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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">G2</font> be   <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">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <br/>  for <font color="Olive" title="b2">V</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b3">G1</font> being    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c1">G2</font>,<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">V</font>  st <font color="Olive" title="b2">V</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">G2</font> &amp;  not <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> &amp; <font color="Maroon" title="c1">G2</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  &amp; (  for <font color="Olive" title="b4">G3</font> being   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Maroon" title="c1">G2</font><br/>  for <font color="Olive" title="b5">w1</font>, <font color="Olive" title="b6">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b4">G3</font>  st <font color="Olive" title="b5">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b6">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">V</font> holds <br/><font color="Olive" title="b5">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">w2</font> ) holds <br/><font color="Olive" title="b3">G1</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">  for <font color="Olive" title="b1">V</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">G1</font> being    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">v</font>,<font color="Olive" title="b1">V</font>  st <font color="Olive" title="b1">V</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">G2</font> &amp;  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> &amp; <font color="Maroon" title="c1">G2</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  &amp; (  for <font color="Olive" title="b3">G3</font> being   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Maroon" title="c1">G2</font><br/>  for <font color="Olive" title="b4">w1</font>, <font color="Olive" title="b5">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b3">G3</font>  st <font color="Olive" title="b4">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b5">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">V</font> holds <br/><font color="Olive" title="b4">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">w2</font> ) holds <br/><font color="Olive" title="b2">G1</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">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">  for <font color="Olive" title="b1">G1</font> being    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">V</font>  st <font color="Maroon" title="c3">V</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">G2</font> &amp;  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> &amp; <font color="Maroon" title="c1">G2</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  &amp; (  for <font color="Olive" title="b2">G3</font> being   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Maroon" title="c1">G2</font><br/>  for <font color="Olive" title="b3">w1</font>, <font color="Olive" title="b4">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b2">G3</font>  st <font color="Olive" title="b3">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Olive" title="b4">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> holds <br/><font color="Olive" title="b3">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w2</font> ) holds <br/><font color="Olive" title="b1">G1</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">G1</font> be    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">V</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">G2</font> &amp;  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> &amp; <font color="Maroon" title="c1">G2</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  &amp; (  for <font color="Olive" title="b1">G3</font> being   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Maroon" title="c1">G2</font><br/>  for <font color="Olive" title="b2">w1</font>, <font color="Olive" title="b3">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">G3</font>  st <font color="Olive" title="b2">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Olive" title="b3">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> holds <br/><font color="Olive" title="b2">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">w2</font> ) implies <font color="Maroon" title="c4">G1</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:110"/><span class="lab"><font color="Green" title="E69">A1</font></span>: 
( <font color="Maroon" title="c3">V</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">G2</font> &amp;  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> )
 <span class="kw">and </span><br/><a NAME="E2:110"/><span class="lab"><font color="Green" title="E70">A2</font></span>: 
<font color="Maroon" title="c1">G2</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> 
 <span class="kw">and </span><br/><a NAME="E3:110"/><span class="lab"><font color="Green" title="E71">A3</font></span>: 
 for <font color="Olive" title="b1">G3</font> being   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Maroon" title="c1">G2</font><br/>  for <font color="Olive" title="b2">w1</font>, <font color="Olive" title="b3">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">G3</font>  st <font color="Olive" title="b2">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Olive" title="b3">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> holds <br/><font color="Olive" title="b2">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">w2</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">G1</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">E</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E5:110"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">E</font> &amp; <font color="Maroon" title="c5">E</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">E</font> )
 <span class="kw">and </span><br/><a NAME="E6:110"/><span class="lab"><font color="Green" title="E72">A4</font></span>: 
 for <font color="Olive" title="b1">v1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> holds <br/> ex <font color="Olive" title="b2">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> &amp; <font color="Olive" title="b2">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> &amp; (  for <font color="Olive" title="b3">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b3">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Olive" title="b2">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">e2</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="ref" href="glib_007.html#D4" target="_self" onmouseover="rs('glib_007/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E7:110"/>
 for <font color="Olive" title="b1">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c4">G1</font>  holds  not <font color="Olive" title="b1">W</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="110_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c6">C</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c4">G1</font><span class="kw"> such that </span><a NAME="E2:110_1"/><span class="lab"><font color="Green" title="E73">A5</font></span>: 
<font color="Maroon" title="c6">C</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</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="E3:110_1"/><span class="lab"><font color="Green" title="E74">A6</font></span>: 
( <font color="Maroon" title="c6">C</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  &amp; <font color="Maroon" title="c6">C</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1"><span class="lab"><font color="Green" title="E73">A5</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">v1</font> = <font color="Maroon" title="c6">C</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ;<br/>
<a NAME="E4:110_1"/><span class="lab"><font color="Green" title="E75">A7</font></span>: 
<font color="Maroon" title="c6">C</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="c6">C</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="#E3:110_1"><span class="lab"><font color="Green" title="E74">A6</font></span></a>, <a class="ref" href="glib_001.html#D24" onmouseover="rs('glib_001/D24')" onmouseout="rh()">GLIB_001:def 24</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:110_1_1"/>
( <font color="Maroon" title="c6">C</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="c2">v</font> or <font color="Maroon" title="c6">C</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="c2">v</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:110_1_1_1"/><span class="lab"><font color="Green" title="E76">A8</font></span>: 
<font color="Maroon" title="c6">C</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="c2">v</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="E2:110_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A9</font></span>: 
 not <font color="Maroon" title="c6">C</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"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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="E3:110_1_1_1"/><span class="lab"><font color="Green" title="E78">A10</font></span>: 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c4">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">G2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="ref" href="glib_007.html#D4" target="_self" onmouseover="rs('glib_007/D4')" onmouseout="rh()">Def4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">v1</font> = <font color="Maroon" title="c6">C</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1_1_1"><span class="lab"><font color="Green" title="E77">A9</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">G3</font> =  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">v1</font></span>)</span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:110_1_1_1_1"/>
( <font color="Maroon" title="c6">C</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="c1">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">v1</font> or  not <font color="Maroon" title="c6">C</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="c1">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">v1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:110_1_1_1_1_1"/>
<font color="Maroon" title="c6">C</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="c1">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">v1</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="E2:110_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A11</font></span>: 
<font color="Maroon" title="c6">C</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>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font>
