2"><span class="kw">suppose </span></a><a NAME="E1:119_1_2_10_1_2"/><span class="lab"><font color="Green" title="E168">A114</font></span>: 
<font color="Maroon" title="c41">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c34">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="c40">R</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span></span><br/><div class="add"><a NAME="E2:119_1_2_10_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="c40">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E115:119_1_2"><span class="lab"><font color="Green" title="E158">A94</font></span></a></span>;<br/><a NAME="E3:119_1_2_10_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E169">A115</font></span>: 
<font color="Maroon" title="c41">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c40">R</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E118:119_1_2"><span class="lab"><font color="Green" title="E161">A97</font></span></a>, <a class="txt" href="#E1:119_1_2_10_1_2"><span class="lab"><font color="Green" title="E168">A114</font></span></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></span>;<br/><a NAME="E4:119_1_2_10_1_2"/>
<font color="Maroon" title="c41">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119_1_2_10_1_2"><span class="lab"><font color="Green" title="E168">A114</font></span></a>, <a class="ref" href="glib_001.html#T89" title="GLIB_001:th.89">GLIB_001:89</a></span>;<br/><span class="kw">hence </span><a NAME="E5:119_1_2_10_1_2"/>
<font color="Maroon" title="c41">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">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="c40">R</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:119_1_2_10_1_2"><span class="lab"><font color="Green" title="E169">A115</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E125:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E167">A116</font></span>: 
<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="c34">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="c34">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="c34">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="c40">R</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_2_11"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c41">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">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="c40">R</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> implies <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#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c34">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="c34">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/><span class="kw">assume </span><a NAME="E1:119_1_2_11"/><span class="lab"><font color="Green" title="E168">A117</font></span>: 
<font color="Maroon" title="c41">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">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="c40">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c34">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="c34">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="E2:119_1_2_11"/>
<font color="Maroon" title="c41">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="c34">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span>,<font color="Maroon" title="c18">y</font>,<font color="Maroon" title="c30">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c34">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E117:119_1_2"><span class="lab"><font color="Green" title="E160">A96</font></span></a>, <a class="txt" href="#E1:119_1_2_11"><span class="lab"><font color="Green" title="E168">A117</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:119_1_2_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E169">A118</font></span>: 
( <font color="Maroon" title="c41">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  or <font color="Maroon" title="c41">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y</font> or <font color="Maroon" title="c41">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">x</font> or <font color="Maroon" title="c41">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></span>;<br/><a NAME="E4:119_1_2_11"/>
<font color="Maroon" title="c41">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119_1_2_11"><span class="lab"><font color="Green" title="E168">A117</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:119_1_2_11"/><span class="kw">then </span>
<font color="Maroon" title="c41">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E110:119_1_2"><span class="lab"><font color="Green" title="E156">A80</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:119_1_2_11"/>
<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#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c34">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="c34">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E26:119_1_2"><span class="lab"><font color="Green" title="E106">A16</font></span></a>, <a class="txt" href="#E15:119_1_2"><span class="lab"><font color="Green" title="E103">A12</font></span></a>, <a class="txt" href="#E68:119_1_2"><span class="lab"><font color="Green" title="E128">A41</font></span></a>, <a class="txt" href="#E3:119_1_2_11"><span class="lab"><font color="Green" title="E169">A118</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E127:119_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c34">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="c40">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="c34">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="c34">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 class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E128:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E168">A119</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c34">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="c40">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="c34">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="c34">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E125:119_1_2"><span class="lab"><font color="Green" title="E167">A116</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><div><span class="lab"><font color="Green" title="E169">A120</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_2_12"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c34">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="c34">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="c34">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="c18">y</font>,<font color="Maroon" title="c1">G</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:119_1_2_12"/><span class="lab"><font color="Green" title="E170">A121</font></span>: 
