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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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp;  not <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  holds <br/> for <font color="Olive" title="b3">S</font> being    <a href="chord.html#M2" title="CHORD:mode.2">VertexSeparator</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>  st <font color="Olive" title="b3">S</font> is  <a href="chord.html#V4" title="CHORD:attr.4">minimal</a>  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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="E1:114"/><span class="lab"><font color="Green" title="E89">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="E2:114"/><span class="lab"><font color="Green" title="E90">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being    <a href="chord.html#M2" title="CHORD:mode.2">VertexSeparator</a> of <font color="Maroon" title="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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="E3:114"/><span class="lab"><font color="Green" title="E91">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H</font> being   <a href="glib_000.html#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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="E4:114"/><span class="lab"><font color="Green" title="E92">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="E5:114"/><span class="lab"><font color="Green" title="E93">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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:114_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="114_1_1"><span class="kw">suppose </span></a><a NAME="E1:114_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:114_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="tarski.html#R2" title="TARSKI: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="E3:114_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:114"><span class="lab"><font color="Green" title="E92">A4</font></span></a>, <a class="txt" href="#E5:114"><span class="lab"><font color="Green" title="E93">A5</font></span></a>, <a class="ref" href="glib_002.html#T9" onmouseover="rs('glib_002/T9')" onmouseout="rh()">GLIB_002:9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2"><span class="kw">suppose </span></a><a NAME="E1:114_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="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#M2" title="HIDDEN:mode.2">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#K4" title="ORDINAL1:func.4">omega</a>  =  <a href="card_1.html#K4" title="CARD_1:func.4">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="tarski.html#R2" title="TARSKI: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="E94">A6</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">x</font>) where <font color="Olive" title="b2">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="b2">x</font> <a href="tarski.html#R2" title="TARSKI: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>  holds <br/><font color="Olive" title="b1">x</font> is  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> </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">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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="tarski.html#R2" title="TARSKI: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="E1:114_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="tarski.html#R2" title="TARSKI: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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font> is  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> </span><br/><a NAME="E2:114_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#K4" title="CARD_1:func.4">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="tarski.html#R2" title="TARSKI: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="E3:114_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:114_1_2"/><span class="lab"><font color="Green" title="E95">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="E5:114_1_2"/><span class="lab"><font color="Green" title="E96">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="tarski.html#R2" title="TARSKI: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" onmouseover="rs('fraenkel/S21')" onmouseout="rh()">FRAENKEL:sch 21</a>(<a class="txt" href="#E4:114_1_2"><span class="lab"><font color="Green" title="E95">A7</font></span></a>);<br/></span><a NAME="E6:114_1_2"/>
<font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI: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="#E4:114"><span class="lab"><font color="Green" title="E92">A4</font></span></a>, <a class="ref" href="glib_002.html#T9" onmouseover="rs('glib_002/T9')" onmouseout="rh()">GLIB_002:9</a></span>;<br/><a NAME="E7:114_1_2"/><span class="kw">then </span>
 <a href="card_1.html#K4" title="CARD_1:func.4">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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:114_1_2"><span class="lab"><font color="Green" title="E96">A8</font></span></a>, <a class="txt" href="#E3:114_1_2"><span class="lab"><font color="Green" title="E94">A6</font></span></a>, <a class="ref" href="membered.html#D6" onmouseover="rs('membered/D6')" onmouseout="rh()">MEMBERED:def 6</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">Ga</font> =  the   <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="E9:114_1_2"/><span class="lab"><font color="Green" title="E97">A9</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="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> <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#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/><a NAME="E10:114_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" onmouseover="rs('xxreal_2/D8')" onmouseout="rh()">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="E12:114_1_2"/><span class="lab"><font color="Green" title="E98">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#K4" title="CARD_1:func.4">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="E13:114_1_2"/><span class="lab"><font color="Green" title="E99">A11</font></span>: 
<font color="Maroon" title="c12">c</font> <a href="tarski.html#R2" title="TARSKI: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="E14:114_1_2"/><span class="lab"><font color="Green" title="E100">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="#E1:114"><span class="lab"><font color="Green" title="E89">A1</font></span></a>, <a class="txt" href="#E2:114"><span class="lab"><font color="Green" title="E90">A2</font></span></a>, <a class="ref" href="chord.html#T75" target="_self" onmouseover="rs('chord/T75')" onmouseout="rh()">Th74</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c14">Gs</font> =  the   <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="E15:114_1_2"/><span class="lab"><font color="Green" title="E101">A13</font></span>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> 
 ;<br/><a NAME="E16:114_1_2"/>
 not <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114"><span class="lab"><font color="Green" title="E89">A1</font></span></a>, <a class="txt" href="#E2:114"><span class="lab"><font color="Green" title="E90">A2</font></span></a>, <a class="ref" href="chord.html#D8" target="_self" onmouseover="rs('chord/D8')" onmouseout="rh()">Def8</a></span>;<br/><a NAME="E17:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">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" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E18:114_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" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_2_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">S</font></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">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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="E1:114_1_2_2"/><span class="lab"><font color="Green" title="E103">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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="E2:114_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="#E14:114_1_2"><span class="lab"><font color="Green" title="E100">A12</font></span></a>, <a class="txt" href="#E1:114_1_2_2"><span class="lab"><font color="Green" title="E103">A15</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:114_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="#E1:114_1_2_2"><span class="lab"><font color="Green" title="E103">A15</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E21:114_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>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">Ga</font> =  the   <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> 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="#E17:114_1_2"><span class="lab"><font color="Green" title="E102">A14</font></span></a>, <a class="ref" href="chord.html#T29" target="_self" onmouseover="rs('chord/T29')" onmouseout="rh()">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="E24:114_1_2"/><span class="lab"><font color="Green" title="E103">A16</font></span>: 
<font color="Maroon" title="c18">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 <span class="kw">and </span><br/><a NAME="E25:114_1_2"/><span class="lab"><font color="Green" title="E104">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="#E5:114"><span class="lab"><font color="Green" title="E93">A5</font></span></a>, <a class="txt" href="#E13:114_1_2"><span class="lab"><font color="Green" title="E99">A11</font></span></a></span>;<br/><a NAME="E26:114_1_2"/><span class="lab"><font color="Green" title="E105">A18</font></span>: 
 not <font color="Maroon" title="c18">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c16">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:114_1_2"><span class="lab"><font color="Green" title="E103">A16</font></span></a>, <a class="txt" href="#E14:114_1_2"><span class="lab"><font color="Green" title="E100">A12</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="kw">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">set </span><font color="Maroon" title="c20">Gay</font> =  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="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="E27:114_1_2"/>
<font color="Maroon" title="c18">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="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" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E28:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A19</font></span>: 
<font color="Maroon" title="c18">y</font> <a href="tarski.html#R2" title="TARSKI: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" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E29:114_1_2"/>
<font color="Maroon" title="c12">c</font> <a href="tarski.html#R2" title="TARSKI: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="#E13:114_1_2"><span class="lab"><font color="Green" title="E99">A11</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><span class="kw">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  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:114_1_2"><span class="lab"><font color="Green" title="E106">A19</font></span></a>, <a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/><a NAME="E31:114_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="tarski.html#R2" title="TARSKI: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="#E1:114"><span class="lab"><font color="Green" title="E89">A1</font></span></a>, <a class="txt" href="#E2:114"><span class="lab"><font color="Green" title="E90">A2</font></span></a>, <a class="txt" href="#E3:114"><span class="lab"><font color="Green" title="E91">A3</font></span></a>, <a class="txt" href="#E4:114"><span class="lab"><font color="Green" title="E92">A4</font></span></a>, <a class="txt" href="#E24:114_1_2"><span class="lab"><font color="Green" title="E103">A16</font></span></a>, <a class="ref" href="chord.html#T82" target="_self" onmouseover="rs('chord/T82')" onmouseout="rh()">Th81</a></span>;<br/><a NAME="E32:114_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1">(<span class="default"><a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c17">Ga</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:114_1_2"><span class="lab"><font color="Green" title="E97">A9</font></span></a>, <a class="txt" href="#E26:114_1_2"><span class="lab"><font color="Green" title="E105">A18</font></span></a></span>;<br/><a NAME="E33:114_1_2"/><span class="kw">then </span>
 the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><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> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T57" target="_self" onmouseover="rs('chord/T57')" onmouseout="rh()">Th56</a></span>;<br/><span class="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  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="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><span class="kw"> such that </span><br/><a NAME="E35:114_1_2"/><span class="lab"><font color="Green" title="E107">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>
