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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P1</font>, <font color="Olive" title="b2">P2</font> being   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">P2</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  &amp;  not <font color="Olive" title="b1">P1</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  &amp;  not <font color="Olive" title="b2">P2</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  &amp; <font color="Olive" title="b1">P1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">P2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  &amp; ( <font color="Olive" title="b1">P1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">P2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  implies <font color="Olive" title="b1">P1</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="Olive" title="b2">P2</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  ) &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">P1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">P2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">P1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> holds <br/><font color="Olive" title="b1">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Olive" title="b2">P2</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">P1</font>, <font color="Maroon" title="c3">P2</font> be   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  &amp;  not <font color="Maroon" title="c2">P1</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  &amp;  not <font color="Maroon" title="c3">P2</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  &amp; <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  &amp; ( <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  implies <font color="Maroon" title="c2">P1</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="c3">P2</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  ) &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> implies <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:20"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 <span class="kw">and </span><br/><a NAME="E2:20"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
 not <font color="Maroon" title="c2">P1</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> 
 <span class="kw">and </span><br/><a NAME="E3:20"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
 not <font color="Maroon" title="c3">P2</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> 
 <span class="kw">and </span><br/><a NAME="E4:20"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <span class="kw">and </span><br/><a NAME="E5:20"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
( <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  implies <font color="Maroon" title="c2">P1</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="c3">P2</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  )
 <span class="kw">and </span><br/><a NAME="E6:20"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> </span><br/>

<span class="kw">thus </span><a NAME="E7:20"/>
<font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font> is  <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="helly.html#T17" target="_self" title="HELLY:th.17">Th17</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>2</sub></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="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive">b<sub>2</sub></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="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font></span>)</span> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">P</font> = <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">m</font>, <font color="Maroon" title="c6">n</font> be   <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#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font> or  not <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> or ( <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E8:20"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">and </span><br/><a NAME="E9:20"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:20"/><span class="lab"><font color="Green" title="E24">A9</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">and </span><br/><a NAME="E11:20"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
( <font color="Maroon" title="c5">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or <font color="Maroon" title="c6">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E12:20"/><span class="lab"><font color="Green" title="E26">A11</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/>
<a NAME="E13:20"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/>
<a NAME="E14:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A12</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E15:20"/>
<font color="Maroon" title="c5">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="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E16:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A13</font></span>: 
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_1"/>
(  ex <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">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> <font color="Maroon" title="c3">P2</font> &amp; <font color="Maroon" title="c6">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="c2">P1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</font> ) or <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">P1</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:20"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">GLIB_001:34</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_1_1"/>
 ex <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">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> <font color="Maroon" title="c3">P2</font> &amp; <font color="Maroon" title="c6">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="c2">P1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</font> )
 ; <a class="txt" 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="c7">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:20_1_1"/><span class="lab"><font color="Green" title="E29">A14</font></span>: 
<font color="Maroon" title="c7">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> <font color="Maroon" title="c3">P2</font>
 <span class="kw">and </span><br/><a NAME="E4:20_1_1"/><span class="lab"><font color="Green" title="E30">A15</font></span>: 
<font color="Maroon" title="c6">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="c2">P1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font>
 ;<br/><a NAME="E5:20_1_1"/><span class="lab"><font color="Green" title="E31">A16</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E3:20_1_1"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="txt" href="#E4:20_1_1"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="glib_001.html#T33" title="GLIB_001:th.33">GLIB_001:33</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">k</font> = <font color="Maroon" title="c7">k</font> as   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_1_1"><span class="lab"><font color="Green" title="E30">A15</font></span></a></span>;<br/><a NAME="E7:20_1_1"/><span class="lab"><font color="Green" title="E32">A17</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_1_1"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E8:20_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A18</font></span>: 
<font color="Maroon" title="c3">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T87" title="GLIB_001:th.87">GLIB_001:87</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_1_1_1"/>
(  ex <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">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> <font color="Maroon" title="c3">P2</font> &amp; <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</font> ) or <font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">P1</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:20"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">GLIB_001:34</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_1_1_1_1"/>
 ex <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">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> <font color="Maroon" title="c3">P2</font> &amp; <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</font> )
 ; <a class="txt" 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="c9">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:20_1_1_1_1"/><span class="lab"><font color="Green" title="E34">A19</font></span>: 
<font color="Maroon" title="c9">l</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="c3">P2</font>
 <span class="kw">and </span><br/><a NAME="E4:20_1_1_1_1"/><span class="lab"><font color="Green" title="E35">A20</font></span>: 
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">l</font>
 ;<br/><a NAME="E5:20_1_1_1_1"/><span class="lab"><font color="Green" title="E36">A21</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E3:20_1_1_1_1"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="txt" href="#E4:20_1_1_1_1"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="ref" href="glib_001.html#T33" title="GLIB_001:th.33">GLIB_001:33</a></span>;<br/><a NAME="E6:20_1_1_1_1"/>
<font color="Maroon" title="c9">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E4:20_1_1"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E4:20_1_1_1_1"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E7:20_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A22</font></span>: 
<font color="Maroon" title="c9">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">l</font> = <font color="Maroon" title="c9">l</font> as   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_1_1_1_1"><span class="lab"><font color="Green" title="E35">A20</font></span></a></span>;<br/><a NAME="E9:20_1_1_1_1"/>
 not <font color="Maroon" title="c10">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 ;<br/><a NAME="E10:20_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A23</font></span>: 
<font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E5:20_1_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E7:20_1_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E5:20_1_1_1_1"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="txt" href="#E7:20_1_1_1_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><a NAME="E11:20_1_1_1_1"/>