<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="c34">Q</font>
 <span class="kw">and </span><br/><a NAME="E2:119_1_2_12"/><span class="lab"><font color="Green" title="E171">A122</font></span>: 
<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="c34">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">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="c34">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="c18">y</font>,<font color="Maroon" title="c1">G</font></span><br/><a NAME="E3:119_1_2_12"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c41">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119_1_2_12"><span class="lab"><font color="Green" title="E170">A121</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E4:119_1_2_12"/><span class="kw">then </span>
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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c42">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c43">ki1</font> = <font color="Maroon" title="c32">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c42">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="E7:119_1_2_12"/><span class="lab"><font color="Green" title="E172">A123</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E8:119_1_2_12"/><span class="lab"><font color="Green" title="E173">A124</font></span>: 
<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="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119_1_2_12"><span class="lab"><font color="Green" title="E170">A121</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E9:119_1_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E174">A125</font></span>: 
<font color="Maroon" title="c42">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="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:119_1_2_12"><span class="lab"><font color="Green" title="E172">A123</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E10:119_1_2_12"/><span class="kw">then </span>
<font color="Maroon" title="c43">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="c24">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E84:119_1_2"><span class="lab"><font color="Green" title="E137">A50</font></span></a>, <a class="txt" href="#E85:119_1_2"><span class="lab"><font color="Green" title="E138">A51</font></span></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></span>;<br/><a NAME="E11:119_1_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E175">A126</font></span>: 
<font color="Maroon" title="c43">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="c24">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E12:119_1_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E176">A127</font></span>: 
<font color="Maroon" title="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">ki1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:119_1_2"><span class="lab"><font color="Green" title="E120">A31</font></span></a>, <a class="ref" href="glib_001.html#T8" title="GLIB_001:th.8">GLIB_001:8</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_2_12_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:119_1_2_12_1"/><span class="lab"><font color="Green" title="E177">A128</font></span>: 
<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="c24">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c32">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="c24">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c32">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T32" title="XREAL_1:th.32">XREAL_1:32</a></span>;<br/><span class="kw">assume </span><a NAME="E2:119_1_2_12_1"/>
<font color="Maroon" title="c43">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="c24">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E3:119_1_2_12_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E87:119_1_2"><span class="lab"><font color="Green" title="E140">A53</font></span></a>, <a class="txt" href="#E8:119_1_2_12"><span class="lab"><font color="Green" title="E173">A124</font></span></a>, <a class="txt" href="#E7:119_1_2_12"><span class="lab"><font color="Green" title="E172">A123</font></span></a>, <a class="txt" href="#E1:119_1_2_12_1"><span class="lab"><font color="Green" title="E177">A128</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E14:119_1_2_12"/><span class="kw">then </span>
<font color="Maroon" title="c43">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="c24">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:119_1_2_12"><span class="lab"><font color="Green" title="E175">A126</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E15:119_1_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E177">A129</font></span>: 
<font color="Maroon" title="c43">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="c24">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></span>;<br/><a NAME="E16:119_1_2_12"/>
<font color="Maroon" title="c43">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="c24">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E87:119_1_2"><span class="lab"><font color="Green" title="E140">A53</font></span></a>, <a class="txt" href="#E2:119_1_2_12"><span class="lab"><font color="Green" title="E171">A122</font></span></a></span>;<br/><a NAME="E17:119_1_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E178">A130</font></span>: 
<font color="Maroon" title="c43">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="c24">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:119_1_2_12"><span class="lab"><font color="Green" title="E177">A129</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E18:119_1_2_12"/><span class="lab"><font color="Green" title="E179">A131</font></span>: 