 ;<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  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="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><span class="kw"> such that </span><br/><a NAME="E37:114_1_2"/><span class="lab"><font color="Green" title="E108">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="E38:114_1_2"/><span class="lab"><font color="Green" title="E109">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" onmouseover="rs('chord/T39')" onmouseout="rh()">Th38</a></span>;<br/><a NAME="E39:114_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="#E35:114_1_2"><span class="lab"><font color="Green" title="E107">A20</font></span></a></span>;<br/><a NAME="E40:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">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="#E37:114_1_2"><span class="lab"><font color="Green" title="E108">A21</font></span></a></span>;<br/><a NAME="E41:114_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="#E35:114_1_2"><span class="lab"><font color="Green" title="E107">A20</font></span></a></span>;<br/><a NAME="E42:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">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="#E37:114_1_2"><span class="lab"><font color="Green" title="E108">A21</font></span></a></span>;<br/><a NAME="E43:114_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="#E13:114_1_2"><span class="lab"><font color="Green" title="E99">A11</font></span></a>, <a class="txt" href="#E24:114_1_2"><span class="lab"><font color="Green" title="E103">A16</font></span></a>, <a class="txt" href="#E14:114_1_2"><span class="lab"><font color="Green" title="E100">A12</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E44:114_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">P</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:114_1_2"><span class="lab"><font color="Green" title="E110">A23</font></span></a>, <a class="txt" href="#E42:114_1_2"><span class="lab"><font color="Green" title="E111">A24</font></span></a>, <a class="ref" href="glib_001.html#T127" onmouseover="rs('glib_001/T127')" onmouseout="rh()">GLIB_001:127</a></span>;<br/><a NAME="E45:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">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#T125" onmouseover="rs('glib_001/T125')" onmouseout="rh()">GLIB_001:125</a></span>;<br/><div><span class="lab"><font color="Green" title="E113">A26</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_2_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not  <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#K2" title="NAT_1:func.2">*</a> 2</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:114_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#K2" title="NAT_1:func.2">*</a> 2</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:114_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" onmouseover="rs('chord/T3')" onmouseout="rh()">Th3</a></span>;<br/><a NAME="E3:114_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">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="#E45:114_1_2"><span class="lab"><font color="Green" title="E112">A25</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E4:114_1_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">P</font>
 ;<br/><a NAME="E5:114_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="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#K1" title="NAT_1:func.1">+</a> 2</span>)</span>, the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D3" onmouseover="rs('glib_001/D3')" onmouseout="rh()">GLIB_001:def 3</a></span>;<br/><a NAME="E6:114_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="#E40:114_1_2"><span class="lab"><font color="Green" title="E110">A23</font></span></a>, <a class="txt" href="#E42:114_1_2"><span class="lab"><font color="Green" title="E111">A24</font></span></a>, <a class="txt" href="#E3:114_1_2_3"><span class="lab"><font color="Green" title="E114">A27</font></span></a>, <a class="ref" href="glib_000.html#T72" onmouseover="rs('glib_000/T72')" onmouseout="rh()">GLIB_000:72</a></span>;<br/><span class="kw">hence </span><a NAME="E7:114_1_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:114_1_2"><span class="lab"><font color="Green" title="E104">A17</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E47:114_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#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</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#K2" title="NAT_1:func.2">*</a> 1</span>)</span> as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">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="E49:114_1_2"/><span class="lab"><font color="Green" title="E114">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#T44" onmouseover="rs('xreal_1/T44')" onmouseout="rh()">XREAL_1:44</a></span>;<br/><div><span class="lab"><font color="Green" title="E115">A29</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_2_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="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></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:114_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:114_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="#E42:114_1_2"><span class="lab"><font color="Green" title="E111">A24</font></span></a>, <a class="txt" href="#E49:114_1_2"><span class="lab"><font color="Green" title="E114">A28</font></span></a>, <a class="ref" href="glib_001.html#D28" onmouseover="rs('glib_001/D28')" onmouseout="rh()">GLIB_001:def 28</a></span>;<br/><a NAME="E3:114_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#K1" title="NAT_1:func.1">+</a> 2
 ;<br/><span class="kw">hence </span><a NAME="E4:114_1_2_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:114_1_2"><span class="lab"><font color="Green" title="E113">A26</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E51:114_1_2"/><span class="lab"><font color="Green" title="E116">A30</font></span>: 
 not <font color="Maroon" title="c18">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><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="#E25:114_1_2"><span class="lab"><font color="Green" title="E104">A17</font></span></a>, <a class="ref" href="chord.html#T52" target="_self" onmouseover="rs('chord/T52')" onmouseout="rh()">Th51</a></span>;<br/><a NAME="E52:114_1_2"/><span class="lab"><font color="Green" title="E117">A31</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="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> <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#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/><a NAME="E53:114_1_2"/><span class="lab"><font color="Green" title="E118">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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:114_1_2"><span class="lab"><font color="Green" title="E114">A28</font></span></a>, <a class="ref" href="glib_001.html#T7" onmouseover="rs('glib_001/T7')" onmouseout="rh()">GLIB_001:7</a></span>;<br/><a NAME="E54:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="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="#E52:114_1_2"><span class="lab"><font color="Green" title="E117">A31</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><span class="kw">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="#E53:114_1_2"><span class="lab"><font color="Green" title="E118">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="E56:114_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="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>, the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:114_1_2"><span class="lab"><font color="Green" title="E114">A28</font></span></a>, <a class="ref" href="glib_001.html#D3" onmouseover="rs('glib_001/D3')" onmouseout="rh()">GLIB_001:def 3</a></span>;<br/><a NAME="E57:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="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="#E42:114_1_2"><span class="lab"><font color="Green" title="E111">A24</font></span></a>, <a class="ref" href="glib_000.html#T72" onmouseover="rs('glib_000/T72')" onmouseout="rh()">GLIB_000:72</a></span>;<br/><a NAME="E58:114_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> 
 ;<br/><a NAME="E59:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">A35</font></span>: 
<font color="Maroon" title="c18">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><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="#E50:114_1_2"><span class="lab"><font color="Green" title="E115">A29</font></span></a>, <a class="ref" href="chord.html#T52" target="_self" onmouseover="rs('chord/T52')" onmouseout="rh()">Th51</a></span>;<br/><a NAME="E60:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">A36</font></span>: 
<font color="Maroon" title="c18">y</font> <a href="tarski.html#R2" title="TARSKI: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="#E24:114_1_2"><span class="lab"><font color="Green" title="E103">A16</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_2_5"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <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></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:114_1_2_5"/><span class="lab"><font color="Green" title="E123">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:114_1_2_5"/>
 not <font color="Maroon" title="c18">y</font> <a href="tarski.html#R2" title="TARSKI: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="#E51:114_1_2"><span class="lab"><font color="Green" title="E116">A30</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:114_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="#E60:114_1_2"><span class="lab"><font color="Green" title="E122">A36</font></span></a>, <a class="txt" href="#E1:114_1_2_5"><span class="lab"><font color="Green" title="E123">A37</font></span></a></span>;<br/><a NAME="E4:114_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A38</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">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#K4" title="CARD_1:func.4">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#T6" onmouseover="rs('trees_1/T6')" onmouseout="rh()">TREES_1:6</a></span>;<br/><a NAME="E5:114_1_2_5"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:114_1_2"><span class="lab"><font color="Green" title="E119">A33</font></span></a>, <a class="txt" href="#E50:114_1_2"><span class="lab"><font color="Green" title="E115">A29</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:114_1_2_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:114_1_2"><span class="lab"><font color="Green" title="E98">A10</font></span></a>, <a class="txt" href="#E4:114_1_2_5"><span class="lab"><font color="Green" title="E124">A38</font></span></a>, <a class="ref" href="xxreal_2.html#D8" onmouseover="rs('xxreal_2/D8')" onmouseout="rh()">XXREAL_2:def 8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c29">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E63:114_1_2"/><span class="lab"><font color="Green" title="E123">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="E64:114_1_2"/><span class="lab"><font color="Green" title="E124">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>
 ;<br/><a NAME="E65:114_1_2"/><span class="lab"><font color="Green" title="E125">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="#E63:114_1_2"><span class="lab"><font color="Green" title="E123">A39</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E66:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">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="#E64:114_1_2"><span class="lab"><font color="Green" title="E124">A40</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="kw">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="#E63:114_1_2"><span class="lab"><font color="Green" title="E123">A39</font></span></a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>( $1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c24">P</font> &amp; $1 is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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">object</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="E68:114_1_2"/><span class="lab"><font color="Green" title="E127">A43</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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="E69:114_1_2"/><span class="lab"><font color="Green" title="E128">A44</font></span>: 
 not <font color="Maroon" title="c30">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c16">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:114_1_2"><span class="lab"><font color="Green" title="E100">A12</font></span></a>, <a class="txt" href="#E65:114_1_2"><span class="lab"><font color="Green" title="E125">A41</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E70:114_1_2"/><span class="lab"><font color="Green" title="E129">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="#E46:114_1_2"><span class="lab"><font color="Green" title="E113">A26</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E71:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A46</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c24">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E72:114_1_2"/>