 <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/><a NAME="E3:110_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A12</font></span>: 
<font color="Maroon" title="c6">C</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_007.html#T65" target="_self" onmouseover="rs('glib_007/T65')" onmouseout="rh()">Th65</a></span>;<br/><a NAME="E4:110_1_1_1_1_1"/>
<font color="Maroon" title="c1">G2</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c4">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#T57" onmouseover="rs('glib_006/T57')" onmouseout="rh()">GLIB_006:57</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">C2</font> = <font color="Maroon" title="c6">C</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1_1_1_1_1"><span class="lab"><font color="Green" title="E79">A11</font></span></a>, <a class="txt" href="#E3:110_1_1_1_1_1"><span class="lab"><font color="Green" title="E80">A12</font></span></a>, <a class="ref" href="glib_001.html#T170" onmouseover="rs('glib_001/T170')" onmouseout="rh()">GLIB_001:170</a></span>;<br/><a NAME="E6:110_1_1_1_1_1"/>
<font color="Maroon" title="c10">C2</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="#E2:110_1"><span class="lab"><font color="Green" title="E73">A5</font></span></a>, <a class="ref" href="glib_006.html#T24" onmouseover="rs('glib_006/T24')" onmouseout="rh()">GLIB_006:24</a></span>;<br/><span class="kw">hence </span><a NAME="E7:110_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="glib_002.html#D2" onmouseover="rs('glib_002/D2')" onmouseout="rh()">GLIB_002:def 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:110_1_1_1_1_2"/>
 not <font color="Maroon" title="c6">C</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="c1">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">v1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:110_1_1_1_1_2"/><span class="lab"><font color="Green" title="E79">A13</font></span>: 
( <font color="Maroon" title="c10">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  &amp;  not <font color="Maroon" title="c10">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">v1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">v2</font> = <font color="Maroon" title="c10">v2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">G1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E79">A13</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">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="E6:110_1_1_1_1_2"/><span class="lab"><font color="Green" title="E80">A14</font></span>: 
<font color="Maroon" title="c12">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="c6">C</font>
 <span class="kw">and </span><br/><a NAME="E7:110_1_1_1_1_2"/><span class="lab"><font color="Green" title="E81">A15</font></span>: 
<font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E79">A13</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="E8:110_1_1_1_1_2"/><span class="lab"><font color="Green" title="E82">A16</font></span>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">n</font> &amp; <font color="Maroon" title="c12">n</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="c6">C</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="110_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:110_1_1_1_1_2_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">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="c12">n</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="c6">C</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="110_1_1_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:110_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E83">A17</font></span>: 
1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">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="E2:110_1_1_1_1_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">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="E3:110_1_1_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E83">A17</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="E4:110_1_1_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E81">A15</font></span></a>, <a class="ref" href="glib_001.html#D6" onmouseover="rs('glib_001/D6')" onmouseout="rh()">GLIB_001:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:110_1_1_1_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E79">A13</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>

<span class="kw">assume </span><a NAME="E2:110_1_1_1_1_2_1"/>
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</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="E3:110_1_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">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> <font color="Maroon" title="c6">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E80">A14</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="E4:110_1_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110_1"><span class="lab"><font color="Green" title="E75">A7</font></span></a>, <a class="txt" href="#E7:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E81">A15</font></span></a>, <a class="ref" href="glib_001.html#D7" onmouseover="rs('glib_001/D7')" onmouseout="rh()">GLIB_001:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:110_1_1_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E79">A13</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="110_1_1_1_1_2_2"><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="E1:110_1_1_1_1_2_2"/>
( <font color="Maroon" title="c11">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> or <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_1_1_1_1_2_2_1"><span class="kw">suppose </span></a><a NAME="E1:110_1_1_1_1_2_2_1"/><span class="lab"><font color="Green" title="E83">A18</font></span>: 
<font color="Maroon" title="c11">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</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="E2:110_1_1_1_1_2_2_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c11">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">v2</font> = <font color="Maroon" title="c11">v2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_1"><span class="lab"><font color="Green" title="E78">A10</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c14">P1</font> = <font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c12">n</font>);<br/><span class="kw">set </span><font color="Maroon" title="c15">P2</font> = <font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c12">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>);<br/><a NAME="E4:110_1_1_1_1_2_2_1"/><span class="lab"><font color="Green" title="E84">A19</font></span>: 
1 is   <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="glib_007.html#E89"><span class="lab"><font color="Green" title="E68">Lm16</font></span></a></span>;<br/><a NAME="E5:110_1_1_1_1_2_2_1"/><span class="lab"><font color="Green" title="E85">A20</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n</font> &amp; <font color="Maroon" title="c12">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="c6">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" onmouseover="rs('abian/T12')" onmouseout="rh()">ABIAN:12</a>, <a class="txt" href="#E6:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E80">A14</font></span></a></span>;<br/><a NAME="E6:110_1_1_1_1_2_2_1"/><span class="lab"><font color="Green" title="E86">A21</font></span>: 
( <font color="Maroon" title="c12">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="c6">C</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</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="c6">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E80">A14</font></span></a></span>;<br/><a NAME="E7:110_1_1_1_1_2_2_1"/><span class="lab"><font color="Green" title="E87">A22</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="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c12">n</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="c12">n</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="#E5:110_1_1_1_1_2_2_1"><span class="lab"><font color="Green" title="E85">A20</font></span></a>, <a class="txt" href="#E4:110_1_1_1_1_2_2_1"><span class="lab"><font color="Green" title="E84">A19</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="E8:110_1_1_1_1_2_2_1"/><span class="lab"><font color="Green" title="E88">A23</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="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c12">n</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">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="c6">C</font></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="#E6:110_1_1_1_1_2_2_1"><span class="lab"><font color="Green" title="E86">A21</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="E9:110_1_1_1_1_2_2_1"/>
(  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c12">n</font>)</span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  or  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c12">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="110_1_1_1_1_2_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:110_1_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c12">n</font>)</span>)</span> <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">  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c12">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">m</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="E3:110_1_1_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E89">A24</font></span>: 
( <font color="Maroon" title="c16">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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c12">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c12">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T87" onmouseover="rs('glib_001/T87')" onmouseout="rh()">GLIB_001:87</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">m1</font> = <font color="Maroon" title="c16">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 as    <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="E5:110_1_1_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E90">A25</font></span>: 
<font color="Maroon" title="c17">m1</font> <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="c16">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="glib_007.html#E87"><span class="lab"><font color="Green" title="E66">Lm14</font></span></a></span>;<br/>
<a NAME="E6:110_1_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c17">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c12">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E89">A24</font></span></a>, <a class="txt" href="glib_007.html#E87"><span class="lab"><font color="Green" title="E66">Lm14</font></span></a></span>;<br/>
<a NAME="E7:110_1_1_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A26</font></span>: 
