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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">P</font> being   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>  st  ex <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">n</font> &amp; <font color="Olive" title="b3">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="Olive" title="b1">P</font> &amp;  ex <font color="Olive" title="b4">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font>,<font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font>,<font color="Maroon" title="c1">G</font> &amp; ( <font color="Olive" title="b1">P</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>  implies ( (  not <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Olive" title="b3">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="Olive" title="b1">P</font> ) &amp; (  not <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Olive" title="b3">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="Olive" title="b1">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 ) &amp; (  not <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or  not <font color="Olive" title="b3">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="Olive" title="b1">P</font> ) ) ) ) holds <br/><font color="Olive" title="b1">P</font> is  <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">P</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"> (  ex <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b2">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">P</font> &amp;  ex <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font>,<font color="Maroon" title="c1">G</font> &amp; ( <font color="Maroon" title="c2">P</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>  implies ( (  not <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Olive" title="b2">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="c2">P</font> ) &amp; (  not <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Olive" title="b2">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">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 ) &amp; (  not <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or  not <font color="Olive" title="b2">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="c2">P</font> ) ) ) ) implies <font color="Maroon" title="c2">P</font> is  <a href="chord.html#V5" title="CHORD:attr.5">chordal</a>  )</span><br/>



<span class="kw">given </span><font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c4">n</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><a NAME="E2:100"/><span class="lab"><font color="Green" title="E82">A1</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">and </span><br/><a NAME="E3:100"/><span class="lab"><font color="Green" title="E83">A2</font></span>: 
<font color="Maroon" title="c4">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">P</font>
 <span class="kw">and </span><br/><a NAME="E4:100"/><span class="lab"><font color="Green" title="E84">A3</font></span>: 
 ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">and </span><br/><a NAME="E5:100"/><span class="lab"><font color="Green" title="E85">A4</font></span>: 
( <font color="Maroon" title="c2">P</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>  implies ( (  not <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Maroon" title="c4">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="c2">P</font> ) &amp; (  not <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Maroon" title="c4">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">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 ) &amp; (  not <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or  not <font color="Maroon" title="c4">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="c2">P</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">P</font> is  <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> </span><br/>

<a NAME="E6:100"/><span class="lab"><font color="Green" title="E86">A5</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">m</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="chord.html#D10" target="_self" title="CHORD:def.10">CHORD:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">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">P</font> &amp; <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp;  ex <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b2">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  holds <br/> not <font color="Olive" title="b2">f</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c1">G</font> ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c4">n</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="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c4">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">P</font> &amp; <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> &amp;  ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b1">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  holds <br/> not <font color="Olive" title="b1">f</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c1">G</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E7:100"/>
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:100"><span class="lab"><font color="Green" title="E82">A1</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"> ( <font color="Maroon" title="c4">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">P</font> &amp; <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> &amp;  ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b1">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  holds <br/> not <font color="Olive" title="b1">f</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c1">G</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E8:100"/>
<font color="Maroon" title="c4">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">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:100"><span class="lab"><font color="Green" title="E83">A2</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"> ( <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> &amp;  ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b1">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  holds <br/> not <font color="Olive" title="b1">f</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c1">G</font> ) )</span><br/>

<a NAME="E9:100"/><span class="lab"><font color="Green" title="E87">A6</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E10:100"/><span class="lab"><font color="Green" title="E88">A7</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:100"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E11:100"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A8</font></span>: 
<font color="Maroon" title="c3">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">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:100"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:100_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="E2:100_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:100"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E3:100"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:100_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</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="E13:100"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A9</font></span>: 