<font color="Maroon" title="c30">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><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="#E63:114_1_2"><span class="lab"><font color="Green" title="E123">A39</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E73:114_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" onmouseover="rs('chord/T52')" onmouseout="rh()">Th51</a></span>;<br/><a NAME="E74:114_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">object</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="#E40:114_1_2"><span class="lab"><font color="Green" title="E110">A23</font></span></a>, <a class="ref" href="chord.html#D3" target="_self" onmouseover="rs('chord/D3')" onmouseout="rh()">Def3</a></span>;<br/><a NAME="E75:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A47</font></span>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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="#E70:114_1_2"><span class="lab"><font color="Green" title="E129">A45</font></span></a>, <a class="txt" href="#E71:114_1_2"><span class="lab"><font color="Green" title="E130">A46</font></span></a>, <a class="txt" href="#E15:114_1_2"><span class="lab"><font color="Green" title="E101">A13</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c31">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E77:114_1_2"/><span class="lab"><font color="Green" title="E132">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="E78:114_1_2"/><span class="lab"><font color="Green" title="E133">A49</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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" onmouseover="rs('nat_1/S6')" onmouseout="rh()">NAT_1:sch 6</a>(<a class="txt" href="#E68:114_1_2"><span class="lab"><font color="Green" title="E127">A43</font></span></a>, <a class="txt" href="#E75:114_1_2"><span class="lab"><font color="Green" title="E131">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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E77:114_1_2"><span class="lab"><font color="Green" title="E132">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#T175" onmouseover="rs('glib_001/T175')" onmouseout="rh()">GLIB_001:175</a></span>;<br/><a NAME="E81:114_1_2"/><span class="lab"><font color="Green" title="E134">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="#E77:114_1_2"><span class="lab"><font color="Green" title="E132">A48</font></span></a>, <a class="ref" href="chord.html#T3" target="_self" onmouseover="rs('chord/T3')" onmouseout="rh()">Th3</a></span>;<br/><a NAME="E82:114_1_2"/><span class="lab"><font color="Green" title="E135">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#T44" onmouseover="rs('xreal_1/T44')" onmouseout="rh()">XREAL_1:44</a></span>;<br/><a NAME="E83:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E136">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="#E38:114_1_2"><span class="lab"><font color="Green" title="E109">A22</font></span></a>, <a class="txt" href="#E81:114_1_2"><span class="lab"><font color="Green" title="E134">A50</font></span></a>, <a class="ref" href="chord.html#T42" target="_self" onmouseover="rs('chord/T42')" onmouseout="rh()">Th41</a></span>;<br/><a NAME="E84:114_1_2"/><span class="lab"><font color="Green" title="E137">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#K1" title="NAT_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="c25">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E81:114_1_2"><span class="lab"><font color="Green" title="E134">A50</font></span></a>, <a class="txt" href="#E82:114_1_2"><span class="lab"><font color="Green" title="E135">A51</font></span></a>, <a class="ref" href="glib_001.html#T36" onmouseover="rs('glib_001/T36')" onmouseout="rh()">GLIB_001:36</a></span>;<br/><div><span class="lab"><font color="Green" title="E138">A54</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_2_6"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c34">Q</font> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 holds <br/> for <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b2">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="Olive" title="b1">i</font>,<font color="Maroon" title="c30">x</font>,<font color="Maroon" title="c1">G</font></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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c35">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="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">object</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="E1:114_1_2_6"/><span class="lab"><font color="Green" title="E139">A55</font></span>: 
<font color="Maroon" title="c35">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c34">Q</font>
 <span class="kw">and </span><br/><a NAME="E2:114_1_2_6"/><span class="lab"><font color="Green" title="E140">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   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="E3:114_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="#E1:114_1_2_6"><span class="lab"><font color="Green" title="E139">A55</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E4:114_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#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</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#K1" title="NAT_1:func.1">+</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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/><a NAME="E7:114_1_2_6"/><span class="lab"><font color="Green" title="E141">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#K5" title="NUMBERS:func.5">0</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E8:114_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="#E1:114_1_2_6"><span class="lab"><font color="Green" title="E139">A55</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E9:114_1_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E142">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="#E7:114_1_2_6"><span class="lab"><font color="Green" title="E141">A57</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E10:114_1_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E143">A59</font></span>: 
<font color="Maroon" title="c37">ki1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c24">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E81:114_1_2"><span class="lab"><font color="Green" title="E134">A50</font></span></a>, <a class="txt" href="#E82:114_1_2"><span class="lab"><font color="Green" title="E135">A51</font></span></a>, <a class="ref" href="glib_001.html#T36" onmouseover="rs('glib_001/T36')" onmouseout="rh()">GLIB_001:36</a></span>;<br/><a NAME="E11:114_1_2_6"/><span class="lab"><font color="Green" title="E144">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="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#K5" title="NUMBERS:func.5">0</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><span class="kw">assume </span><a NAME="E12:114_1_2_6"/><span class="lab"><font color="Green" title="E145">A61</font></span>: 
 ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E13:114_1_2_6"/>
<font color="Maroon" title="c36">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="#E84:114_1_2"><span class="lab"><font color="Green" title="E137">A53</font></span></a>, <a class="txt" href="#E9:114_1_2_6"><span class="lab"><font color="Green" title="E142">A58</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E14:114_1_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E146">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="#E11:114_1_2_6"><span class="lab"><font color="Green" title="E144">A60</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E15:114_1_2_6"/><span class="lab"><font color="Green" title="E147">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#K1" title="NAT_1:func.1">+</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="#E81:114_1_2"><span class="lab"><font color="Green" title="E134">A50</font></span></a>, <a class="txt" href="#E82:114_1_2"><span class="lab"><font color="Green" title="E135">A51</font></span></a>, <a class="txt" href="#E9:114_1_2_6"><span class="lab"><font color="Green" title="E142">A58</font></span></a>, <a class="ref" href="glib_001.html#T36" onmouseover="rs('glib_001/T36')" onmouseout="rh()">GLIB_001:36</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_2_6_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c36">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:114_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#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:114_1_2_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c32">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c32">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c36">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:114_1_2_6_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:114_1_2"><span class="lab"><font color="Green" title="E133">A49</font></span></a>, <a class="txt" href="#E12:114_1_2_6"><span class="lab"><font color="Green" title="E145">A61</font></span></a>, <a class="txt" href="#E15:114_1_2_6"><span class="lab"><font color="Green" title="E147">A63</font></span></a>, <a class="txt" href="#E10:114_1_2_6"><span class="lab"><font color="Green" title="E143">A59</font></span></a>, <a class="txt" href="#E14:114_1_2_6"><span class="lab"><font color="Green" title="E146">A62</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E17:114_1_2_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:114_1_2_6"><span class="lab"><font color="Green" title="E140">A56</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><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="E86:114_1_2"/><span class="lab"><font color="Green" title="E139">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="#E81:114_1_2"><span class="lab"><font color="Green" title="E134">A50</font></span></a>, <a class="txt" href="#E82:114_1_2"><span class="lab"><font color="Green" title="E135">A51</font></span></a>, <a class="ref" href="glib_001.html#T37" onmouseover="rs('glib_001/T37')" onmouseout="rh()">GLIB_001:37</a></span>;<br/><a NAME="E87:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">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="#E77:114_1_2"><span class="lab"><font color="Green" title="E132">A48</font></span></a></span>;<br/><a NAME="E88:114_1_2"/><span class="lab"><font color="Green" title="E141">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="#E24:114_1_2"><span class="lab"><font color="Green" title="E103">A16</font></span></a>, <a class="txt" href="#E59:114_1_2"><span class="lab"><font color="Green" title="E121">A35</font></span></a>, <a class="txt" href="#E64:114_1_2"><span class="lab"><font color="Green" title="E124">A40</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E89:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E142">A67</font></span>: 
 not <font color="Maroon" title="c30">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="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" onmouseover="rs('tarski/D1')" onmouseout="rh()">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  the   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:114_1_2"><span class="lab"><font color="Green" title="E103">A16</font></span></a>, <a class="txt" href="#E65:114_1_2"><span class="lab"><font color="Green" title="E125">A41</font></span></a>, <a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/><a NAME="E91:114_1_2"/>
 the   <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> is  <a href="chord.html#V2" title="CHORD:attr.2">complete</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114"><span class="lab"><font color="Green" title="E89">A1</font></span></a>, <a class="txt" href="#E2:114"><span class="lab"><font color="Green" title="E90">A2</font></span></a>, <a class="txt" href="#E3:114"><span class="lab"><font color="Green" title="E91">A3</font></span></a>, <a class="ref" href="chord.html#T98" target="_self" onmouseover="rs('chord/T98')" onmouseout="rh()">Th97</a></span>;<br/><a NAME="E92:114_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="#E88:114_1_2"><span class="lab"><font color="Green" title="E141">A66</font></span></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">object</a> <span class="kw"> such that </span><br/><a NAME="E94:114_1_2"/><span class="lab"><font color="Green" title="E143">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>, the   <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/><a NAME="E95:114_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="#E94:114_1_2"><span class="lab"><font color="Green" title="E143">A68</font></span></a>, <a class="ref" href="glib_000.html#T72" onmouseover="rs('glib_000/T72')" onmouseout="rh()">GLIB_000:72</a></span>;<br/><a NAME="E96:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E144">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> 
 ;<br/><a NAME="E97:114_1_2"/><span class="lab"><font color="Green" title="E145">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="#E81:114_1_2"><span class="lab"><font color="Green" title="E134">A50</font></span></a>, <a class="txt" href="#E82:114_1_2"><span class="lab"><font color="Green" title="E135">A51</font></span></a>, <a class="ref" href="glib_001.html#T37" onmouseover="rs('glib_001/T37')" onmouseout="rh()">GLIB_001:37</a></span>;<br/><a NAME="E98:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E146">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="E99:114_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="#E14:114_1_2"><span class="lab"><font color="Green" title="E100">A12</font></span></a>, <a class="txt" href="#E54:114_1_2"><span class="lab"><font color="Green" title="E119">A33</font></span></a>, <a class="txt" href="#E50:114_1_2"><span class="lab"><font color="Green" title="E115">A29</font></span></a>, <a class="txt" href="#E65:114_1_2"><span class="lab"><font color="Green" title="E125">A41</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E100:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E147">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="#E66:114_1_2"><span class="lab"><font color="Green" title="E126">A42</font></span></a>, <a class="ref" href="chord.html#T52" target="_self" onmouseover="rs('chord/T52')" onmouseout="rh()">Th51</a></span>;<br/><a NAME="E101:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E148">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="#E77:114_1_2"><span class="lab"><font color="Green" title="E132">A48</font></span></a>, <a class="txt" href="#E86:114_1_2"><span class="lab"><font color="Green" title="E139">A64</font></span></a>, <a class="txt" href="#E97:114_1_2"><span class="lab"><font color="Green" title="E145">A70</font></span></a>, <a class="ref" href="chord.html#D3" target="_self" onmouseover="rs('chord/D3')" onmouseout="rh()">Def3</a></span>;<br/><a NAME="E102:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E149">A74</font></span>: 