<font color="Maroon" title="c3">P2</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="c3">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E5:20_1_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E7:20_1_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E5:20_1_1_1_1"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="txt" href="#E7:20_1_1_1_1"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><span class="kw">hence </span><a NAME="E12:20_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E5:20_1_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E5:20_1_1_1_1"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="txt" href="#E10:20_1_1_1_1"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></span>; <a class="txt" 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="20_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_1_1_1_2"/><span class="lab"><font color="Green" title="E34">A24</font></span>: 
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">P1</font>
 ; <a class="txt" 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">set </span><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c2">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font>;<br/><a NAME="E2:20_1_1_1_2"/><span class="lab"><font color="Green" title="E35">A25</font></span>: 
<font color="Maroon" title="c2">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_1_2"><span class="lab"><font color="Green" title="E34">A24</font></span></a>, <a class="ref" href="glib_001.html#T32" title="GLIB_001:th.32">GLIB_001:32</a></span>;<br/><a NAME="E3:20_1_1_1_2"/><span class="lab"><font color="Green" title="E36">A26</font></span>: 
<font color="Maroon" title="c5">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="c2">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_1_2"><span class="lab"><font color="Green" title="E34">A24</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E4:20_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T87" title="GLIB_001:th.87">GLIB_001:87</a></span>;<br/><a NAME="E5:20_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A27</font></span>: 
<font color="Maroon" title="c2">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E5:20_1_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E8:20_1_1"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E2:20_1_1_1_2"><span class="lab"><font color="Green" title="E35">A25</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_1_1_1_2_1"/>
( <font color="Maroon" title="c2">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  or <font color="Maroon" title="c2">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E5:20_1_1_1_2"><span class="lab"><font color="Green" title="E37">A27</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E38">A28</font></span>: 
<font color="Maroon" title="c2">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E39">A29</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_2_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:20_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c5">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="c2">P1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:20_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">P1</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">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E38">A28</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><span class="kw">hence </span><a NAME="E3:20_1_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E1:20_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E38">A28</font></span></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E40">A30</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_2_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:20_1_1_1_2_1_1_2"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:20_1_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">P2</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="c3">P2</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="#E1:20"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E5:20_1_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E7:20_1_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E2:20_1_1_1_2"><span class="lab"><font color="Green" title="E35">A25</font></span></a>, <a class="txt" href="#E1:20_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E38">A28</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><span class="kw">hence </span><a NAME="E3:20_1_1_1_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:20_1_1_1_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E5:20_1_1_1_2_1_1"/><span class="kw">then </span>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E40">A30</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:20_1_1_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E4:20_1_1"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E2:20_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E39">A29</font></span></a></span>; <a class="txt" 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="20_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E38">A31</font></span>: 
<font color="Maroon" title="c2">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:20_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A32</font></span>: 
<font color="Maroon" title="c2">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P1</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">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><div><span class="lab"><font color="Green" title="E40">A33</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_2_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:20_1_1_1_2_1_2_1"/>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:20_1_1_1_2_1_2_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:20_1_1_1_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P1</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="#E3:20_1_1_1_2"><span class="lab"><font color="Green" title="E36">A26</font></span></a>, <a class="txt" href="#E2:20_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E39">A32</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><span class="kw">hence </span><a NAME="E4:20_1_1_1_2_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E1:20_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E38">A31</font></span></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></span>; <a class="txt" 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="hs2(this)" href="javascript:()" title="20_1_1_1_2_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:20_1_1_1_2_1_2_2"/>
<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c3">P2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:20_1_1_1_2_1_2_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="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="glib_001.html#T28" title="GLIB_001:th.28">GLIB_001:28</a></span>;<br/><span class="kw">hence </span><a NAME="E3:20_1_1_1_2_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:20"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E4:20_1_1"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E3:20_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E40">A33</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E5:20_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A34</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20_1_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E6:20_1_1_1_2_1_2"/>
<font color="Maroon" title="c3">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P2</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="#E5:20"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E5:20_1_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E8:20_1_1"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E1:20_1_1_1_2"><span class="lab"><font color="Green" title="E34">A24</font></span></a>, <a class="txt" href="#E1:20_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E38">A31</font></span></a>, <a class="ref" href="glib_001.html#T32" title="GLIB_001:th.32">GLIB_001:32</a></span>;<br/><a NAME="E7:20_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">P2</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="c3">P2</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="#E5:20_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E41">A34</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><span class="kw">hence </span><a NAME="E8:20_1_1_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></span>; <a class="txt" 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="20_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_1_2"/><span class="lab"><font color="Green" title="E29">A35</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">P1</font>
 ; <a class="txt" 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="E2:20_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A36</font></span>: 
<font color="Maroon" title="c6">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="c2">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E3:20_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="c5">m</font> &amp; <font color="Maroon" title="c5">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="c2">P1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:20_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A37</font></span>: <font color="Maroon" title="c2">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T32" title="GLIB_001:th.32">GLIB_001:32</a></span>
<br/>.= 
<font color="Maroon" title="c2">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E1:20_1_2"><span class="lab"><font color="Green" title="E29">A35</font></span></a>, <a class="ref" href="glib_001.html#T32" title="GLIB_001:th.32">GLIB_001:32</a></span>
;<br/><a NAME="E6:20_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E2:20_1_2"><span class="lab"><font color="Green" title="E30">A36</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><a NAME="E7:20_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">P1</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">P1</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="#E8:20"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E2:20_1_2"><span class="lab"><font color="Green" title="E30">A36</font></span></a>, <a class="txt" href="#E5:20_1_2"><span class="lab"><font color="Green" title="E31">A37</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><span class="kw">hence </span><a NAME="E8:20_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></span>; <a class="txt" 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>

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