<font color="Maroon" title="c6">C</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> <font color="Maroon" title="c17">m1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110_1_1_1_1_2_2_1"><span class="lab"><font color="Green" title="E84">A19</font></span></a>, <a class="txt" href="#E5:110_1_1_1_1_2_2_1"><span class="lab"><font color="Green" title="E85">A20</font></span></a>, <a class="txt" href="#E3:110_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E89">A24</font></span></a>, <a class="txt" href="#E5:110_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E90">A25</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="E8:110_1_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c12">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span> <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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">k</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="E10:110_1_1_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E92">A27</font></span>: 
( <font color="Maroon" title="c18">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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c12">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c12">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T87" onmouseover="rs('glib_001/T87')" onmouseout="rh()">GLIB_001:87</a></span>;<br/>
<a NAME="E11:110_1_1_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E93">A28</font></span>: 
<font color="Maroon" title="c18">k</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c12">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="110_1_1_1_1_2_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:110_1_1_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c12">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</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="E2:110_1_1_1_1_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A29</font></span>: 
<font color="Maroon" title="c18">k</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="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c12">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:110_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E92">A27</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/>
<font color="Maroon" title="c8">v1</font> = 
<font color="Maroon" title="c6">C</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="c6">C</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110_1"><span class="lab"><font color="Green" title="E75">A7</font></span></a>, <a class="ref" href="glib_001.html#D7" onmouseover="rs('glib_001/D7')" onmouseout="rh()">GLIB_001:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c12">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span> <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="#E6:110_1_1_1_1_2_2_1"><span class="lab"><font color="Green" title="E86">A21</font></span></a>, <a class="ref" href="glib_001.html#T37" onmouseover="rs('glib_001/T37')" onmouseout="rh()">GLIB_001:37</a></span>
<br/>.= 
<font color="Maroon" title="c2">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:110_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E92">A27</font></span></a>, <a class="txt" href="#E2:110_1_1_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E94">A29</font></span></a>, <a class="ref" href="glib_001.html#D7" onmouseover="rs('glib_001/D7')" onmouseout="rh()">GLIB_001:def 7</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:110_1_1_1_1_2_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_1_1_1"><span class="lab"><font color="Green" title="E76">A8</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">reconsider </span><font color="Maroon" title="c19">k1</font> = <font color="Maroon" title="c18">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/>
<a NAME="E13:110_1_1_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E94">A30</font></span>: 
<font color="Maroon" title="c19">k1</font> <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="c18">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="glib_007.html#E87"><span class="lab"><font color="Green" title="E66">Lm14</font></span></a></span>;<br/>
<a NAME="E14:110_1_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c19">k1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c12">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:110_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E92">A27</font></span></a>, <a class="txt" href="glib_007.html#E87"><span class="lab"><font color="Green" title="E66">Lm14</font></span></a></span>;<br/>
<a NAME="E15:110_1_1_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:110_1_1_1_1_2_2_1"><span class="lab"><font color="Green" title="E86">A21</font></span></a>, <a class="txt" href="#E10:110_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E92">A27</font></span></a>, <a class="txt" href="#E13:110_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E94">A30</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="E16:110_1_1_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A31</font></span>: 
<font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">k1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E90">A25</font></span></a>, <a class="txt" href="#E7:110_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E91">A26</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">nk</font> = <font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">k1</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="glib_007.html#E88"><span class="lab"><font color="Green" title="E67">Lm15</font></span></a></span>;<br/>
<a NAME="E18:110_1_1_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E96">A32</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="110_1_1_1_1_2_2_1_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:110_1_1_1_1_2_2_1_1_2"/><span class="lab"><font color="Green" title="E97">A33</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">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="E2:110_1_1_1_1_2_2_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">k</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="110_1_1_1_1_2_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:110_1_1_1_1_2_2_1_1_2_1"/>
1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c18">k</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="E2:110_1_1_1_1_2_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_1_1_1_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E97">A33</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="E3:110_1_1_1_1_2_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A34</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c12">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</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="c2">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:110_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E92">A27</font></span></a>, <a class="ref" href="glib_001.html#D6" onmouseover="rs('glib_001/D6')" onmouseout="rh()">GLIB_001:def 6</a></span>;<br/>
<a NAME="E4:110_1_1_1_1_2_2_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c12">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</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="c13">v2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E81">A15</font></span></a>, <a class="txt" href="#E6:110_1_1_1_1_2_2_1"><span class="lab"><font color="Green" title="E86">A21</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="E5:110_1_1_1_1_2_2_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_1_1_2_2_1_1_2_1"><span class="lab"><font color="Green" title="E98">A34</font></span></a>, <a class="txt" href="#E1:110_1_1_1_1_2_2_1"><span class="lab"><font color="Green" title="E83">A18</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">hence </span><a NAME="E3:110_1_1_1_1_2_2_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T49" onmouseover="rs('nat_d/T49')" onmouseout="rh()">NAT_D:49</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="E19:110_1_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c16">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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c12">n</font>)</span>)</span></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="#E3:110_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E89">A24</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E20:110_1_1_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A35</font></span>: 
<font color="Maroon" title="c16">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">n</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="#E7:110_1_1_1_1_2_2_1"><span class="lab"><font color="Green" title="E87">A22</font></span></a></span>;<br/>
<a NAME="E21:110_1_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c12">n</font> <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="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:110_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E96">A32</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="E22:110_1_1_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A36</font></span>: 
<font color="Maroon" title="c16">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">nk</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:110_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E97">A35</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="E23:110_1_1_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E99">A37</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">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="E24:110_1_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">n</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="c6">C</font></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="#E10:110_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E92">A27</font></span></a>, <a class="txt" href="#E8:110_1_1_1_1_2_2_1"><span class="lab"><font color="Green" title="E88">A23</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="E25:110_1_1_1_1_2_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <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="c6">C</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/>
<a NAME="E26:110_1_1_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <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="c6">C</font>
 
;<br/>
<a NAME="E27:110_1_1_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">k1</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="c6">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:110_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E99">A37</font></span></a>, <a class="ref" href="xreal_1.html#T233" onmouseover="rs('xreal_1/T233')" onmouseout="rh()">XREAL_1:233</a></span>;<br/>