 not <font color="Maroon" title="c2">P</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T126" title="GLIB_001:th.126">GLIB_001:126</a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b1">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  holds <br/> not <font color="Olive" title="b1">f</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c1">G</font> ) )</span>
<div class="add"><a NAME="E1:100_2"/><span class="lab"><font color="Green" title="E91">A10</font></span>: 
<font color="Maroon" title="c2">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span>
 ;<br/><a NAME="E2:100_2"/><span class="lab"><font color="Green" title="E92">A11</font></span>: 
<font color="Maroon" title="c2">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 ;<br/><span class="kw">assume </span><a NAME="E3:100_2"/><span class="lab"><font color="Green" title="E93">A12</font></span>: 
<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</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="E4:100_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A13</font></span>: 
<font color="Maroon" title="c4">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="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:100"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E9:100"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="txt" href="#E6:100"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="txt" href="#E10:100"><span class="lab"><font color="Green" title="E88">A7</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="E5:100_2"/>
<font color="Maroon" title="c3">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="#E3:100"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E9:100"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="txt" href="#E6:100"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="txt" href="#E10:100"><span class="lab"><font color="Green" title="E88">A7</font></span></a>, <a class="txt" href="#E3:100_2"><span class="lab"><font color="Green" title="E93">A12</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="E6:100_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">P</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:100_2"><span class="lab"><font color="Green" title="E93">A12</font></span></a>, <a class="txt" href="#E4:100_2"><span class="lab"><font color="Green" title="E94">A13</font></span></a>, <a class="txt" href="#E2:100_2"><span class="lab"><font color="Green" title="E92">A11</font></span></a>, <a class="txt" href="#E1:100_2"><span class="lab"><font color="Green" title="E91">A10</font></span></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></span>;<br/><span class="kw">hence </span><a NAME="E7:100_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:100"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E5:100"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E9:100"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="txt" href="#E6:100"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="txt" href="#E10:100"><span class="lab"><font color="Green" title="E88">A7</font></span></a>, <a class="txt" href="#E13:100"><span class="lab"><font color="Green" title="E90">A9</font></span></a>, <a class="txt" href="#E3:100_2"><span class="lab"><font color="Green" title="E93">A12</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a>, <a class="ref" href="glib_001.html#D31" title="GLIB_001:def.31">GLIB_001:def 31</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>

<span class="kw">thus </span><a NAME="E15:100"/>
 ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:100"><span class="lab"><font color="Green" title="E84">A3</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">  for <font color="Olive" title="b1">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  holds <br/> not <font color="Olive" title="b1">f</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c1">G</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  implies  not <font color="Maroon" title="c5">f</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c1">G</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E16:100"/><span class="lab"><font color="Green" title="E91">A14</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <span class="kw">and </span><br/><a NAME="E17:100"/><span class="lab"><font color="Green" title="E92">A15</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<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"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E19:100"/><span class="lab"><font color="Green" title="E93">A16</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E20:100"/><span class="lab"><font color="Green" title="E94">A17</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:100"><span class="lab"><font color="Green" title="E91">A14</font></span></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">i</font> = <font color="Maroon" title="c6">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">k</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1;<br/>
<a NAME="E22:100"/><span class="lab"><font color="Green" title="E95">A18</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:100"><span class="lab"><font color="Green" title="E93">A16</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E23:100"/><span class="kw">then </span>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/>
<a NAME="E24:100"/><span class="kw">then </span>
2 <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">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">i</font></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" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E25:100"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A19</font></span>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/>
<a NAME="E26:100"/>
<font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:100"><span class="lab"><font color="Green" title="E93">A16</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E27:100"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A20</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:100"><span class="lab"><font color="Green" title="E95">A18</font></span></a>, <a class="ref" href="glib_001.html#D15" title="GLIB_001:def.15">GLIB_001:def 15</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">k</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:100"><span class="lab"><font color="Green" title="E96">A19</font></span></a></span>;<br/>
<a NAME="E29:100"/>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:100"><span class="lab"><font color="Green" title="E93">A16</font></span></a>, <a class="ref" href="glib_001.html#T78" title="GLIB_001:th.78">GLIB_001:78</a></span>;<br/>
<a NAME="E30:100"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A21</font></span>: 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</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="E31:100"/>
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">i</font>
 
;<br/>
<a NAME="E32:100"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A22</font></span>: 
<font color="Maroon" title="c9">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="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:100"><span class="lab"><font color="Green" title="E98">A21</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E33:100"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A23</font></span>: 
<font color="Maroon" title="c9">k</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="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></span>;<br/>