<font color="Maroon" title="c24">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="c24">P</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:119_1_2"><span class="lab"><font color="Green" title="E120">A31</font></span></a>, <a class="ref" href="glib_001.html#T8" title="GLIB_001:th.8">GLIB_001:8</a></span>;<br/><span class="kw">assume </span><a NAME="E19:119_1_2_12"/><span class="lab"><font color="Green" title="E180">A132</font></span>: 
 ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c34">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="c18">y</font>,<font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E20:119_1_2_12"/>
<font color="Maroon" title="c34">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c42">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="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">ki1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E84:119_1_2"><span class="lab"><font color="Green" title="E137">A50</font></span></a>, <a class="txt" href="#E85:119_1_2"><span class="lab"><font color="Green" title="E138">A51</font></span></a>, <a class="txt" href="#E9:119_1_2_12"><span class="lab"><font color="Green" title="E174">A125</font></span></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></span>;<br/><span class="kw">hence </span><a NAME="E21:119_1_2_12"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:119_1_2"><span class="lab"><font color="Green" title="E112">A22</font></span></a>, <a class="txt" href="#E45:119_1_2"><span class="lab"><font color="Green" title="E114">A24</font></span></a>, <a class="txt" href="#E19:119_1_2_12"><span class="lab"><font color="Green" title="E180">A132</font></span></a>, <a class="txt" href="#E17:119_1_2_12"><span class="lab"><font color="Green" title="E178">A130</font></span></a>, <a class="txt" href="#E12:119_1_2_12"><span class="lab"><font color="Green" title="E176">A127</font></span></a>, <a class="txt" href="#E18:119_1_2_12"><span class="lab"><font color="Green" title="E179">A131</font></span></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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E130:119_1_2"/><span class="lab"><font color="Green" title="E170">A133</font></span>: 
<font color="Maroon" title="c34">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="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E89:119_1_2"><span class="lab"><font color="Green" title="E142">A64</font></span></a></span>;<br/><a NAME="E131:119_1_2"/>
<font color="Maroon" title="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E103:119_1_2"><span class="lab"><font color="Green" title="E150">A72</font></span></a>, <a class="txt" href="#E80:119_1_2"><span class="lab"><font color="Green" title="E135">A48</font></span></a>, <a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></span>;<br/><a NAME="E132:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E171">A134</font></span>: 
not <font color="Maroon" title="c34">Q</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E130:119_1_2"><span class="lab"><font color="Green" title="E170">A133</font></span></a>, <a class="txt" href="#E101:119_1_2"><span class="lab"><font color="Green" title="E149">A71</font></span></a>, <a class="ref" href="glib_001.html#T128" title="GLIB_001:th.128">GLIB_001:128</a></span>;<br/><a NAME="E133:119_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c34">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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D26" title="GLIB_001:def.26">GLIB_001:def 26</a></span>;<br/><a NAME="E134:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E172">A135</font></span>: 
<font color="Maroon" title="c34">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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c41">C</font> = <font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">R</font>;<br/><a NAME="E135:119_1_2"/><span class="lab"><font color="Green" title="E173">A136</font></span>: 
<font color="Maroon" title="c40">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="c34">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E118:119_1_2"><span class="lab"><font color="Green" title="E161">A97</font></span></a></span>;<br/><a NAME="E136:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E174">A137</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">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="c34">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="c40">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E115:119_1_2"><span class="lab"><font color="Green" title="E158">A94</font></span></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">GLIB_001:34</a></span>;<br/><a NAME="E137:119_1_2"/><span class="lab"><font color="Green" title="E175">A138</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">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="c34">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="c40">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E115:119_1_2"><span class="lab"><font color="Green" title="E158">A94</font></span></a>, <a class="txt" href="#E135:119_1_2"><span class="lab"><font color="Green" title="E173">A136</font></span></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">GLIB_001:34</a></span>;<br/><a NAME="E138:119_1_2"/><span class="lab"><font color="Green" title="E176">A139</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">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="c34">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="c40">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E115:119_1_2"><span class="lab"><font color="Green" title="E158">A94</font></span></a>, <a class="txt" href="#E135:119_1_2"><span class="lab"><font color="Green" title="E173">A136</font></span></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">GLIB_001:34</a></span>;<br/><a NAME="E139:119_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">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="c34">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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E116:119_1_2"><span class="lab"><font color="Green" title="E159">A95</font></span></a>, <a class="txt" href="#E135:119_1_2"><span class="lab"><font color="Green" title="E173">A136</font></span></a>, <a class="ref" href="chord.html#T28" target="_self" title="CHORD:th.28">Th28</a></span>;<br/><a NAME="E140:119_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E134:119_1_2"><span class="lab"><font color="Green" title="E172">A135</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E141:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E177">A140</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E142:119_1_2"/><span class="lab"><font color="Green" title="E178">A141</font></span>: 