<a NAME="E28:110_1_1_1_1_2_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c16">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c20">nk</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="c6">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1"><span class="lab"><font color="Green" title="E74">A6</font></span></a>, <a class="txt" href="#E16:110_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E95">A31</font></span></a>, <a class="ref" href="glib_001.html#D28" onmouseover="rs('glib_001/D28')" onmouseout="rh()">GLIB_001:def 28</a>, <a class="txt" href="#E22:110_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E98">A36</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E29:110_1_1_1_1_2_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:110_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E93">A28</font></span></a>, <a class="txt" href="#E13:110_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E94">A30</font></span></a>, <a class="txt" href="#E8:110_1_1_1_1_2_2_1"><span class="lab"><font color="Green" title="E88">A23</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="110_1_1_1_1_2_2_1_2"><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="E1:110_1_1_1_1_2_2_1_2"/><span class="kw">then </span>
(  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c12">n</font>)</span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  or  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c12">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_1_1_1_1_2_2_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:110_1_1_1_1_2_2_1_2_1"/>
 not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c12">n</font>)</span>)</span> <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"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">P3</font> = <font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c12">n</font>) as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="ref" href="glib_007.html#T64" target="_self" onmouseover="rs('glib_007/T64')" onmouseout="rh()">Th64</a></span>;<br/><a NAME="E3:110_1_1_1_1_2_2_1_2_1"/>
<font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c12">n</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1,<font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1_1_1_1_2_2_1"><span class="lab"><font color="Green" title="E85">A20</font></span></a>, <a class="txt" href="#E4:110_1_1_1_1_2_2_1"><span class="lab"><font color="Green" title="E84">A19</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="E4:110_1_1_1_1_2_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c12">n</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c13">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E81">A15</font></span></a>, <a class="ref" href="glib_001.html#D6" onmouseover="rs('glib_001/D6')" onmouseout="rh()">GLIB_001:def 6</a></span>;<br/><a NAME="E5:110_1_1_1_1_2_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">P3</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c13">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T19" onmouseover="rs('glib_001/T19')" onmouseout="rh()">GLIB_001:19</a></span>;<br/><a NAME="E6:110_1_1_1_1_2_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D5" onmouseover="rs('glib_002/D5')" onmouseout="rh()">GLIB_002:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E7:110_1_1_1_1_2_2_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E79">A13</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="110_1_1_1_1_2_2_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:110_1_1_1_1_2_2_1_2_2"/>
 not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c12">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span> <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"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">P3</font> = <font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c12">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>) as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="ref" href="glib_007.html#T64" target="_self" onmouseover="rs('glib_007/T64')" onmouseout="rh()">Th64</a></span>;<br/><a NAME="E3:110_1_1_1_1_2_2_1_2_2"/>
<font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c12">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c6">C</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="c6">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:110_1_1_1_1_2_2_1"><span class="lab"><font color="Green" title="E86">A21</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="E4:110_1_1_1_1_2_2_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c12">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c13">v2</font>,<font color="Maroon" title="c8">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E81">A15</font></span></a>, <a class="txt" href="#E4:110_1"><span class="lab"><font color="Green" title="E75">A7</font></span></a>, <a class="ref" href="glib_001.html#D7" onmouseover="rs('glib_001/D7')" onmouseout="rh()">GLIB_001:def 7</a></span>;<br/><a NAME="E5:110_1_1_1_1_2_2_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">P3</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c13">v2</font>,<font color="Maroon" title="c8">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T19" onmouseover="rs('glib_001/T19')" onmouseout="rh()">GLIB_001:19</a></span>;<br/><a NAME="E6:110_1_1_1_1_2_2_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">P3</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c13">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T23" onmouseover="rs('glib_001/T23')" onmouseout="rh()">GLIB_001:23</a></span>;<br/><span class="kw">hence </span><a NAME="E7:110_1_1_1_1_2_2_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E79">A13</font></span></a>, <a class="ref" href="glib_002.html#D5" onmouseover="rs('glib_002/D5')" onmouseout="rh()">GLIB_002: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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_1_1_1_1_2_2_2"><span class="kw">suppose </span></a><a NAME="E1:110_1_1_1_1_2_2_2"/><span class="lab"><font color="Green" title="E83">A38</font></span>: 
<font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</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">reconsider </span><font color="Maroon" title="c13">m</font> = <font color="Maroon" title="c12">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 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="#E8:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E82">A16</font></span></a>, <a class="txt" href="glib_007.html#E79"><span class="lab"><font color="Green" title="E60">Lm13</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c14">P1</font> = <font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c13">m</font>);<br/><span class="kw">set </span><font color="Maroon" title="c15">P2</font> = <font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>);<br/><a NAME="E3:110_1_1_1_1_2_2_2"/>
<font color="Maroon" title="c12">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="c6">C</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E80">A14</font></span></a>, <a class="ref" href="xreal_1.html#T13" onmouseover="rs('xreal_1/T13')" onmouseout="rh()">XREAL_1:13</a></span>;<br/><a NAME="E4:110_1_1_1_1_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A39</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font> &amp; <font color="Maroon" title="c13">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="c6">C</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="E5:110_1_1_1_1_2_2_2"/><span class="lab"><font color="Green" title="E85">A40</font></span>: 
( <font color="Maroon" title="c12">n</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="c6">C</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</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="c6">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E82">A16</font></span></a>, <a class="ref" href="glib_001.html#T1" onmouseover="rs('glib_001/T1')" onmouseout="rh()">GLIB_001:1</a></span>;<br/><a NAME="E6:110_1_1_1_1_2_2_2"/><span class="lab"><font color="Green" title="E86">A41</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="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c13">m</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="c13">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E84">A39</font></span></a>, <a class="txt" href="glib_007.html#E89"><span class="lab"><font color="Green" title="E68">Lm16</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="E7:110_1_1_1_1_2_2_2"/><span class="lab"><font color="Green" title="E87">A42</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="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></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="#E5:110_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E85">A40</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="E8:110_1_1_1_1_2_2_2"/><span class="lab"><font color="Green" title="E88">A43</font></span>: 
(  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c13">m</font>)</span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  &amp;  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="110_1_1_1_1_2_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:110_1_1_1_1_2_2_2_1"/>
( <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c13">m</font>)</span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  or <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span> <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"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_1_1_1_1_2_2_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:110_1_1_1_1_2_2_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c13">m</font>)</span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  or <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_1_1_1_1_2_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:110_1_1_1_1_2_2_2_1_1_1"/>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c13">m</font>)</span>)</span> <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"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">k</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="E3:110_1_1_1_1_2_2_2_1_1_1"/><span class="lab"><font color="Green" title="E89">A44</font></span>: 
( <font color="Maroon" title="c16">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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c13">m</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c13">m</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T87" onmouseover="rs('glib_001/T87')" onmouseout="rh()">GLIB_001:87</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">k1</font> = <font color="Maroon" title="c16">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 as    <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="E5:110_1_1_1_1_2_2_2_1_1_1"/>