<a NAME="E34:100"/><span class="lab"><font color="Green" title="E101">A24</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E35:100"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A25</font></span>: 
<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</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="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:100"><span class="lab"><font color="Green" title="E99">A22</font></span></a>, <a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_3"><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="E1:100_3"/>
( ( <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</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> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> ) or ( <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</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> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:100"><span class="lab"><font color="Green" title="E92">A15</font></span></a>, <a class="txt" href="#E20:100"><span class="lab"><font color="Green" title="E94">A17</font></span></a>, <a class="txt" href="#E27:100"><span class="lab"><font color="Green" title="E97">A20</font></span></a>, <a class="txt" href="#E35:100"><span class="lab"><font color="Green" title="E102">A25</font></span></a>, <a class="ref" href="glib_000.html#T15" title="GLIB_000:th.15">GLIB_000:15</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_3_1"><span class="kw">suppose </span></a><a NAME="E1:100_3_1"/><span class="lab"><font color="Green" title="E103">A26</font></span>: 
( <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</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> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</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"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_3_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="E1:100_3_1_1"/>
( <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> or <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> or <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">m</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:100_3_1_1_1"/>
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</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">hence </span><a NAME="E2:100_3_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:100"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="txt" href="#E11:100"><span class="lab"><font color="Green" title="E89">A8</font></span></a>, <a class="txt" href="#E34:100"><span class="lab"><font color="Green" title="E101">A24</font></span></a>, <a class="txt" href="#E1:100_3_1"><span class="lab"><font color="Green" title="E103">A26</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</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="100_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:100_3_1_1_2"/><span class="lab"><font color="Green" title="E104">A27</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</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"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_3_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="E1:100_3_1_1_2_1"/>
( <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font> or <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> or <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">n</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_3_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:100_3_1_1_2_1_1"/><span class="lab"><font color="Green" title="E105">A28</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</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="E2:100_3_1_1_2_1_1"/><span class="lab"><font color="Green" title="E106">A29</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><a NAME="E3:100_3_1_1_2_1_1"/>
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:100"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E6:100"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="txt" href="#E1:100_3_1"><span class="lab"><font color="Green" title="E103">A26</font></span></a>, <a class="txt" href="#E1:100_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E105">A28</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><a NAME="E4:100_3_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:100_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E106">A29</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E5:100_3_1_1_2_1_1"/>
contradiction
 <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>; <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="100_3_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:100_3_1_1_2_1_2"/>
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</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">hence </span><a NAME="E2:100_3_1_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:100"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E1:100_3_1_1_2"><span class="lab"><font color="Green" title="E104">A27</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="100_3_1_1_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:100_3_1_1_2_1_3"/><span class="lab"><font color="Green" title="E105">A30</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">n</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="E2:100_3_1_1_2_1_3"/>
<font color="Maroon" title="c9">k</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="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:100"><span class="lab"><font color="Green" title="E99">A22</font></span></a>, <a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></span>;<br/><a NAME="E3:100_3_1_1_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">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="#E6:100"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="txt" href="#E1:100_3_1"><span class="lab"><font color="Green" title="E103">A26</font></span></a>, <a class="txt" href="#E1:100_3_1_1_2_1_3"><span class="lab"><font color="Green" title="E105">A30</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="E4:100_3_1_1_2_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:100"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="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="100_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:100_3_1_1_3"/>
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">m</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">hence </span><a NAME="E2:100_3_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:100"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="txt" href="#E34:100"><span class="lab"><font color="Green" title="E101">A24</font></span></a>, <a class="txt" href="#E32:100"><span class="lab"><font color="Green" title="E99">A22</font></span></a>, <a class="txt" href="#E1:100_3_1"><span class="lab"><font color="Green" title="E103">A26</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_3_2"><span class="kw">suppose </span></a><a NAME="E1:100_3_2"/><span class="lab"><font color="Green" title="E103">A31</font></span>: 
( <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</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> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</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"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_3_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="E1:100_3_2_1"/>