<font color="Maroon" title="c40">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="c34">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E115:119_1_2"><span class="lab"><font color="Green" title="E158">A94</font></span></a>, <a class="txt" href="#E121:119_1_2"><span class="lab"><font color="Green" title="E164">A100</font></span></a></span>;<br/><a NAME="E143:119_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E179">A142</font></span>: 
not <font color="Maroon" title="c40">R</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E104:119_1_2"><span class="lab"><font color="Green" title="E151">A73</font></span></a>, <a class="txt" href="#E135:119_1_2"><span class="lab"><font color="Green" title="E173">A136</font></span></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></span>;<br/><a NAME="E144:119_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">R</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E105:119_1_2"><span class="lab"><font color="Green" title="E152">A74</font></span></a>, <a class="txt" href="#E135:119_1_2"><span class="lab"><font color="Green" title="E173">A136</font></span></a>, <a class="txt" href="#E142:119_1_2"><span class="lab"><font color="Green" title="E178">A141</font></span></a>, <a class="txt" href="#E122:119_1_2"><span class="lab"><font color="Green" title="E165">A101</font></span></a>, <a class="txt" href="#E123:119_1_2"><span class="lab"><font color="Green" title="E166">A102</font></span></a>, <a class="txt" href="#E128:119_1_2"><span class="lab"><font color="Green" title="E168">A119</font></span></a>, <a class="ref" href="chord.html#T27" target="_self" title="CHORD:th.27">Th27</a></span>;<br/><a NAME="E145:119_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">R</font> is  <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E141:119_1_2"><span class="lab"><font color="Green" title="E177">A140</font></span></a>, <a class="ref" href="chord.html#D11" target="_self" title="CHORD:def.11">Def11</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c42">m</font>, <font color="Maroon" title="c43">n</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E147:119_1_2"/><span class="lab"><font color="Green" title="E180">A143</font></span>: 
<font color="Maroon" title="c42">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c43">n</font>
 <span class="kw">and </span><br/><a NAME="E148:119_1_2"/><span class="lab"><font color="Green" title="E181">A144</font></span>: 
<font color="Maroon" title="c43">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">R</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E149:119_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">n</font>
 <span class="kw">and </span><br/><a NAME="E150:119_1_2"/><span class="lab"><font color="Green" title="E182">A145</font></span>: 
 ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">n</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">and </span><br/><a NAME="E151:119_1_2"/><span class="lab"><font color="Green" title="E183">A146</font></span>: 
( <font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">R</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>  implies ( (  not <font color="Maroon" title="c42">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Maroon" title="c43">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="c34">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">R</font></span>)</span> ) &amp; (  not <font color="Maroon" title="c42">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">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="c42">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or  not <font color="Maroon" title="c43">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="c34">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">R</font></span>)</span> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T84" target="_self" title="CHORD:th.84">Th84</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c44">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E153:119_1_2"/><span class="lab"><font color="Green" title="E184">A147</font></span>: 
<font color="Maroon" title="c44">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">n</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E150:119_1_2"><span class="lab"><font color="Green" title="E182">A145</font></span></a></span>;<br/><a NAME="E154:119_1_2"/><span class="lab"><font color="Green" title="E185">A148</font></span>: 
<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="c34">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">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="c34">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="c40">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E135:119_1_2"><span class="lab"><font color="Green" title="E173">A136</font></span></a>, <a class="ref" href="glib_001.html#T29" title="GLIB_001:th.29">GLIB_001:29</a></span>;<br/><a NAME="E155:119_1_2"/><span class="lab"><font color="Green" title="E186">A149</font></span>: 
( <font color="Maroon" title="c43">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font> or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">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="119_1_2_13"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_13_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:119_1_2_13_1"/>
( <font color="Maroon" title="c43">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font> or <font color="Maroon" title="c43">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_13_1_1"><span class="kw">suppose </span></a><a NAME="E1:119_1_2_13_1_1"/>
<font color="Maroon" title="c43">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c43">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font> or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6 )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:119_1_2_13_1_1"/>