<font color="Maroon" title="c34">Q</font> is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a> 
 ;<br/><a NAME="E103:114_1_2"/><span class="lab"><font color="Green" title="E150">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#T98" onmouseover="rs('glib_001/T98')" onmouseout="rh()">GLIB_001:98</a></span>;<br/><a NAME="E104:114_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="tarski.html#R2" title="TARSKI: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#T88" onmouseover="rs('glib_001/T88')" onmouseout="rh()">GLIB_001:88</a></span>;<br/><a NAME="E105:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E151">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="#E52:114_1_2"><span class="lab"><font color="Green" title="E117">A31</font></span></a>, <a class="txt" href="#E69:114_1_2"><span class="lab"><font color="Green" title="E128">A44</font></span></a>, <a class="txt" href="#E89:114_1_2"><span class="lab"><font color="Green" title="E142">A67</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div><span class="lab"><font color="Green" title="E152">A77</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_2_7"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="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></span></a><div class="add"><a NAME="E1:114_1_2_7"/><span class="lab"><font color="Green" title="E153">A78</font></span>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" onmouseover="rs('abian/T12')" onmouseout="rh()">ABIAN:12</a></span>;<br/><a NAME="E2:114_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E154">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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E77:114_1_2"><span class="lab"><font color="Green" title="E132">A48</font></span></a>, <a class="ref" href="glib_001.html#T36" onmouseover="rs('glib_001/T36')" onmouseout="rh()">GLIB_001:36</a></span>;<br/><span class="kw">assume </span><a NAME="E3:114_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E4:114_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="#E77:114_1_2"><span class="lab"><font color="Green" title="E132">A48</font></span></a>, <a class="txt" href="#E1:114_1_2_7"><span class="lab"><font color="Green" title="E153">A78</font></span></a>, <a class="ref" href="glib_001.html#T37" onmouseover="rs('glib_001/T37')" onmouseout="rh()">GLIB_001:37</a></span>;<br/><span class="kw">hence </span><a NAME="E5:114_1_2_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:114_1_2"><span class="lab"><font color="Green" title="E109">A22</font></span></a>, <a class="txt" href="#E77:114_1_2"><span class="lab"><font color="Green" title="E132">A48</font></span></a>, <a class="txt" href="#E2:114_1_2_7"><span class="lab"><font color="Green" title="E154">A79</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E153">A80</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_2_8"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  holds <br/><font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c16">A</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c40">v</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c40">v</font> <a href="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c16">A</font> )</span><br/><span class="kw">assume </span><a NAME="E1:114_1_2_8"/><span class="lab"><font color="Green" title="E154">A81</font></span>: 
<font color="Maroon" title="c40">v</font> <a href="tarski.html#R2" title="TARSKI: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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c40">v</font> <a href="tarski.html#R2" title="TARSKI: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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:114_1_2_8"/><span class="lab"><font color="Green" title="E155">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="E4:114_1_2_8"/><span class="lab"><font color="Green" title="E156">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="#E1:114_1_2_8"><span class="lab"><font color="Green" title="E154">A81</font></span></a>, <a class="ref" href="glib_001.html#T87" onmouseover="rs('glib_001/T87')" onmouseout="rh()">GLIB_001:87</a></span>;<br/><a NAME="E5:114_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" onmouseover="rs('abian/T12')" onmouseout="rh()">ABIAN:12</a></span>;<br/><a NAME="E6:114_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#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</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#K1" title="NAT_1:func.1">+</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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/><a NAME="E9:114_1_2_8"/><span class="lab"><font color="Green" title="E157">A84</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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="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#K5" title="NUMBERS:func.5">0</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E10:114_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="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#K5" title="NUMBERS:func.5">0</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E11:114_1_2_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E158">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="#E3:114_1_2_8"><span class="lab"><font color="Green" title="E155">A82</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E12:114_1_2_8"/><span class="kw">then </span>
<font color="Maroon" title="c32">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="#E84:114_1_2"><span class="lab"><font color="Green" title="E137">A53</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E13:114_1_2_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E159">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="#E9:114_1_2_8"><span class="lab"><font color="Green" title="E157">A84</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E14:114_1_2_8"/><span class="lab"><font color="Green" title="E160">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#K1" title="NAT_1:func.1">+</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="#E81:114_1_2"><span class="lab"><font color="Green" title="E134">A50</font></span></a>, <a class="txt" href="#E82:114_1_2"><span class="lab"><font color="Green" title="E135">A51</font></span></a>, <a class="txt" href="#E11:114_1_2_8"><span class="lab"><font color="Green" title="E158">A85</font></span></a>, <a class="ref" href="glib_001.html#T36" onmouseover="rs('glib_001/T36')" onmouseout="rh()">GLIB_001:36</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_2_8_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c40">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y</font></span></a><div class="add"><a NAME="E1:114_1_2_8_1"/><span class="lab"><font color="Green" title="E161">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" onmouseover="rs('abian/T12')" onmouseout="rh()">ABIAN:12</a></span>;<br/><span class="kw">assume </span><a NAME="E2:114_1_2_8_1"/><span class="lab"><font color="Green" title="E162">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:114_1_2_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E163">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="#E42:114_1_2"><span class="lab"><font color="Green" title="E111">A24</font></span></a>, <a class="txt" href="#E4:114_1_2_8"><span class="lab"><font color="Green" title="E156">A83</font></span></a>, <a class="txt" href="#E14:114_1_2_8"><span class="lab"><font color="Green" title="E160">A87</font></span></a>, <a class="txt" href="#E13:114_1_2_8"><span class="lab"><font color="Green" title="E159">A86</font></span></a>, <a class="ref" href="glib_001.html#D28" onmouseover="rs('glib_001/D28')" onmouseout="rh()">GLIB_001:def 28</a></span>;<br/><div><span class="lab"><font color="Green" title="E164">A91</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_2_8_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c41">n</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:114_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:114_1_2_8_1_1"/><span class="kw">then </span>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:114_1_2_8_1_1"/><span class="kw">then </span>
2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c41">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114_1_2_8_1"><span class="lab"><font color="Green" title="E161">A88</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E4:114_1_2_8_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:114_1_2_8_1"><span class="lab"><font color="Green" title="E163">A90</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E5:114_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" onmouseover="rs('abian/T12')" onmouseout="rh()">ABIAN:12</a></span>;<br/><span class="kw">hence </span><a NAME="E6:114_1_2_8_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:114_1_2"><span class="lab"><font color="Green" title="E111">A24</font></span></a>, <a class="txt" href="#E106:114_1_2"><span class="lab"><font color="Green" title="E152">A77</font></span></a>, <a class="txt" href="#E86:114_1_2"><span class="lab"><font color="Green" title="E139">A64</font></span></a>, <a class="txt" href="#E4:114_1_2_8"><span class="lab"><font color="Green" title="E156">A83</font></span></a>, <a class="txt" href="#E2:114_1_2_8_1"><span class="lab"><font color="Green" title="E162">A89</font></span></a>, <a class="txt" href="#E4:114_1_2_8_1"><span class="lab"><font color="Green" title="E164">A91</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E16:114_1_2_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E161">A92</font></span>: 
 not <font color="Maroon" title="c40">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="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" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E17:114_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="#E81:114_1_2"><span class="lab"><font color="Green" title="E134">A50</font></span></a>, <a class="txt" href="#E82:114_1_2"><span class="lab"><font color="Green" title="E135">A51</font></span></a>, <a class="txt" href="#E103:114_1_2"><span class="lab"><font color="Green" title="E150">A75</font></span></a>, <a class="ref" href="glib_001.html#T94" onmouseover="rs('glib_001/T94')" onmouseout="rh()">GLIB_001:94</a></span>;<br/><a NAME="E18:114_1_2_8"/><span class="kw">then </span>
<font color="Maroon" title="c40">v</font> <a href="tarski.html#R2" title="TARSKI: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="#E1:114_1_2_8"><span class="lab"><font color="Green" title="E154">A81</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E19:114_1_2_8"/>
<font color="Maroon" title="c40">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c16">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:114_1_2"><span class="lab"><font color="Green" title="E117">A31</font></span></a>, <a class="txt" href="#E16:114_1_2_8"><span class="lab"><font color="Green" title="E161">A92</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E108:114_1_2"/>
<font color="Maroon" title="c34">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="tarski.html#R2" title="TARSKI: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="#E103:114_1_2"><span class="lab"><font color="Green" title="E150">A75</font></span></a>, <a class="ref" href="glib_001.html#T88" onmouseover="rs('glib_001/T88')" onmouseout="rh()">GLIB_001:88</a></span>;<br/><a NAME="E109:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E154">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="#E52:114_1_2"><span class="lab"><font color="Green" title="E117">A31</font></span></a>, <a class="txt" href="#E69:114_1_2"><span class="lab"><font color="Green" title="E128">A44</font></span></a>, <a class="txt" href="#E89:114_1_2"><span class="lab"><font color="Green" title="E142">A67</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E110:114_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="#E57:114_1_2"><span class="lab"><font color="Green" title="E120">A34</font></span></a>, <a class="txt" href="#E97:114_1_2"><span class="lab"><font color="Green" title="E145">A70</font></span></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="E112:114_1_2"/><span class="lab"><font color="Green" title="E155">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="E113:114_1_2"/><span class="lab"><font color="Green" title="E156">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="E114:114_1_2"/><span class="lab"><font color="Green" title="E157">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="E115:114_1_2"/><span class="lab"><font color="Green" title="E158">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="E116:114_1_2"/><span class="lab"><font color="Green" title="E159">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="E117:114_1_2"/><span class="lab"><font color="Green" title="E160">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="E118:114_1_2"/><span class="lab"><font color="Green" title="E161">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="#E42:114_1_2"><span class="lab"><font color="Green" title="E111">A24</font></span></a>, <a class="txt" href="#E50:114_1_2"><span class="lab"><font color="Green" title="E115">A29</font></span></a>, <a class="txt" href="#E88:114_1_2"><span class="lab"><font color="Green" title="E141">A66</font></span></a>, <a class="txt" href="#E96:114_1_2"><span class="lab"><font color="Green" title="E144">A69</font></span></a>, <a class="txt" href="#E106:114_1_2"><span class="lab"><font color="Green" title="E152">A77</font></span></a>, <a class="txt" href="#E86:114_1_2"><span class="lab"><font color="Green" title="E139">A64</font></span></a>, <a class="txt" href="#E97:114_1_2"><span class="lab"><font color="Green" title="E145">A70</font></span></a>, <a class="txt" href="#E105:114_1_2"><span class="lab"><font color="Green" title="E151">A76</font></span></a>, <a class="txt" href="#E109:114_1_2"><span class="lab"><font color="Green" title="E154">A93</font></span></a>, <a class="txt" href="#E87:114_1_2"><span class="lab"><font color="Green" title="E140">A65</font></span></a>, <a class="ref" href="chord.html#T48" target="_self" onmouseover="rs('chord/T48')" onmouseout="rh()">Th47</a></span>;<br/><a NAME="E119:114_1_2"/><span class="lab"><font color="Green" title="E162">A101</font></span>: 