<font color="Maroon" title="c17">k1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c13">m</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_1_1_2_2_2_1_1_1"><span class="lab"><font color="Green" title="E89">A44</font></span></a>, <a class="txt" href="glib_007.html#E87"><span class="lab"><font color="Green" title="E66">Lm14</font></span></a></span>;<br/><a NAME="E6:110_1_1_1_1_2_2_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c13">m</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k1</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="c6">C</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> <font color="Maroon" title="c17">k1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E84">A39</font></span></a>, <a class="txt" href="glib_007.html#E89"><span class="lab"><font color="Green" title="E68">Lm16</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="E7:110_1_1_1_1_2_2_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c13">m</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_007.html#E87"><span class="lab"><font color="Green" title="E66">Lm14</font></span></a></span>;<br/><a NAME="E8:110_1_1_1_1_2_2_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c13">m</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_007.html#E87"><span class="lab"><font color="Green" title="E66">Lm14</font></span></a></span>;<br/><a NAME="E9:110_1_1_1_1_2_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A45</font></span>: 
<font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E83">A38</font></span></a>, <a class="txt" href="#E7:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E81">A15</font></span></a>, <a class="txt" href="#E3:110_1_1_1_1_2_2_2_1_1_1"><span class="lab"><font color="Green" title="E89">A44</font></span></a></span>;<br/><a NAME="E10:110_1_1_1_1_2_2_2_1_1_1"/>
<font color="Maroon" title="c12">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T15" onmouseover="rs('xreal_1/T15')" onmouseout="rh()">XREAL_1:15</a></span>;<br/><a NAME="E11:110_1_1_1_1_2_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A46</font></span>: 
<font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">n</font>
 ;<br/><a NAME="E12:110_1_1_1_1_2_2_2_1_1_1"/>
<font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_1_1_2_2_2_1_1_1"><span class="lab"><font color="Green" title="E89">A44</font></span></a>, <a class="txt" href="#E6:110_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E86">A41</font></span></a></span>;<br/><a NAME="E13:110_1_1_1_1_2_2_2_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">n</font> &amp; <font color="Maroon" title="c12">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="c6">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E80">A14</font></span></a>, <a class="txt" href="#E11:110_1_1_1_1_2_2_2_1_1_1"><span class="lab"><font color="Green" title="E91">A46</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="E14:110_1_1_1_1_2_2_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">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> <font color="Maroon" title="c6">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1"><span class="lab"><font color="Green" title="E73">A5</font></span></a>, <a class="txt" href="#E9:110_1_1_1_1_2_2_2_1_1_1"><span class="lab"><font color="Green" title="E90">A45</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/><span class="kw">hence </span><a NAME="E15:110_1_1_1_1_2_2_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E82">A16</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="110_1_1_1_1_2_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:110_1_1_1_1_2_2_2_1_1_2"/>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span> <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"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">k</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="E3:110_1_1_1_1_2_2_2_1_1_2"/><span class="lab"><font color="Green" title="E89">A47</font></span>: 
( <font color="Maroon" title="c16">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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T87" onmouseover="rs('glib_001/T87')" onmouseout="rh()">GLIB_001:87</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">k1</font> = <font color="Maroon" title="c16">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 as    <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="E5:110_1_1_1_1_2_2_2_1_1_2"/>
<font color="Maroon" title="c17">k1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_1_1_2_2_2_1_1_2"><span class="lab"><font color="Green" title="E89">A47</font></span></a>, <a class="txt" href="glib_007.html#E87"><span class="lab"><font color="Green" title="E66">Lm14</font></span></a></span>;<br/><a NAME="E6:110_1_1_1_1_2_2_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k1</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="c6">C</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"><font color="Maroon" title="c12">n</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> <font color="Maroon" title="c17">k1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E85">A40</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="E7:110_1_1_1_1_2_2_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">C</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"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <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="glib_007.html#E87"><span class="lab"><font color="Green" title="E66">Lm14</font></span></a></span>;<br/><a NAME="E8:110_1_1_1_1_2_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A48</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">C</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"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">k</font></span>)</span> <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="glib_007.html#E87"><span class="lab"><font color="Green" title="E66">Lm14</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">nk</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/><a NAME="E10:110_1_1_1_1_2_2_2_1_1_2"/><span class="lab"><font color="Green" title="E91">A49</font></span>: 
<font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">nk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E83">A38</font></span></a>, <a class="txt" href="#E7:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E81">A15</font></span></a>, <a class="txt" href="#E3:110_1_1_1_1_2_2_2_1_1_2"><span class="lab"><font color="Green" title="E89">A47</font></span></a>, <a class="txt" href="#E8:110_1_1_1_1_2_2_2_1_1_2"><span class="lab"><font color="Green" title="E90">A48</font></span></a></span>;<br/><a NAME="E11:110_1_1_1_1_2_2_2_1_1_2"/>
( <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">nk</font> &amp; <font color="Maroon" title="c18">nk</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="c6">C</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="110_1_1_1_1_2_2_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:110_1_1_1_1_2_2_2_1_1_2_1"/>
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E2:110_1_1_1_1_2_2_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c16">k</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="E3:110_1_1_1_1_2_2_2_1_1_2_1"/>
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">nk</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="c18">nk</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="c6">C</font></span><br/>

<a NAME="E4:110_1_1_1_1_2_2_2_1_1_2_1"/>
<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="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:110_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E87">A42</font></span></a></span>;<br/>
<a NAME="E5:110_1_1_1_1_2_2_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c6">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_1_1_2_2_2_1_1_2"><span class="lab"><font color="Green" title="E89">A47</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:110_1_1_1_1_2_2_2_1_1_2_1"/>
<font color="Maroon" title="c18">nk</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="c6">C</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"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E12:110_1_1_1_1_2_2_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:110_1_1_1_1_2_2_2_1_1_2"><span class="lab"><font color="Green" title="E91">A49</font></span></a>, <a class="txt" href="#E2:110_1"><span class="lab"><font color="Green" title="E73">A5</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/><span class="kw">hence </span><a NAME="E13:110_1_1_1_1_2_2_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E82">A16</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><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">P1</font> = <font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c13">m</font>), <font color="Maroon" title="c17">P2</font> = <font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>) as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="ref" href="glib_007.html#T64" target="_self" onmouseover="rs('glib_007/T64')" onmouseout="rh()">Th64</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c18">w1</font> = <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font>;<br/><span class="kw">set </span><font color="Maroon" title="c19">w2</font> = <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c20">e1</font> = <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c21">e2</font> = <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><a NAME="E10:110_1_1_1_1_2_2_2"/>
<font color="Maroon" title="c12">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <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="c6">C</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E80">A14</font></span></a>, <a class="ref" href="xreal_1.html#T15" onmouseover="rs('xreal_1/T15')" onmouseout="rh()">XREAL_1:15</a></span>;<br/><a NAME="E11:110_1_1_1_1_2_2_2"/><span class="kw">then </span>
( <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font>,<font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c4">G1</font> &amp; <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</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="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c4">G1</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>, <a class="txt" href="#E8:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E82">A16</font></span></a></span>;<br/><a NAME="E12:110_1_1_1_1_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A50</font></span>: 
( <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> &amp; <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</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="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E83">A38</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</a>, <a class="txt" href="#E7:110_1_1_1_1_2"><span class="lab"><font color="Green" title="E81">A15</font></span></a></span>;<br/><a NAME="E13:110_1_1_1_1_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A51</font></span>: 