( <font color="Maroon" title="c43">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font> or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_13_1_2"><span class="kw">suppose </span></a><a NAME="E1:119_1_2_13_1_2"/>
<font color="Maroon" title="c43">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c43">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font> or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6 )</span><br/><div class="add"><a NAME="E2:119_1_2_13_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E187">A150</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">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="c43">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_13_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:119_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="c34">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="c43">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="c34">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="c43">n</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:119_1_2_13_1_2"><span class="lab"><font color="Green" title="E187">A150</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_13_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:119_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="c34">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="c43">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c43">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font> or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6 )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:119_1_2_13_1_2_1_1"/>
( <font color="Maroon" title="c43">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font> or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_13_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:119_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="c34">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="c43">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c43">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font> or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6 )</span><br/><div class="add"><a NAME="E2:119_1_2_13_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E188">A151</font></span>: 
<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="c34">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="c43">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_13_1_2_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:119_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="c34">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="c43">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="c34">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="c43">n</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:119_1_2_13_1_2_1_2"><span class="lab"><font color="Green" title="E188">A151</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_13_1_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:119_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="c34">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="c43">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c43">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font> or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6 )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:119_1_2_13_1_2_1_2_1_1"/>
( <font color="Maroon" title="c43">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font> or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_13_1_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:119_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="c34">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="c43">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c43">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font> or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6 )</span><br/><div class="add"><a NAME="E2:119_1_2_13_1_2_1_2_1_2"/><span class="kw">then </span>
<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="c34">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="c43">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:119_1_2_13_1_2_1_2_1_2"/>
( <font color="Maroon" title="c43">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font> or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E115:119_1_2"><span class="lab"><font color="Green" title="E158">A94</font></span></a>, <a class="txt" href="#E154:119_1_2"><span class="lab"><font color="Green" title="E185">A148</font></span></a>, <a class="txt" href="#E148:119_1_2"><span class="lab"><font color="Green" title="E181">A144</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E156:119_1_2"/><span class="lab"><font color="Green" title="E187">A152</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c42">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_14"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:119_1_2_14"/>
( <font color="Maroon" title="c43">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font> or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 or <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E155:119_1_2"><span class="lab"><font color="Green" title="E186">A149</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_14_1"><span class="kw">suppose </span></a><a NAME="E1:119_1_2_14_1"/><span class="lab"><font color="Green" title="E188">A153</font></span>: 
<font color="Maroon" title="c43">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:119_1_2_14_1"/><span class="lab"><font color="Green" title="E189">A154</font></span>: 
<font color="Maroon" title="c42">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c42">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E3:119_1_2_14_1"/>
<font color="Maroon" title="c42">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#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="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E147:119_1_2"><span class="lab"><font color="Green" title="E180">A143</font></span></a>, <a class="txt" href="#E1:119_1_2_14_1"><span class="lab"><font color="Green" title="E188">A153</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:119_1_2_14_1"/><span class="kw">then </span>