<font color="Maroon" title="c40">R</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E112:114_1_2"><span class="lab"><font color="Green" title="E155">A94</font></span></a>, <a class="ref" href="glib_001.html#T126" onmouseover="rs('glib_001/T126')" onmouseout="rh()">GLIB_001:126</a></span>;<br/><a NAME="E120:114_1_2"/><span class="lab"><font color="Green" title="E163">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_2_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:114_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:114_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> 
 
;<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">object</a> <span class="kw"> such that </span><br/><a NAME="E4:114_1_2_9"/><span class="lab"><font color="Green" title="E164">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" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E5:114_1_2_9"/><span class="lab"><font color="Green" title="E165">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="#E4:114_1_2_9"><span class="lab"><font color="Green" title="E164">A103</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E6:114_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="#E4:114_1_2_9"><span class="lab"><font color="Green" title="E164">A103</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E8:114_1_2_9"/><span class="lab"><font color="Green" title="E166">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="E9:114_1_2_9"/><span class="lab"><font color="Green" title="E167">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="E10:114_1_2_9"/><span class="lab"><font color="Green" title="E168">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="#E112:114_1_2"><span class="lab"><font color="Green" title="E155">A94</font></span></a>, <a class="ref" href="glib_001.html#T99" onmouseover="rs('glib_001/T99')" onmouseout="rh()">GLIB_001:99</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_9_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:114_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="#E8:114_1_2_9"><span class="lab"><font color="Green" title="E166">A105</font></span></a>, <a class="txt" href="#E9:114_1_2_9"><span class="lab"><font color="Green" title="E167">A106</font></span></a>, <a class="ref" href="chord.html#T13" target="_self" onmouseover="rs('chord/T13')" onmouseout="rh()">Th13</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_9_1_1"><span class="kw">suppose </span></a><a NAME="E1:114_1_2_9_1_1"/><span class="lab"><font color="Green" title="E169">A108</font></span>: 
<font color="Maroon" title="c42">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:114_1_2_9_1_1"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E112:114_1_2"><span class="lab"><font color="Green" title="E155">A94</font></span></a></span>;<br/><a NAME="E3:114_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c40">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#K1" title="NAT_1:func.1">+</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" onmouseover="rs('glib_001/D3')" onmouseout="rh()">GLIB_001:def 3</a></span>;<br/><a NAME="E4:114_1_2_9_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">y</font> <a href="tarski.html#R2" title="TARSKI: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="#E116:114_1_2"><span class="lab"><font color="Green" title="E159">A98</font></span></a>, <a class="txt" href="#E5:114_1_2_9"><span class="lab"><font color="Green" title="E165">A104</font></span></a>, <a class="txt" href="#E10:114_1_2_9"><span class="lab"><font color="Green" title="E168">A107</font></span></a>, <a class="txt" href="#E1:114_1_2_9_1_1"><span class="lab"><font color="Green" title="E169">A108</font></span></a>, <a class="ref" href="glib_001.html#T105" onmouseover="rs('glib_001/T105')" onmouseout="rh()">GLIB_001:105</a></span>;<br/><a NAME="E5:114_1_2_9_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c16">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E107:114_1_2"><span class="lab"><font color="Green" title="E153">A80</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:114_1_2_9_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:114_1_2"><span class="lab"><font color="Green" title="E103">A16</font></span></a>, <a class="txt" href="#E14:114_1_2"><span class="lab"><font color="Green" title="E100">A12</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_9_1_2"><span class="kw">suppose </span></a><a NAME="E1:114_1_2_9_1_2"/><span class="lab"><font color="Green" title="E169">A109</font></span>: 
<font color="Maroon" title="c42">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:114_1_2_9_1_2"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E112:114_1_2"><span class="lab"><font color="Green" title="E155">A94</font></span></a></span>;<br/><a NAME="E3:114_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c40">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#K1" title="NAT_1:func.1">+</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" onmouseover="rs('glib_001/D3')" onmouseout="rh()">GLIB_001:def 3</a></span>;<br/><a NAME="E4:114_1_2_9_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">y</font> <a href="tarski.html#R2" title="TARSKI: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="#E116:114_1_2"><span class="lab"><font color="Green" title="E159">A98</font></span></a>, <a class="txt" href="#E5:114_1_2_9"><span class="lab"><font color="Green" title="E165">A104</font></span></a>, <a class="txt" href="#E10:114_1_2_9"><span class="lab"><font color="Green" title="E168">A107</font></span></a>, <a class="txt" href="#E1:114_1_2_9_1_2"><span class="lab"><font color="Green" title="E169">A109</font></span></a>, <a class="ref" href="glib_001.html#T105" onmouseover="rs('glib_001/T105')" onmouseout="rh()">GLIB_001:105</a></span>;<br/><a NAME="E5:114_1_2_9_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c16">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E107:114_1_2"><span class="lab"><font color="Green" title="E153">A80</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:114_1_2_9_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:114_1_2"><span class="lab"><font color="Green" title="E103">A16</font></span></a>, <a class="txt" href="#E14:114_1_2"><span class="lab"><font color="Green" title="E100">A12</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_9_1_3"><span class="kw">suppose </span></a><a NAME="E1:114_1_2_9_1_3"/><span class="lab"><font color="Green" title="E169">A110</font></span>: 
<font color="Maroon" title="c42">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:114_1_2_9_1_3"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E112:114_1_2"><span class="lab"><font color="Green" title="E155">A94</font></span></a></span>;<br/><a NAME="E3:114_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c40">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#K1" title="NAT_1:func.1">+</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" onmouseover="rs('glib_001/D3')" onmouseout="rh()">GLIB_001:def 3</a></span>;<br/><a NAME="E4:114_1_2_9_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c30">x</font> <a href="tarski.html#R2" title="TARSKI: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="#E117:114_1_2"><span class="lab"><font color="Green" title="E160">A99</font></span></a>, <a class="txt" href="#E5:114_1_2_9"><span class="lab"><font color="Green" title="E165">A104</font></span></a>, <a class="txt" href="#E10:114_1_2_9"><span class="lab"><font color="Green" title="E168">A107</font></span></a>, <a class="txt" href="#E1:114_1_2_9_1_3"><span class="lab"><font color="Green" title="E169">A110</font></span></a>, <a class="ref" href="glib_001.html#T105" onmouseover="rs('glib_001/T105')" onmouseout="rh()">GLIB_001:105</a></span>;<br/><a NAME="E5:114_1_2_9_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c30">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c16">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E107:114_1_2"><span class="lab"><font color="Green" title="E153">A80</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:114_1_2_9_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:114_1_2"><span class="lab"><font color="Green" title="E100">A12</font></span></a>, <a class="txt" href="#E65:114_1_2"><span class="lab"><font color="Green" title="E125">A41</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_2_10"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">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> holds <br/><font color="Olive" title="b1">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></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">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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="E1:114_1_2_10"/><span class="lab"><font color="Green" title="E164">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="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="114_1_2_10_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:114_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="#E1:114_1_2_10"><span class="lab"><font color="Green" title="E164">A111</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_10_1_1"><span class="kw">suppose </span></a><a NAME="E1:114_1_2_10_1_1"/><span class="lab"><font color="Green" title="E165">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="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="E2:114_1_2_10_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E166">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#T88" onmouseover="rs('glib_001/T88')" onmouseout="rh()">GLIB_001:88</a></span>;<br/><a NAME="E3:114_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="#E112:114_1_2"><span class="lab"><font color="Green" title="E155">A94</font></span></a>, <a class="txt" href="#E118:114_1_2"><span class="lab"><font color="Green" title="E161">A100</font></span></a>, <a class="txt" href="#E1:114_1_2_10_1_1"><span class="lab"><font color="Green" title="E165">A112</font></span></a>, <a class="ref" href="glib_001.html#T87" onmouseover="rs('glib_001/T87')" onmouseout="rh()">GLIB_001:87</a></span>;<br/><span class="kw">hence </span><a NAME="E4:114_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="#E2:114_1_2_10_1_1"><span class="lab"><font color="Green" title="E166">A113</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_10_1_2"><span class="kw">suppose </span></a><a NAME="E1:114_1_2_10_1_2"/><span class="lab"><font color="Green" title="E165">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="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="E2:114_1_2_10_1_2"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="#E112:114_1_2"><span class="lab"><font color="Green" title="E155">A94</font></span></a></span>;<br/><a NAME="E3:114_1_2_10_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E166">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="#E115:114_1_2"><span class="lab"><font color="Green" title="E158">A97</font></span></a>, <a class="txt" href="#E1:114_1_2_10_1_2"><span class="lab"><font color="Green" title="E165">A114</font></span></a>, <a class="ref" href="glib_001.html#T87" onmouseover="rs('glib_001/T87')" onmouseout="rh()">GLIB_001:87</a></span>;<br/><a NAME="E4:114_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="#E1:114_1_2_10_1_2"><span class="lab"><font color="Green" title="E165">A114</font></span></a>, <a class="ref" href="glib_001.html#T88" onmouseover="rs('glib_001/T88')" onmouseout="rh()">GLIB_001:88</a></span>;<br/><span class="kw">hence </span><a NAME="E5:114_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="#E3:114_1_2_10_1_2"><span class="lab"><font color="Green" title="E166">A115</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E122:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E164">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>