( <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="ref" href="glib_007.html#D4" target="_self" onmouseover="rs('glib_007/D4')" onmouseout="rh()">Def4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">w1</font> = <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font>, <font color="Maroon" title="c23">w2</font> = <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E69">A1</font></span></a></span>;<br/><a NAME="E15:110_1_1_1_1_2_2_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c13">m</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="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c13">m</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="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E84">A39</font></span></a>, <a class="txt" href="glib_007.html#E89"><span class="lab"><font color="Green" title="E68">Lm16</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="E16:110_1_1_1_1_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A52</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c13">m</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="c8">v1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c13">m</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="c22">w1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D6" onmouseover="rs('glib_001/D6')" onmouseout="rh()">GLIB_001:def 6</a></span>;<br/><a NAME="E17:110_1_1_1_1_2_2_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">w2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</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="c6">C</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="c6">C</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E85">A40</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="E18:110_1_1_1_1_2_2_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">w2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</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="c8">v1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110_1"><span class="lab"><font color="Green" title="E75">A7</font></span></a>, <a class="ref" href="glib_001.html#D7" onmouseover="rs('glib_001/D7')" onmouseout="rh()">GLIB_001:def 7</a></span>;<br/><a NAME="E19:110_1_1_1_1_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A53</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></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="c8">v1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">w2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T22" onmouseover="rs('glib_001/T22')" onmouseout="rh()">GLIB_001:22</a></span>;<br/><a NAME="E20:110_1_1_1_1_2_2_2"/>
<font color="Maroon" title="c22">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="txt" href="#E8:110_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E88">A43</font></span></a>, <a class="txt" href="#E16:110_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E91">A52</font></span></a>, <a class="ref" href="glib_007.html#T69" target="_self" onmouseover="rs('glib_007/T69')" onmouseout="rh()">Th69</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">w1</font> = <font color="Maroon" title="c22">w1</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">G2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">v1</font></span>)</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="E22:110_1_1_1_1_2_2_2"/>
<font color="Maroon" title="c23">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">v1</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="110_1_1_1_1_2_2_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:110_1_1_1_1_2_2_2_2"/>
 not <font color="Maroon" title="c23">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">v1</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="E2:110_1_1_1_1_2_2_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></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="#E1:110"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="txt" href="#E19:110_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E92">A53</font></span></a>, <a class="ref" href="glib_007.html#T69" target="_self" onmouseover="rs('glib_007/T69')" onmouseout="rh()">Th69</a></span>;<br/>
<a NAME="E3:110_1_1_1_1_2_2_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span>)</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#T92" onmouseover="rs('glib_001/T92')" onmouseout="rh()">GLIB_001:92</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:110_1_1_1_1_2_2_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:110_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E88">A43</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c25">w2</font> = <font color="Maroon" title="c23">w2</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">G2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">v1</font></span>)</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="E24:110_1_1_1_1_2_2_2"/><span class="lab"><font color="Green" title="E93">A54</font></span>: 
<font color="Maroon" title="c24">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">w2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:110_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E90">A51</font></span></a>, <a class="txt" href="#E3:110"><span class="lab"><font color="Green" title="E71">A3</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c26">e3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E26:110_1_1_1_1_2_2_2"/>
( <font color="Maroon" title="c26">e3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> &amp; <font color="Maroon" title="c26">e3</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c24">w1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> )
 <span class="kw">and </span><br/><a NAME="E27:110_1_1_1_1_2_2_2"/><span class="lab"><font color="Green" title="E94">A55</font></span>: 
 for <font color="Olive" title="b1">e5</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e5</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c24">w1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Olive" title="b1">e5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">e3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:110"><span class="lab"><font color="Green" title="E72">A4</font></span></a>, <a class="txt" href="#E13:110_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E90">A51</font></span></a></span>;<br/><a NAME="E28:110_1_1_1_1_2_2_2"/>
( <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">e3</font> &amp; <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</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="c26">e3</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:110_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E93">A54</font></span></a>, <a class="txt" href="#E12:110_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E89">A50</font></span></a>, <a class="txt" href="#E27:110_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E94">A55</font></span></a></span>;<br/><a NAME="E29:110_1_1_1_1_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A56</font></span>: 
<font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c27">n2</font> = <font color="Maroon" title="c13">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>  ;<br/><a NAME="E31:110_1_1_1_1_2_2_2"/>
<font color="Maroon" title="c12">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="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E32:110_1_1_1_1_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A57</font></span>: 
<font color="Maroon" title="c27">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E33:110_1_1_1_1_2_2_2"/><span class="lab"><font color="Green" title="E97">A58</font></span>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E34:110_1_1_1_1_2_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E85">A40</font></span></a>, <a class="ref" href="xreal_1.html#T13" onmouseover="rs('xreal_1/T13')" onmouseout="rh()">XREAL_1:13</a></span>;<br/><a NAME="E35:110_1_1_1_1_2_2_2"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">n2</font> &amp; <font color="Maroon" title="c27">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c12">n</font> <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="c6">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:110_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E96">A57</font></span></a>, <a class="txt" href="#E33:110_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E97">A58</font></span></a></span>;<br/><a NAME="E36:110_1_1_1_1_2_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">n2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1"><span class="lab"><font color="Green" title="E74">A6</font></span></a>, <a class="ref" href="glib_001.html#T138" onmouseover="rs('glib_001/T138')" onmouseout="rh()">GLIB_001:138</a></span>;<br/><span class="kw">hence </span><a NAME="E37:110_1_1_1_1_2_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:110_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E95">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></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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:110_1_1_2"/><span class="lab"><font color="Green" title="E76">A59</font></span>: 
<font color="Maroon" title="c6">C</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="c2">v</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="E2:110_1_1_2"/><span class="lab"><font color="Green" title="E77">A60</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 5
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="110_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:110_1_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 5
 ; <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="E2:110_1_1_2_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<a NAME="E3:110_1_1_2_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E4:110_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A61</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1"><span class="lab"><font color="Green" title="E73">A5</font></span></a>, <a class="ref" href="glib_001.html#T126" onmouseover="rs('glib_001/T126')" onmouseout="rh()">GLIB_001:126</a>, <a class="ref" href="chord.html#T7" onmouseover="rs('chord/T7')" onmouseout="rh()">CHORD:7</a></span>;<br/>
<a NAME="E5:110_1_1_2_1"/><span class="kw">then </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="c6">C</font>
 
;<br/>
<a NAME="E6:110_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">C</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="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1,<font color="Maroon" title="c6">C</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="c4">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="glib_007.html#E89"><span class="lab"><font color="Green" title="E68">Lm16</font></span></a>, <a class="ref" href="glib_001.html#D3" onmouseover="rs('glib_001/D3')" onmouseout="rh()">GLIB_001:def 3</a></span>;<br/>