<font color="Maroon" title="c42">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="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:119_1_2_14_1"><span class="lab"><font color="Green" title="E189">A154</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E5:119_1_2_14_1"/><span class="kw">then </span>
<font color="Maroon" title="c42">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E156:119_1_2"><span class="lab"><font color="Green" title="E187">A152</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E6:119_1_2_14_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E190">A155</font></span>: 
<font color="Maroon" title="c34">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T33" title="GLIB_001:th.33">GLIB_001:33</a></span>;<br/><a NAME="E7:119_1_2_14_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><a NAME="E8:119_1_2_14_1"/><span class="kw">then </span>
<font color="Maroon" title="c43">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119_1_2_14_1"><span class="lab"><font color="Green" title="E188">A153</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E9:119_1_2_14_1"/><span class="kw">then </span>
<font color="Maroon" title="c34">Q</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> <span class="p1">(<span class="default"><font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T33" title="GLIB_001:th.33">GLIB_001:33</a></span>;<br/><span class="kw">hence </span><a NAME="E10:119_1_2_14_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E86:119_1_2"><span class="lab"><font color="Green" title="E139">A52</font></span></a>, <a class="txt" href="#E147:119_1_2"><span class="lab"><font color="Green" title="E180">A143</font></span></a>, <a class="txt" href="#E153:119_1_2"><span class="lab"><font color="Green" title="E184">A147</font></span></a>, <a class="txt" href="#E1:119_1_2_14_1"><span class="lab"><font color="Green" title="E188">A153</font></span></a>, <a class="txt" href="#E6:119_1_2_14_1"><span class="lab"><font color="Green" title="E190">A155</font></span></a>, <a class="ref" href="chord.html#T41" target="_self" title="CHORD:th.41">Th41</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_14_2"><span class="kw">suppose </span></a><a NAME="E1:119_1_2_14_2"/><span class="lab"><font color="Green" title="E188">A156</font></span>: 
<font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:119_1_2_14_2"/><span class="kw">then </span>
<font color="Maroon" title="c42">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E147:119_1_2"><span class="lab"><font color="Green" title="E180">A143</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E3:119_1_2_14_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E189">A157</font></span>: 
<font color="Maroon" title="c42">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E156:119_1_2"><span class="lab"><font color="Green" title="E187">A152</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E4:119_1_2_14_2"/><span class="kw">then </span>
<font color="Maroon" title="c44">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c34">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font>,<font color="Maroon" title="c18">y</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E119:119_1_2"><span class="lab"><font color="Green" title="E162">A98</font></span></a>, <a class="txt" href="#E138:119_1_2"><span class="lab"><font color="Green" title="E176">A139</font></span></a>, <a class="txt" href="#E153:119_1_2"><span class="lab"><font color="Green" title="E184">A147</font></span></a>, <a class="txt" href="#E1:119_1_2_14_2"><span class="lab"><font color="Green" title="E188">A156</font></span></a>, <a class="ref" href="glib_001.html#T33" title="GLIB_001:th.33">GLIB_001:33</a></span>;<br/><span class="kw">hence </span><a NAME="E5:119_1_2_14_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E129:119_1_2"><span class="lab"><font color="Green" title="E169">A120</font></span></a>, <a class="txt" href="#E147:119_1_2"><span class="lab"><font color="Green" title="E180">A143</font></span></a>, <a class="txt" href="#E1:119_1_2_14_2"><span class="lab"><font color="Green" title="E188">A156</font></span></a>, <a class="txt" href="#E3:119_1_2_14_2"><span class="lab"><font color="Green" title="E189">A157</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_14_3"><span class="kw">suppose </span></a><a NAME="E1:119_1_2_14_3"/><span class="lab"><font color="Green" title="E188">A158</font></span>: 
<font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:119_1_2_14_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c42">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 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="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E147:119_1_2"><span class="lab"><font color="Green" title="E180">A143</font></span></a>, <a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></span>;<br/><a NAME="E3:119_1_2_14_3"/><span class="kw">then </span>
<font color="Maroon" title="c42">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="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4
 ;<br/><a NAME="E4:119_1_2_14_3"/><span class="kw">then </span>
<font color="Maroon" title="c42">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="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E5:119_1_2_14_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E189">A159</font></span>: 
<font color="Maroon" title="c42">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E156:119_1_2"><span class="lab"><font color="Green" title="E187">A152</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E6:119_1_2_14_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E190">A160</font></span>: 