 ;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_2_11"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">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> holds <br/><font color="Olive" title="b1">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></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">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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="E1:114_1_2_11"/><span class="lab"><font color="Green" title="E165">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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="E2:114_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="#E114:114_1_2"><span class="lab"><font color="Green" title="E157">A96</font></span></a>, <a class="txt" href="#E1:114_1_2_11"><span class="lab"><font color="Green" title="E165">A117</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:114_1_2_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E166">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" onmouseover="rs('enumset1/D2')" onmouseout="rh()">ENUMSET1:def 2</a></span>;<br/><a NAME="E4:114_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="#E1:114_1_2_11"><span class="lab"><font color="Green" title="E165">A117</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:114_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="#E107:114_1_2"><span class="lab"><font color="Green" title="E153">A80</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:114_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="#E24:114_1_2"><span class="lab"><font color="Green" title="E103">A16</font></span></a>, <a class="txt" href="#E14:114_1_2"><span class="lab"><font color="Green" title="E100">A12</font></span></a>, <a class="txt" href="#E65:114_1_2"><span class="lab"><font color="Green" title="E125">A41</font></span></a>, <a class="txt" href="#E3:114_1_2_11"><span class="lab"><font color="Green" title="E166">A118</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E124:114_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>
 ;<br/><a NAME="E125:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E165">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="#E122:114_1_2"><span class="lab"><font color="Green" title="E164">A116</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E166">A120</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_2_12"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c34">Q</font> &amp; <font color="Olive" title="b1">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> holds <br/> for <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b2">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="Olive" title="b1">i</font>,<font color="Maroon" title="c18">y</font>,<font color="Maroon" title="c1">G</font></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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c41">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="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">object</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="E1:114_1_2_12"/><span class="lab"><font color="Green" title="E167">A121</font></span>: 
<font color="Maroon" title="c41">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c34">Q</font>
 <span class="kw">and </span><br/><a NAME="E2:114_1_2_12"/><span class="lab"><font color="Green" title="E168">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   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="E3:114_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="#E1:114_1_2_12"><span class="lab"><font color="Green" title="E167">A121</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E4:114_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#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</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#K1" title="NAT_1:func.1">+</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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/><a NAME="E7:114_1_2_12"/><span class="lab"><font color="Green" title="E169">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#K5" title="NUMBERS:func.5">0</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E8:114_1_2_12"/><span class="lab"><font color="Green" title="E170">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="#E1:114_1_2_12"><span class="lab"><font color="Green" title="E167">A121</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E9:114_1_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E171">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="#E7:114_1_2_12"><span class="lab"><font color="Green" title="E169">A123</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E10:114_1_2_12"/><span class="kw">then </span>
<font color="Maroon" title="c43">ki1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c24">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E81:114_1_2"><span class="lab"><font color="Green" title="E134">A50</font></span></a>, <a class="txt" href="#E82:114_1_2"><span class="lab"><font color="Green" title="E135">A51</font></span></a>, <a class="ref" href="glib_001.html#T36" onmouseover="rs('glib_001/T36')" onmouseout="rh()">GLIB_001:36</a></span>;<br/><a NAME="E11:114_1_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E172">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#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E12:114_1_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E173">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="tarski.html#R2" title="TARSKI: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="#E52:114_1_2"><span class="lab"><font color="Green" title="E117">A31</font></span></a>, <a class="ref" href="glib_001.html#T7" onmouseover="rs('glib_001/T7')" onmouseout="rh()">GLIB_001:7</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_2_12_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="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></span></a><div class="add"><a NAME="E1:114_1_2_12_1"/><span class="lab"><font color="Green" title="E174">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#T30" onmouseover="rs('xreal_1/T30')" onmouseout="rh()">XREAL_1:30</a></span>;<br/><span class="kw">assume </span><a NAME="E2:114_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E3:114_1_2_12_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E84:114_1_2"><span class="lab"><font color="Green" title="E137">A53</font></span></a>, <a class="txt" href="#E8:114_1_2_12"><span class="lab"><font color="Green" title="E170">A124</font></span></a>, <a class="txt" href="#E7:114_1_2_12"><span class="lab"><font color="Green" title="E169">A123</font></span></a>, <a class="txt" href="#E1:114_1_2_12_1"><span class="lab"><font color="Green" title="E174">A128</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E14:114_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="#E11:114_1_2_12"><span class="lab"><font color="Green" title="E172">A126</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E15:114_1_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E174">A129</font></span>: 
<font color="Maroon" title="c43">ki1</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="c24">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T4" target="_self" onmouseover="rs('chord/T4')" onmouseout="rh()">Th4</a></span>;<br/><a NAME="E16:114_1_2_12"/>
<font color="Maroon" title="c43">ki1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="#E84:114_1_2"><span class="lab"><font color="Green" title="E137">A53</font></span></a>, <a class="txt" href="#E2:114_1_2_12"><span class="lab"><font color="Green" title="E168">A122</font></span></a></span>;<br/><a NAME="E17:114_1_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E175">A130</font></span>: 
<font color="Maroon" title="c43">ki1</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>  <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="#E15:114_1_2_12"><span class="lab"><font color="Green" title="E174">A129</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E18:114_1_2_12"/><span class="lab"><font color="Green" title="E176">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="tarski.html#R2" title="TARSKI: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="#E52:114_1_2"><span class="lab"><font color="Green" title="E117">A31</font></span></a>, <a class="ref" href="glib_001.html#T7" onmouseover="rs('glib_001/T7')" onmouseout="rh()">GLIB_001:7</a></span>;<br/><span class="kw">assume </span><a NAME="E19:114_1_2_12"/><span class="lab"><font color="Green" title="E177">A132</font></span>: 
 ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E20:114_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#K1" title="NAT_1:func.1">+</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="#E81:114_1_2"><span class="lab"><font color="Green" title="E134">A50</font></span></a>, <a class="txt" href="#E82:114_1_2"><span class="lab"><font color="Green" title="E135">A51</font></span></a>, <a class="txt" href="#E9:114_1_2_12"><span class="lab"><font color="Green" title="E171">A125</font></span></a>, <a class="ref" href="glib_001.html#T36" onmouseover="rs('glib_001/T36')" onmouseout="rh()">GLIB_001:36</a></span>;<br/><span class="kw">hence </span><a NAME="E21:114_1_2_12"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:114_1_2"><span class="lab"><font color="Green" title="E109">A22</font></span></a>, <a class="txt" href="#E42:114_1_2"><span class="lab"><font color="Green" title="E111">A24</font></span></a>, <a class="txt" href="#E19:114_1_2_12"><span class="lab"><font color="Green" title="E177">A132</font></span></a>, <a class="txt" href="#E17:114_1_2_12"><span class="lab"><font color="Green" title="E175">A130</font></span></a>, <a class="txt" href="#E12:114_1_2_12"><span class="lab"><font color="Green" title="E173">A127</font></span></a>, <a class="txt" href="#E18:114_1_2_12"><span class="lab"><font color="Green" title="E176">A131</font></span></a>, <a class="ref" href="chord.html#T19" target="_self" onmouseover="rs('chord/T19')" onmouseout="rh()">Th19</a>, <a class="ref" href="chord.html#T40" target="_self" onmouseover="rs('chord/T40')" onmouseout="rh()">Th39</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E127:114_1_2"/><span class="lab"><font color="Green" title="E167">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="#E86:114_1_2"><span class="lab"><font color="Green" title="E139">A64</font></span></a></span>;<br/><a NAME="E128:114_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="#E100:114_1_2"><span class="lab"><font color="Green" title="E147">A72</font></span></a>, <a class="txt" href="#E77:114_1_2"><span class="lab"><font color="Green" title="E132">A48</font></span></a>, <a class="ref" href="chord.html#D3" target="_self" onmouseover="rs('chord/D3')" onmouseout="rh()">Def3</a></span>;<br/><a NAME="E129:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E168">A134</font></span>: 
<font color="Maroon" title="c34">Q</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E127:114_1_2"><span class="lab"><font color="Green" title="E167">A133</font></span></a>, <a class="txt" href="#E98:114_1_2"><span class="lab"><font color="Green" title="E146">A71</font></span></a>, <a class="ref" href="glib_001.html#T127" onmouseover="rs('glib_001/T127')" onmouseout="rh()">GLIB_001:127</a></span>;<br/><a NAME="E130:114_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#K5" title="NUMBERS:func.5">0</a> 
 ;<br/><a NAME="E131:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E169">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#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">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="E132:114_1_2"/><span class="lab"><font color="Green" title="E170">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="#E115:114_1_2"><span class="lab"><font color="Green" title="E158">A97</font></span></a></span>;<br/><a NAME="E133:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E171">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E112:114_1_2"><span class="lab"><font color="Green" title="E155">A94</font></span></a>, <a class="ref" href="glib_001.html#T33" onmouseover="rs('glib_001/T33')" onmouseout="rh()">GLIB_001:33</a></span>;<br/><a NAME="E134:114_1_2"/><span class="lab"><font color="Green" title="E172">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E112:114_1_2"><span class="lab"><font color="Green" title="E155">A94</font></span></a>, <a class="txt" href="#E132:114_1_2"><span class="lab"><font color="Green" title="E170">A136</font></span></a>, <a class="ref" href="glib_001.html#T33" onmouseover="rs('glib_001/T33')" onmouseout="rh()">GLIB_001:33</a></span>;<br/><a NAME="E135:114_1_2"/><span class="lab"><font color="Green" title="E173">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E112:114_1_2"><span class="lab"><font color="Green" title="E155">A94</font></span></a>, <a class="txt" href="#E132:114_1_2"><span class="lab"><font color="Green" title="E170">A136</font></span></a>, <a class="ref" href="glib_001.html#T33" onmouseover="rs('glib_001/T33')" onmouseout="rh()">GLIB_001:33</a></span>;<br/><a NAME="E136:114_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#K1" title="NAT_1:func.1">+</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E113:114_1_2"><span class="lab"><font color="Green" title="E156">A95</font></span></a>, <a class="txt" href="#E132:114_1_2"><span class="lab"><font color="Green" title="E170">A136</font></span></a>, <a class="ref" href="chord.html#T28" target="_self" onmouseover="rs('chord/T28')" onmouseout="rh()">Th28</a></span>;<br/><a NAME="E137:114_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#K1" title="NAT_1:func.1">+</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E131:114_1_2"><span class="lab"><font color="Green" title="E169">A135</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E138:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E174">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" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E139:114_1_2"/><span class="lab"><font color="Green" title="E175">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="#E112:114_1_2"><span class="lab"><font color="Green" title="E155">A94</font></span></a>, <a class="txt" href="#E118:114_1_2"><span class="lab"><font color="Green" title="E161">A100</font></span></a></span>;<br/><a NAME="E140:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E176">A142</font></span>: 