<a NAME="E7:110_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">C</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="c6">C</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c6">C</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="c6">C</font></span>)</span>,<font color="Maroon" title="c4">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110_1_1_2_1"><span class="lab"><font color="Green" title="E78">A61</font></span></a>, <a class="ref" href="glib_001.html#D6" onmouseover="rs('glib_001/D6')" onmouseout="rh()">GLIB_001:def 6</a></span>;<br/>
<a NAME="E8:110_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">C</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="c6">C</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c6">C</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c4">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D7" onmouseover="rs('glib_001/D7')" onmouseout="rh()">GLIB_001:def 7</a></span>;<br/>
<a NAME="E9:110_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">C</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="c2">v</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_1_1_2"><span class="lab"><font color="Green" title="E76">A59</font></span></a>, <a class="txt" href="#E4:110_1"><span class="lab"><font color="Green" title="E75">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:110_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="ref" href="glib_007.html#D4" target="_self" onmouseover="rs('glib_007/D4')" onmouseout="rh()">Def4</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="E3:110_1_1_2"/><span class="lab"><font color="Green" title="E78">A62</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="c6">C</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="110_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:110_1_1_2_2"/><span class="lab"><font color="Green" title="E79">A63</font></span>: 
1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</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="E2:110_1_1_2_2"/>
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="c6">C</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="E3:110_1_1_2_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_1_1_2_2"><span class="lab"><font color="Green" title="E79">A63</font></span></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="E4:110_1_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1_1_2"><span class="lab"><font color="Green" title="E77">A60</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">m</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 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="glib_007.html#E79"><span class="lab"><font color="Green" title="E60">Lm13</font></span></a></span>;<br/><a NAME="E5:110_1_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="c6">C</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <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="c6">C</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T15" onmouseover="rs('xreal_1/T15')" onmouseout="rh()">XREAL_1:15</a></span>;<br/><a NAME="E6:110_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A64</font></span>: 
<font color="Maroon" title="c8">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="c6">C</font>
 ;<br/><span class="kw">set </span><font color="Maroon" title="c9">w1</font> = <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3;<br/><span class="kw">set </span><font color="Maroon" title="c10">w2</font> = <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font>;<br/><span class="kw">set </span><font color="Maroon" title="c11">e1</font> = <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2;<br/><span class="kw">set </span><font color="Maroon" title="c12">e2</font> = <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c13">P</font> = <font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (3,<font color="Maroon" title="c8">m</font>);<br/><a NAME="E7:110_1_1_2"/><span class="lab"><font color="Green" title="E80">A65</font></span>: 
3 is   <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="glib_007.html#E89"><span class="lab"><font color="Green" title="E68">Lm16</font></span></a></span>;<br/><a NAME="E8:110_1_1_2"/><span class="lab"><font color="Green" title="E81">A66</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="c6">C</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="c6">C</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T13" onmouseover="rs('xreal_1/T13')" onmouseout="rh()">XREAL_1:13</a></span>;<br/><a NAME="E9:110_1_1_2"/>
5 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="c6">C</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1_1_2"><span class="lab"><font color="Green" title="E77">A60</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E10:110_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A67</font></span>: 
( 3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font> &amp; <font color="Maroon" title="c8">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="c6">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:110_1_1_2"><span class="lab"><font color="Green" title="E81">A66</font></span></a></span>;<br/><a NAME="E11:110_1_1_2"/><span class="kw">then </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="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (3,<font color="Maroon" title="c8">m</font>)</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3 <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="c6">C</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> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:110_1_1_2"><span class="lab"><font color="Green" title="E80">A65</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="E12:110_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A68</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="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (3,<font color="Maroon" title="c8">m</font>)</span>)</span></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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font>
 ;<br/><a NAME="E13:110_1_1_2"/>
 not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (3,<font color="Maroon" title="c8">m</font>)</span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="110_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:110_1_1_2_3"/>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (3,<font color="Maroon" title="c8">m</font>)</span>)</span> <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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">k</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="E3:110_1_1_2_3"/><span class="lab"><font color="Green" title="E84">A69</font></span>: 
( <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (3,<font color="Maroon" title="c8">m</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (3,<font color="Maroon" title="c8">m</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T87" onmouseover="rs('glib_001/T87')" onmouseout="rh()">GLIB_001:87</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">k1</font> = <font color="Maroon" title="c14">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 as    <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="E5:110_1_1_2_3"/><span class="lab"><font color="Green" title="E85">A70</font></span>: 
<font color="Maroon" title="c15">k1</font> <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="c14">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="glib_007.html#E87"><span class="lab"><font color="Green" title="E66">Lm14</font></span></a></span>;<br/>
<a NAME="E6:110_1_1_2_3"/>
<font color="Maroon" title="c15">k1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (3,<font color="Maroon" title="c8">m</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_2_3"><span class="lab"><font color="Green" title="E84">A69</font></span></a>, <a class="txt" href="glib_007.html#E87"><span class="lab"><font color="Green" title="E66">Lm14</font></span></a></span>;<br/>
<a NAME="E7:110_1_1_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (3,<font color="Maroon" title="c8">m</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k1</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="c6">C</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> <font color="Maroon" title="c15">k1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:110_1_1_2"><span class="lab"><font color="Green" title="E82">A67</font></span></a>, <a class="txt" href="#E7:110_1_1_2"><span class="lab"><font color="Green" title="E80">A65</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="E8:110_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A71</font></span>: 
<font color="Maroon" title="c6">C</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> <font color="Maroon" title="c15">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">C</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="c6">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110_1"><span class="lab"><font color="Green" title="E75">A7</font></span></a>, <a class="txt" href="#E1:110_1_1_2"><span class="lab"><font color="Green" title="E76">A59</font></span></a>, <a class="txt" href="#E3:110_1_1_2_3"><span class="lab"><font color="Green" title="E84">A69</font></span></a>, <a class="txt" href="#E5:110_1_1_2_3"><span class="lab"><font color="Green" title="E85">A70</font></span></a>, <a class="ref" href="glib_001.html#D7" onmouseover="rs('glib_001/D7')" onmouseout="rh()">GLIB_001:def 7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">k3</font> = 3 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">k1</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="glib_007.html#E88"><span class="lab"><font color="Green" title="E67">Lm15</font></span></a>, <a class="txt" href="glib_007.html#E89"><span class="lab"><font color="Green" title="E68">Lm16</font></span></a></span>;<br/>
<a NAME="E10:110_1_1_2_3"/>
<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="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (3,<font color="Maroon" title="c8">m</font>)</span>)</span></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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:110_1_1_2"><span class="lab"><font color="Green" title="E83">A68</font></span></a></span>;<br/>
<a NAME="E11:110_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A72</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="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (3,<font color="Maroon" title="c8">m</font>)</span>)</span></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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T15" onmouseover="rs('xreal_1/T15')" onmouseout="rh()">XREAL_1:15</a></span>;<br/>
<a NAME="E12:110_1_1_2_3"/>
<font color="Maroon" title="c16">k3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
;<br/>
<a NAME="E13:110_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c16">k3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="glib_007.html#E87"><span class="lab"><font color="Green" title="E66">Lm14</font></span></a></span>;<br/>