<font color="Maroon" title="c44">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c34">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font>,<font color="Maroon" title="c30">x</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E120:119_1_2"><span class="lab"><font color="Green" title="E163">A99</font></span></a>, <a class="txt" href="#E137:119_1_2"><span class="lab"><font color="Green" title="E175">A138</font></span></a>, <a class="txt" href="#E153:119_1_2"><span class="lab"><font color="Green" title="E184">A147</font></span></a>, <a class="txt" href="#E1:119_1_2_14_3"><span class="lab"><font color="Green" title="E188">A158</font></span></a>, <a class="ref" href="glib_001.html#T33" title="GLIB_001:th.33">GLIB_001:33</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_14_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:119_1_2_14_3_1"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">m</font> or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c42">m</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E156:119_1_2"><span class="lab"><font color="Green" title="E187">A152</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_14_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:119_1_2_14_3_1_1"/>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:119_1_2_14_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E105:119_1_2"><span class="lab"><font color="Green" title="E152">A74</font></span></a>, <a class="txt" href="#E115:119_1_2"><span class="lab"><font color="Green" title="E158">A94</font></span></a>, <a class="txt" href="#E135:119_1_2"><span class="lab"><font color="Green" title="E173">A136</font></span></a>, <a class="txt" href="#E142:119_1_2"><span class="lab"><font color="Green" title="E178">A141</font></span></a>, <a class="txt" href="#E122:119_1_2"><span class="lab"><font color="Green" title="E165">A101</font></span></a>, <a class="txt" href="#E143:119_1_2"><span class="lab"><font color="Green" title="E179">A142</font></span></a>, <a class="txt" href="#E123:119_1_2"><span class="lab"><font color="Green" title="E166">A102</font></span></a>, <a class="txt" href="#E128:119_1_2"><span class="lab"><font color="Green" title="E168">A119</font></span></a>, <a class="txt" href="#E154:119_1_2"><span class="lab"><font color="Green" title="E185">A148</font></span></a>, <a class="txt" href="#E151:119_1_2"><span class="lab"><font color="Green" title="E183">A146</font></span></a>, <a class="txt" href="#E1:119_1_2_14_3"><span class="lab"><font color="Green" title="E188">A158</font></span></a>, <a class="ref" href="chord.html#T27" target="_self" title="CHORD:th.27">Th27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_14_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:119_1_2_14_3_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c42">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:119_1_2_14_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E88:119_1_2"><span class="lab"><font color="Green" title="E141">A54</font></span></a>, <a class="txt" href="#E5:119_1_2_14_3"><span class="lab"><font color="Green" title="E189">A159</font></span></a>, <a class="txt" href="#E6:119_1_2_14_3"><span class="lab"><font color="Green" title="E190">A160</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_14_4"><span class="kw">suppose </span></a><a NAME="E1:119_1_2_14_4"/><span class="lab"><font color="Green" title="E188">A161</font></span>: 
<font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:119_1_2_14_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c42">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 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="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E147:119_1_2"><span class="lab"><font color="Green" title="E180">A143</font></span></a>, <a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></span>;<br/><a NAME="E3:119_1_2_14_4"/><span class="kw">then </span>
<font color="Maroon" title="c42">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="c34">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="E4:119_1_2_14_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E189">A162</font></span>: 
<font color="Maroon" title="c42">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="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_14_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:119_1_2_14_4_1"/>
( <font color="Maroon" title="c42">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c42">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:119_1_2_14_4"><span class="lab"><font color="Green" title="E189">A162</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_14_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:119_1_2_14_4_1_1"/><span class="lab"><font color="Green" title="E190">A163</font></span>: 
<font color="Maroon" title="c42">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_2_14_4_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:119_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="c42">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:119_1_2_14_4_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c42">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E105:119_1_2"><span class="lab"><font color="Green" title="E152">A74</font></span></a>, <a class="txt" href="#E115:119_1_2"><span class="lab"><font color="Green" title="E158">A94</font></span></a>, <a class="txt" href="#E135:119_1_2"><span class="lab"><font color="Green" title="E173">A136</font></span></a>, <a class="txt" href="#E142:119_1_2"><span class="lab"><font color="Green" title="E178">A141</font></span></a>, <a class="txt" href="#E122:119_1_2"><span class="lab"><font color="Green" title="E165">A101</font></span></a>, <a class="txt" href="#E143:119_1_2"><span class="lab"><font color="Green" title="E179">A142</font></span></a>, <a class="txt" href="#E123:119_1_2"><span class="lab"><font color="Green" title="E166">A102</font></span></a>, <a class="txt" href="#E128:119_1_2"><span class="lab"><font color="Green" title="E168">A119</font></span></a>, <a class="txt" href="#E154:119_1_2"><span class="lab"><font color="Green" title="E185">A148</font></span></a>, <a class="txt" href="#E151:119_1_2"><span class="lab"><font color="Green" title="E183">A146</font></span></a>, <a class="txt" href="#E1:119_1_2_14_4"><span class="lab"><font color="Green" title="E188">A161</font></span></a>, <a class="ref" href="chord.html#T27" target="_self" title="CHORD:th.27">Th27</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:119_1_2_14_4_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c42">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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T3" target="_self" title="CHORD:th.3">Th3</a></span>;<br/><a