<font color="Maroon" title="c40">R</font> is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E101:114_1_2"><span class="lab"><font color="Green" title="E148">A73</font></span></a>, <a class="txt" href="#E132:114_1_2"><span class="lab"><font color="Green" title="E170">A136</font></span></a></span>;<br/><a NAME="E141:114_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="#E102:114_1_2"><span class="lab"><font color="Green" title="E149">A74</font></span></a>, <a class="txt" href="#E132:114_1_2"><span class="lab"><font color="Green" title="E170">A136</font></span></a>, <a class="txt" href="#E139:114_1_2"><span class="lab"><font color="Green" title="E175">A141</font></span></a>, <a class="txt" href="#E119:114_1_2"><span class="lab"><font color="Green" title="E162">A101</font></span></a>, <a class="txt" href="#E120:114_1_2"><span class="lab"><font color="Green" title="E163">A102</font></span></a>, <a class="txt" href="#E125:114_1_2"><span class="lab"><font color="Green" title="E165">A119</font></span></a>, <a class="ref" href="chord.html#T27" target="_self" onmouseover="rs('chord/T27')" onmouseout="rh()">Th27</a></span>;<br/><a NAME="E142:114_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="#E138:114_1_2"><span class="lab"><font color="Green" title="E174">A140</font></span></a>, <a class="ref" href="chord.html#D11" target="_self" onmouseover="rs('chord/D11')" onmouseout="rh()">Def11</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c42">m</font>, <font color="Maroon" title="c43">n</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E144:114_1_2"/><span class="lab"><font color="Green" title="E177">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="E145:114_1_2"/><span class="lab"><font color="Green" title="E178">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="E146:114_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="E147:114_1_2"/><span class="lab"><font color="Green" title="E179">A145</font></span>: 
 ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon" title="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="E148:114_1_2"/><span class="lab"><font color="Green" title="E180">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" onmouseover="rs('chord/T84')" onmouseout="rh()">Th83</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c44">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E150:114_1_2"/><span class="lab"><font color="Green" title="E181">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="#E147:114_1_2"><span class="lab"><font color="Green" title="E179">A145</font></span></a></span>;<br/><a NAME="E151:114_1_2"/><span class="lab"><font color="Green" title="E182">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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="#E132:114_1_2"><span class="lab"><font color="Green" title="E170">A136</font></span></a>, <a class="ref" href="glib_001.html#T28" onmouseover="rs('glib_001/T28')" onmouseout="rh()">GLIB_001:28</a></span>;<br/><a NAME="E152:114_1_2"/><span class="lab"><font color="Green" title="E183">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 6 )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_2_13"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_13_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:114_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="114_1_2_13_1_1"><span class="kw">suppose </span></a><a NAME="E1:114_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 6 )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:114_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 6 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_13_1_2"><span class="kw">suppose </span></a><a NAME="E1:114_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 6 )</span><br/><div class="add"><a NAME="E2:114_1_2_13_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E184">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#K1" title="NAT_1:func.1">+</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" onmouseover="rs('chord/T4')" onmouseout="rh()">Th4</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_13_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:114_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#K1" title="NAT_1:func.1">+</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#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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:114_1_2_13_1_2"><span class="lab"><font color="Green" title="E184">A150</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_13_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:114_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#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 6 )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:114_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 6 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_13_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:114_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#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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 6 )</span><br/><div class="add"><a NAME="E2:114_1_2_13_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E185">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#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <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> <font color="Maroon" title="c43">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T4" target="_self" onmouseover="rs('chord/T4')" onmouseout="rh()">Th4</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_13_1_2_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:114_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</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="#E2:114_1_2_13_1_2_1_2"><span class="lab"><font color="Green" title="E185">A151</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_13_1_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:114_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#K1" title="NAT_1:func.1">+</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 6 )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:114_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 6 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_13_1_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:114_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#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 6 )</span><br/><div class="add"><a NAME="E2:114_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#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <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> <font color="Maroon" title="c43">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T4" target="_self" onmouseover="rs('chord/T4')" onmouseout="rh()">Th4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:114_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 6 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E112:114_1_2"><span class="lab"><font color="Green" title="E155">A94</font></span></a>, <a class="txt" href="#E151:114_1_2"><span class="lab"><font color="Green" title="E182">A148</font></span></a>, <a class="txt" href="#E145:114_1_2"><span class="lab"><font color="Green" title="E178">A144</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></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="E153:114_1_2"/><span class="lab"><font color="Green" title="E184">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" onmouseover="rs('abian/T12')" onmouseout="rh()">ABIAN:12</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_14"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:114_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 6 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E152:114_1_2"><span class="lab"><font color="Green" title="E183">A149</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_14_1"><span class="kw">suppose </span></a><a NAME="E1:114_1_2_14_1"/><span class="lab"><font color="Green" title="E185">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:114_1_2_14_1"/><span class="lab"><font color="Green" title="E186">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#K5" title="NUMBERS:func.5">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#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E3:114_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="#E144:114_1_2"><span class="lab"><font color="Green" title="E177">A143</font></span></a>, <a class="txt" href="#E1:114_1_2_14_1"><span class="lab"><font color="Green" title="E185">A153</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E4:114_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="#E2:114_1_2_14_1"><span class="lab"><font color="Green" title="E186">A154</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E5:114_1_2_14_1"/><span class="kw">then </span>
<font color="Maroon" title="c42">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E153:114_1_2"><span class="lab"><font color="Green" title="E184">A152</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E6:114_1_2_14_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E187">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#T32" onmouseover="rs('glib_001/T32')" onmouseout="rh()">GLIB_001:32</a></span>;<br/><a NAME="E7:114_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" onmouseover="rs('abian/T12')" onmouseout="rh()">ABIAN:12</a></span>;<br/><a NAME="E8:114_1_2_14_1"/><span class="kw">then </span>
<font color="Maroon" title="c43">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114_1_2_14_1"><span class="lab"><font color="Green" title="E185">A153</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E9:114_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#T32" onmouseover="rs('glib_001/T32')" onmouseout="rh()">GLIB_001:32</a></span>;<br/><span class="kw">hence </span><a NAME="E10:114_1_2_14_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E83:114_1_2"><span class="lab"><font color="Green" title="E136">A52</font></span></a>, <a class="txt" href="#E144:114_1_2"><span class="lab"><font color="Green" title="E177">A143</font></span></a>, <a class="txt" href="#E150:114_1_2"><span class="lab"><font color="Green" title="E181">A147</font></span></a>, <a class="txt" href="#E1:114_1_2_14_1"><span class="lab"><font color="Green" title="E185">A153</font></span></a>, <a class="txt" href="#E6:114_1_2_14_1"><span class="lab"><font color="Green" title="E187">A155</font></span></a>, <a class="ref" href="chord.html#T41" target="_self" onmouseover="rs('chord/T41')" onmouseout="rh()">Th40</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_14_2"><span class="kw">suppose </span></a><a NAME="E1:114_1_2_14_2"/><span class="lab"><font color="Green" title="E185">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#K1" title="NAT_1:func.1">+</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:114_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="#E144:114_1_2"><span class="lab"><font color="Green" title="E177">A143</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E3:114_1_2_14_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E186">A157</font></span>: 
<font color="Maroon" title="c42">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E153:114_1_2"><span class="lab"><font color="Green" title="E184">A152</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E4:114_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="#E116:114_1_2"><span class="lab"><font color="Green" title="E159">A98</font></span></a>, <a class="txt" href="#E135:114_1_2"><span class="lab"><font color="Green" title="E173">A139</font></span></a>, <a class="txt" href="#E150:114_1_2"><span class="lab"><font color="Green" title="E181">A147</font></span></a>, <a class="txt" href="#E1:114_1_2_14_2"><span class="lab"><font color="Green" title="E185">A156</font></span></a>, <a class="ref" href="glib_001.html#T32" onmouseover="rs('glib_001/T32')" onmouseout="rh()">GLIB_001:32</a></span>;<br/><span class="kw">hence </span><a NAME="E5:114_1_2_14_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E126:114_1_2"><span class="lab"><font color="Green" title="E166">A120</font></span></a>, <a class="txt" href="#E144:114_1_2"><span class="lab"><font color="Green" title="E177">A143</font></span></a>, <a class="txt" href="#E1:114_1_2_14_2"><span class="lab"><font color="Green" title="E185">A156</font></span></a>, <a class="txt" href="#E3:114_1_2_14_2"><span class="lab"><font color="Green" title="E186">A157</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_14_3"><span class="kw">suppose </span></a><a NAME="E1:114_1_2_14_3"/><span class="lab"><font color="Green" title="E185">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#K1" title="NAT_1:func.1">+</a> 4