<a NAME="E14:110_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c16">k3</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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (3,<font color="Maroon" title="c8">m</font>)</span>)</span></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="txt" href="#E3:110_1_1_2_3"><span class="lab"><font color="Green" title="E84">A69</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="E15:110_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c16">k3</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="c6">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:110_1_1_2_3"><span class="lab"><font color="Green" title="E87">A72</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="E16:110_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A73</font></span>: 
<font color="Maroon" title="c16">k3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:110_1_1_2_3"><span class="lab"><font color="Green" title="E86">A71</font></span></a>, <a class="txt" href="#E3:110_1"><span class="lab"><font color="Green" title="E74">A6</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="E17:110_1_1_2_3"/>
3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:110_1_1_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:110_1_1_2_3"><span class="lab"><font color="Green" title="E88">A73</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">P</font> = <font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (3,<font color="Maroon" title="c8">m</font>) as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="ref" href="glib_007.html#T64" target="_self" onmouseover="rs('glib_007/T64')" onmouseout="rh()">Th64</a></span>;<br/><a NAME="E15:110_1_1_2"/>
( <font color="Maroon" title="c6">C</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="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1,<font color="Maroon" title="c6">C</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="c4">G1</font> &amp; <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font>,<font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c4">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_2"><span class="lab"><font color="Green" title="E78">A62</font></span></a>, <a class="txt" href="#E6:110_1_1_2"><span class="lab"><font color="Green" title="E79">A64</font></span></a>, <a class="txt" href="glib_007.html#E89"><span class="lab"><font color="Green" title="E68">Lm16</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="E16:110_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c6">C</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="c6">C</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3,<font color="Maroon" title="c4">G1</font> &amp; <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font>,<font color="Maroon" title="c6">C</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="c6">C</font></span>)</span>,<font color="Maroon" title="c4">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D6" onmouseover="rs('glib_001/D6')" onmouseout="rh()">GLIB_001:def 6</a></span>;<br/><a NAME="E17:110_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c6">C</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="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> &amp; <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font>,<font color="Maroon" title="c6">C</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c4">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_1_1_2"><span class="lab"><font color="Green" title="E76">A59</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</a>, <a class="ref" href="glib_001.html#D7" onmouseover="rs('glib_001/D7')" onmouseout="rh()">GLIB_001:def 7</a></span>;<br/><a NAME="E18:110_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A74</font></span>: 
( <font color="Maroon" title="c6">C</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="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> &amp; <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_1_1_2"><span class="lab"><font color="Green" title="E76">A59</font></span></a>, <a class="txt" href="#E4:110_1"><span class="lab"><font color="Green" title="E75">A7</font></span></a></span>;<br/><a NAME="E19:110_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A75</font></span>: 
( <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="ref" href="glib_007.html#D4" target="_self" onmouseover="rs('glib_007/D4')" onmouseout="rh()">Def4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">w1</font> = <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3, <font color="Maroon" title="c16">w2</font> = <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E69">A1</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c17">G3</font> =  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c15">w1</font></span>)</span>;<br/><a NAME="E21:110_1_1_2"/>
<font color="Maroon" title="c6">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (3,<font color="Maroon" title="c8">m</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c15">w1</font>,<font color="Maroon" title="c16">w2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:110_1_1_2"><span class="lab"><font color="Green" title="E82">A67</font></span></a>, <a class="txt" href="glib_007.html#E89"><span class="lab"><font color="Green" title="E68">Lm16</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="E22:110_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c15">w1</font>,<font color="Maroon" title="c16">w2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T19" onmouseover="rs('glib_001/T19')" onmouseout="rh()">GLIB_001:19</a></span>;<br/><a NAME="E23:110_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c15">w1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D5" onmouseover="rs('glib_002/D5')" onmouseout="rh()">GLIB_002:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">w2</font> = <font color="Maroon" title="c16">w2</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">G2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c15">w1</font></span>)</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="E25:110_1_1_2"/>
<font color="Maroon" title="c15">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c15">w1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#T9" onmouseover="rs('glib_002/T9')" onmouseout="rh()">GLIB_002:9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">w1</font> = <font color="Maroon" title="c15">w1</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">G2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c15">w1</font></span>)</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="E27:110_1_1_2"/><span class="lab"><font color="Green" title="E86">A76</font></span>: 
<font color="Maroon" title="c19">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">w2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="txt" href="#E19:110_1_1_2"><span class="lab"><font color="Green" title="E85">A75</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c20">e3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E29:110_1_1_2"/>
( <font color="Maroon" title="c20">e3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> &amp; <font color="Maroon" title="c20">e3</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c19">w1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> )
 <span class="kw">and </span><br/><a NAME="E30:110_1_1_2"/><span class="lab"><font color="Green" title="E87">A77</font></span>: 
 for <font color="Olive" title="b1">e5</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e5</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c19">w1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Maroon" title="c20">e3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e5</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:110"><span class="lab"><font color="Green" title="E72">A4</font></span></a>, <a class="txt" href="#E19:110_1_1_2"><span class="lab"><font color="Green" title="E85">A75</font></span></a></span>;<br/><a NAME="E31:110_1_1_2"/><span class="lab"><font color="Green" title="E88">A78</font></span>: 
( <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">e3</font> &amp; <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">e3</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:110_1_1_2"><span class="lab"><font color="Green" title="E84">A74</font></span></a>, <a class="txt" href="#E27:110_1_1_2"><span class="lab"><font color="Green" title="E86">A76</font></span></a>, <a class="txt" href="#E30:110_1_1_2"><span class="lab"><font color="Green" title="E87">A77</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">n2</font> = 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="txt" href="glib_007.html#E89"><span class="lab"><font color="Green" title="E68">Lm16</font></span></a></span>;<br/><a NAME="E33:110_1_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="c6">C</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">C</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T13" onmouseover="rs('xreal_1/T13')" onmouseout="rh()">XREAL_1:13</a></span>;<br/><a NAME="E34:110_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A79</font></span>: 
<font color="Maroon" title="c8">m</font> <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="c6">C</font>
 ;<br/><a NAME="E35:110_1_1_2"/>
( 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 3 &amp; 3 <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="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:110_1_1_2"><span class="lab"><font color="Green" title="E82">A67</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="E36:110_1_1_2"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">n2</font> &amp; <font color="Maroon" title="c21">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c8">m</font> <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="c6">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:110_1_1_2"><span class="lab"><font color="Green" title="E89">A79</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="E37:110_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">n2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1"><span class="lab"><font color="Green" title="E73">A5</font></span></a>, <a class="ref" href="glib_001.html#T138" onmouseover="rs('glib_001/T138')" onmouseout="rh()">GLIB_001:138</a></span>;<br/><span class="kw">hence </span><a NAME="E38:110_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:110_1_1_2"><span class="lab"><font color="Green" title="E88">A78</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>
<span class="kw">hence </span><a NAME="E8:110"/>
<font color="Maroon" title="c4">G1</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D2" onmouseover="rs('glib_002/D2')" onmouseout="rh()">GLIB_002:def 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>