NAME="E4:119_1_2_14_4_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c42">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E105:119_1_2"><span class="lab"><font color="Green" title="E152">A74</font></span></a>, <a class="txt" href="#E115:119_1_2"><span class="lab"><font color="Green" title="E158">A94</font></span></a>, <a class="txt" href="#E135:119_1_2"><span class="lab"><font color="Green" title="E173">A136</font></span></a>, <a class="txt" href="#E142:119_1_2"><span class="lab"><font color="Green" title="E178">A141</font></span></a>, <a class="txt" href="#E122:119_1_2"><span class="lab"><font color="Green" title="E165">A101</font></span></a>, <a class="txt" href="#E143:119_1_2"><span class="lab"><font color="Green" title="E179">A142</font></span></a>, <a class="txt" href="#E123:119_1_2"><span class="lab"><font color="Green" title="E166">A102</font></span></a>, <a class="txt" href="#E128:119_1_2"><span class="lab"><font color="Green" title="E168">A119</font></span></a>, <a class="txt" href="#E154:119_1_2"><span class="lab"><font color="Green" title="E185">A148</font></span></a>, <a class="txt" href="#E151:119_1_2"><span class="lab"><font color="Green" title="E183">A146</font></span></a>, <a class="txt" href="#E1:119_1_2_14_4"><span class="lab"><font color="Green" title="E188">A161</font></span></a>, <a class="ref" href="chord.html#T27" target="_self" title="CHORD:th.27">Th27</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:119_1_2_14_4_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:119_1_2_14_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E191">A164</font></span>: 
<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="c42">m</font>
 ;<br/><a NAME="E4:119_1_2_14_4_1_1"/><span class="lab"><font color="Green" title="E192">A165</font></span>: 
<font color="Maroon" title="c42">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="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119_1_2_14_4_1_1"><span class="lab"><font color="Green" title="E190">A163</font></span></a>, <a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></span>;<br/><a NAME="E5:119_1_2_14_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c42">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E156:119_1_2"><span class="lab"><font color="Green" title="E187">A152</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E6:119_1_2_14_4_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c40">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T33" title="GLIB_001:th.33">GLIB_001:33</a></span>;<br/><span class="kw">hence </span><a NAME="E7:119_1_2_14_4_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E86:119_1_2"><span class="lab"><font color="Green" title="E139">A52</font></span></a>, <a class="txt" href="#E121:119_1_2"><span class="lab"><font color="Green" title="E164">A100</font></span></a>, <a class="txt" href="#E136:119_1_2"><span class="lab"><font color="Green" title="E174">A137</font></span></a>, <a class="txt" href="#E153:119_1_2"><span class="lab"><font color="Green" title="E184">A147</font></span></a>, <a class="txt" href="#E1:119_1_2_14_4"><span class="lab"><font color="Green" title="E188">A161</font></span></a>, <a class="txt" href="#E4:119_1_2_14_4_1_1"><span class="lab"><font color="Green" title="E192">A165</font></span></a>, <a class="txt" href="#E3:119_1_2_14_4_1_1"><span class="lab"><font color="Green" title="E191">A164</font></span></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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2_14_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:119_1_2_14_4_1_2"/><span class="lab"><font color="Green" title="E190">A166</font></span>: 
<font color="Maroon" title="c42">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:119_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="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E132:119_1_2"><span class="lab"><font color="Green" title="E171">A134</font></span></a>, <a class="ref" href="glib_001.html#T126" title="GLIB_001:th.126">GLIB_001:126</a></span>;<br/><a NAME="E3:119_1_2_14_4_1_2"/><span class="kw">then </span>
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="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:119_1_2_14_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E191">A167</font></span>: 
<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="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E5:119_1_2_14_4_1_2"/>
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="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E103:119_1_2"><span class="lab"><font color="Green" title="E150">A72</font></span></a>, <a class="txt" href="#E80:119_1_2"><span class="lab"><font color="Green" title="E135">A48</font></span></a>, <a class="txt" href="#E89:119_1_2"><span class="lab"><font color="Green" title="E142">A64</font></span></a>, <a class="txt" href="#E100:119_1_2"><span class="lab"><font color="Green" title="E148">A70</font></span></a>, <a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E6:119_1_2_14_4_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E129:119_1_2"><span class="lab"><font color="Green" title="E169">A120</font></span></a>, <a class="txt" href="#E119:119_1_2"><span class="lab"><font color="Green" title="E162">A98</font></span></a>, <a class="txt" href="#E121:119_1_2"><span class="lab"><font color="Green" title="E164">A100</font></span></a>, <a class="txt" href="#E136:119_1_2"><span class="lab"><font color="Green" title="E174">A137</font></span></a>, <a class="txt" href="#E138:119_1_2"><span class="lab"><font color="Green" title="E176">A139</font></span></a>, <a class="txt" href="#E153:119_1_2"><span class="lab"><font color="Green" title="E184">A147</font></span></a>, <a class="txt" href="#E1:119_1_2_14_4"><span class="lab"><font color="Green" title="E188">A161</font></span></a>, <a class="txt" href="#E1:119_1_2_14_4_1_2"><span class="lab"><font color="Green" title="E190">A166</font></span></a>, <a class="txt" href="#E4:119_1_2_14_4_1_2"><span class="lab"><font color="Green" title="E191">A167</font></span></a>, <a class="ref" href="glib_000.html#T17" title="GLIB_000:th.17">GLIB_000:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