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:114_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 4
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E144:114_1_2"><span class="lab"><font color="Green" title="E177">A143</font></span></a>, <a class="ref" href="chord.html#T4" target="_self" onmouseover="rs('chord/T4')" onmouseout="rh()">Th4</a></span>;<br/><a NAME="E3:114_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#K1" title="NAT_1:func.1">+</a> 4
 ;<br/><a NAME="E4:114_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#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E5:114_1_2_14_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E186">A159</font></span>: 
<font color="Maroon" title="c42">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E153:114_1_2"><span class="lab"><font color="Green" title="E184">A152</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E6:114_1_2_14_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E187">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="#E117:114_1_2"><span class="lab"><font color="Green" title="E160">A99</font></span></a>, <a class="txt" href="#E134:114_1_2"><span class="lab"><font color="Green" title="E172">A138</font></span></a>, <a class="txt" href="#E150:114_1_2"><span class="lab"><font color="Green" title="E181">A147</font></span></a>, <a class="txt" href="#E1:114_1_2_14_3"><span class="lab"><font color="Green" title="E185">A158</font></span></a>, <a class="ref" href="glib_001.html#T32" onmouseover="rs('glib_001/T32')" onmouseout="rh()">GLIB_001:32</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_14_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:114_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="#E153:114_1_2"><span class="lab"><font color="Green" title="E184">A152</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_14_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:114_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:114_1_2_14_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E102:114_1_2"><span class="lab"><font color="Green" title="E149">A74</font></span></a>, <a class="txt" href="#E112:114_1_2"><span class="lab"><font color="Green" title="E155">A94</font></span></a>, <a class="txt" href="#E132:114_1_2"><span class="lab"><font color="Green" title="E170">A136</font></span></a>, <a class="txt" href="#E139:114_1_2"><span class="lab"><font color="Green" title="E175">A141</font></span></a>, <a class="txt" href="#E119:114_1_2"><span class="lab"><font color="Green" title="E162">A101</font></span></a>, <a class="txt" href="#E140:114_1_2"><span class="lab"><font color="Green" title="E176">A142</font></span></a>, <a class="txt" href="#E120:114_1_2"><span class="lab"><font color="Green" title="E163">A102</font></span></a>, <a class="txt" href="#E125:114_1_2"><span class="lab"><font color="Green" title="E165">A119</font></span></a>, <a class="txt" href="#E151:114_1_2"><span class="lab"><font color="Green" title="E182">A148</font></span></a>, <a class="txt" href="#E148:114_1_2"><span class="lab"><font color="Green" title="E180">A146</font></span></a>, <a class="txt" href="#E1:114_1_2_14_3"><span class="lab"><font color="Green" title="E185">A158</font></span></a>, <a class="ref" href="chord.html#T27" target="_self" onmouseover="rs('chord/T27')" onmouseout="rh()">Th27</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_14_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:114_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:114_1_2_14_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E85:114_1_2"><span class="lab"><font color="Green" title="E138">A54</font></span></a>, <a class="txt" href="#E5:114_1_2_14_3"><span class="lab"><font color="Green" title="E186">A159</font></span></a>, <a class="txt" href="#E6:114_1_2_14_3"><span class="lab"><font color="Green" title="E187">A160</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_14_4"><span class="kw">suppose </span></a><a NAME="E1:114_1_2_14_4"/><span class="lab"><font color="Green" title="E185">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#K1" title="NAT_1:func.1">+</a> 6
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:114_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 6
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E144:114_1_2"><span class="lab"><font color="Green" title="E177">A143</font></span></a>, <a class="ref" href="chord.html#T4" target="_self" onmouseover="rs('chord/T4')" onmouseout="rh()">Th4</a></span>;<br/><a NAME="E3:114_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#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4
 ;<br/><a NAME="E4:114_1_2_14_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E186">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#K1" title="NAT_1:func.1">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_14_4_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:114_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:114_1_2_14_4"><span class="lab"><font color="Green" title="E186">A162</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_14_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:114_1_2_14_4_1_1"/><span class="lab"><font color="Green" title="E187">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#K1" title="NAT_1:func.1">+</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="114_1_2_14_4_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c42">m</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:114_1_2_14_4_1_1_1"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c42">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:114_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#K2" title="NAT_1:func.2">*</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E102:114_1_2"><span class="lab"><font color="Green" title="E149">A74</font></span></a>, <a class="txt" href="#E112:114_1_2"><span class="lab"><font color="Green" title="E155">A94</font></span></a>, <a class="txt" href="#E132:114_1_2"><span class="lab"><font color="Green" title="E170">A136</font></span></a>, <a class="txt" href="#E139:114_1_2"><span class="lab"><font color="Green" title="E175">A141</font></span></a>, <a class="txt" href="#E119:114_1_2"><span class="lab"><font color="Green" title="E162">A101</font></span></a>, <a class="txt" href="#E140:114_1_2"><span class="lab"><font color="Green" title="E176">A142</font></span></a>, <a class="txt" href="#E120:114_1_2"><span class="lab"><font color="Green" title="E163">A102</font></span></a>, <a class="txt" href="#E125:114_1_2"><span class="lab"><font color="Green" title="E165">A119</font></span></a>, <a class="txt" href="#E151:114_1_2"><span class="lab"><font color="Green" title="E182">A148</font></span></a>, <a class="txt" href="#E148:114_1_2"><span class="lab"><font color="Green" title="E180">A146</font></span></a>, <a class="txt" href="#E1:114_1_2_14_4"><span class="lab"><font color="Green" title="E185">A161</font></span></a>, <a class="ref" href="chord.html#T27" target="_self" onmouseover="rs('chord/T27')" onmouseout="rh()">Th27</a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:114_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" onmouseover="rs('chord/T3')" onmouseout="rh()">Th3</a></span>;<br/><a NAME="E4:114_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="#E102:114_1_2"><span class="lab"><font color="Green" title="E149">A74</font></span></a>, <a class="txt" href="#E112:114_1_2"><span class="lab"><font color="Green" title="E155">A94</font></span></a>, <a class="txt" href="#E132:114_1_2"><span class="lab"><font color="Green" title="E170">A136</font></span></a>, <a class="txt" href="#E139:114_1_2"><span class="lab"><font color="Green" title="E175">A141</font></span></a>, <a class="txt" href="#E119:114_1_2"><span class="lab"><font color="Green" title="E162">A101</font></span></a>, <a class="txt" href="#E140:114_1_2"><span class="lab"><font color="Green" title="E176">A142</font></span></a>, <a class="txt" href="#E120:114_1_2"><span class="lab"><font color="Green" title="E163">A102</font></span></a>, <a class="txt" href="#E125:114_1_2"><span class="lab"><font color="Green" title="E165">A119</font></span></a>, <a class="txt" href="#E151:114_1_2"><span class="lab"><font color="Green" title="E182">A148</font></span></a>, <a class="txt" href="#E148:114_1_2"><span class="lab"><font color="Green" title="E180">A146</font></span></a>, <a class="txt" href="#E1:114_1_2_14_4"><span class="lab"><font color="Green" title="E185">A161</font></span></a>, <a class="ref" href="chord.html#T27" target="_self" onmouseover="rs('chord/T27')" onmouseout="rh()">Th27</a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:114_1_2_14_4_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" onmouseover="rs('abian/T12')" onmouseout="rh()">ABIAN:12</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:114_1_2_14_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E188">A164</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c42">m</font>
 ;<br/><a NAME="E4:114_1_2_14_4_1_1"/><span class="lab"><font color="Green" title="E189">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="#E1:114_1_2_14_4_1_1"><span class="lab"><font color="Green" title="E187">A163</font></span></a>, <a class="ref" href="chord.html#T4" target="_self" onmouseover="rs('chord/T4')" onmouseout="rh()">Th4</a></span>;<br/><a NAME="E5:114_1_2_14_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c42">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E153:114_1_2"><span class="lab"><font color="Green" title="E184">A152</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E6:114_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#T32" onmouseover="rs('glib_001/T32')" onmouseout="rh()">GLIB_001:32</a></span>;<br/><span class="kw">hence </span><a NAME="E7:114_1_2_14_4_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E83:114_1_2"><span class="lab"><font color="Green" title="E136">A52</font></span></a>, <a class="txt" href="#E118:114_1_2"><span class="lab"><font color="Green" title="E161">A100</font></span></a>, <a class="txt" href="#E133:114_1_2"><span class="lab"><font color="Green" title="E171">A137</font></span></a>, <a class="txt" href="#E150:114_1_2"><span class="lab"><font color="Green" title="E181">A147</font></span></a>, <a class="txt" href="#E1:114_1_2_14_4"><span class="lab"><font color="Green" title="E185">A161</font></span></a>, <a class="txt" href="#E4:114_1_2_14_4_1_1"><span class="lab"><font color="Green" title="E189">A165</font></span></a>, <a class="txt" href="#E3:114_1_2_14_4_1_1"><span class="lab"><font color="Green" title="E188">A164</font></span></a>, <a class="ref" href="chord.html#T41" target="_self" onmouseover="rs('chord/T41')" onmouseout="rh()">Th40</a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_14_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:114_1_2_14_4_1_2"/><span class="lab"><font color="Green" title="E187">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#K1" title="NAT_1:func.1">+</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:114_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="#E129:114_1_2"><span class="lab"><font color="Green" title="E168">A134</font></span></a>, <a class="ref" href="glib_001.html#T125" onmouseover="rs('glib_001/T125')" onmouseout="rh()">GLIB_001:125</a></span>;<br/><a NAME="E3:114_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" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E4:114_1_2_14_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E188">A167</font></span>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c34">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E5:114_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="#E100:114_1_2"><span class="lab"><font color="Green" title="E147">A72</font></span></a>, <a class="txt" href="#E77:114_1_2"><span class="lab"><font color="Green" title="E132">A48</font></span></a>, <a class="txt" href="#E86:114_1_2"><span class="lab"><font color="Green" title="E139">A64</font></span></a>, <a class="txt" href="#E97:114_1_2"><span class="lab"><font color="Green" title="E145">A70</font></span></a>, <a class="ref" href="chord.html#D3" target="_self" onmouseover="rs('chord/D3')" onmouseout="rh()">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E6:114_1_2_14_4_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E126:114_1_2"><span class="lab"><font color="Green" title="E166">A120</font></span></a>, <a class="txt" href="#E116:114_1_2"><span class="lab"><font color="Green" title="E159">A98</font></span></a>, <a class="txt" href="#E118:114_1_2"><span class="lab"><font color="Green" title="E161">A100</font></span></a>, <a class="txt" href="#E133:114_1_2"><span class="lab"><font color="Green" title="E171">A137</font></span></a>, <a class="txt" href="#E135:114_1_2"><span class="lab"><font color="Green" title="E173">A139</font></span></a>, <a class="txt" href="#E150:114_1_2"><span class="lab"><font color="Green" title="E181">A147</font></span></a>, <a class="txt" href="#E1:114_1_2_14_4"><span class="lab"><font color="Green" title="E185">A161</font></span></a>, <a class="txt" href="#E1:114_1_2_14_4_1_2"><span class="lab"><font color="Green" title="E187">A166</font></span></a>, <a class="txt" href="#E4:114_1_2_14_4_1_2"><span class="lab"><font color="Green" title="E188">A167</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