( <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font> or <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> or <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">n</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_3_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:100_3_2_1_1"/><span class="lab"><font color="Green" title="E104">A32</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</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="E2:100_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A33</font></span>: 
<font color="Maroon" title="c4">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="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:100"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E6:100"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="txt" href="#E34:100"><span class="lab"><font color="Green" title="E101">A24</font></span></a>, <a class="txt" href="#E1:100_3_2"><span class="lab"><font color="Green" title="E103">A31</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><a NAME="E3:100_3_2_1_1"/><span class="lab"><font color="Green" title="E106">A34</font></span>: 
<font color="Maroon" title="c9">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="#E3:100"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E6:100"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="txt" href="#E34:100"><span class="lab"><font color="Green" title="E101">A24</font></span></a>, <a class="txt" href="#E1:100_3_2"><span class="lab"><font color="Green" title="E103">A31</font></span></a>, <a class="txt" href="#E1:100_3_2_1_1"><span class="lab"><font color="Green" title="E104">A32</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_3_2_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="E1:100_3_2_1_1_1"/>
( <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> or <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> or <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">m</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_3_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:100_3_2_1_1_1_1"/>
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</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">hence </span><a NAME="E2:100_3_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:100"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="txt" href="#E11:100"><span class="lab"><font color="Green" title="E89">A8</font></span></a>, <a class="txt" href="#E1:100_3_2"><span class="lab"><font color="Green" title="E103">A31</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</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="100_3_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:100_3_2_1_1_1_2"/><span class="lab"><font color="Green" title="E107">A35</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</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="E2:100_3_2_1_1_1_2"/><span class="lab"><font color="Green" title="E108">A36</font></span>: 
<font color="Maroon" title="c2">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span>
 ;<br/><a NAME="E3:100_3_2_1_1_1_2"/>
<font color="Maroon" title="c2">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 ;<br/><a NAME="E4:100_3_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">P</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100_3_2"><span class="lab"><font color="Green" title="E103">A31</font></span></a>, <a class="txt" href="#E3:100_3_2_1_1"><span class="lab"><font color="Green" title="E106">A34</font></span></a>, <a class="txt" href="#E2:100_3_2_1_1"><span class="lab"><font color="Green" title="E105">A33</font></span></a>, <a class="txt" href="#E2:100_3_2_1_1_1_2"><span class="lab"><font color="Green" title="E108">A36</font></span></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></span>;<br/><span class="kw">hence </span><a NAME="E5:100_3_2_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:100"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E5:100"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E6:100"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="txt" href="#E13:100"><span class="lab"><font color="Green" title="E90">A9</font></span></a>, <a class="txt" href="#E1:100_3_2"><span class="lab"><font color="Green" title="E103">A31</font></span></a>, <a class="txt" href="#E1:100_3_2_1_1"><span class="lab"><font color="Green" title="E104">A32</font></span></a>, <a class="txt" href="#E3:100_3_2_1_1"><span class="lab"><font color="Green" title="E106">A34</font></span></a>, <a class="txt" href="#E1:100_3_2_1_1_1_2"><span class="lab"><font color="Green" title="E107">A35</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a>, <a class="ref" href="glib_001.html#D31" title="GLIB_001:def.31">GLIB_001:def 31</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="100_3_2_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:100_3_2_1_1_1_3"/><span class="lab"><font color="Green" title="E107">A37</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">m</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="E2:100_3_2_1_1_1_3"/><span class="lab"><font color="Green" title="E108">A38</font></span>: 
<font color="Maroon" title="c2">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span>
 ;<br/><a NAME="E3:100_3_2_1_1_1_3"/><span class="lab"><font color="Green" title="E109">A39</font></span>: 
<font color="Maroon" title="c9">k</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> <font color="Maroon" title="c4">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100_3_2_1_1"><span class="lab"><font color="Green" title="E104">A32</font></span></a>, <a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></span>;<br/><a NAME="E4:100_3_2_1_1_1_3"/><span class="lab"><font color="Green" title="E110">A40</font></span>: 
<font color="Maroon" title="c2">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 ;<br/><a NAME="E5:100_3_2_1_1_1_3"/><span class="lab"><font color="Green" title="E111">A41</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <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="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:100"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="txt" href="#E33:100"><span class="lab"><font color="Green" title="E100">A23</font></span></a>, <a class="txt" href="#E1:100_3_2"><span class="lab"><font color="Green" title="E103">A31</font></span></a>, <a class="txt" href="#E1:100_3_2_1_1_1_3"><span class="lab"><font color="Green" title="E107">A37</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="E6:100_3_2_1_1_1_3"/>
<font color="Maroon" title="c3">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="#E9:100"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="txt" href="#E33:100"><span class="lab"><font color="Green" title="E100">A23</font></span></a>, <a class="txt" href="#E1:100_3_2"><span class="lab"><font color="Green" title="E103">A31</font></span></a>, <a class="txt" href="#E1:100_3_2_1_1_1_3"><span class="lab"><font color="Green" title="E107">A37</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="E7:100_3_2_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">P</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100_3_2"><span class="lab"><font color="Green" title="E103">A31</font></span></a>, <a class="txt" href="#E5:100_3_2_1_1_1_3"><span class="lab"><font color="Green" title="E111">A41</font></span></a>, <a class="txt" href="#E4:100_3_2_1_1_1_3"><span class="lab"><font color="Green" title="E110">A40</font></span></a>, <a class="txt" href="#E2:100_3_2_1_1_1_3"><span class="lab"><font color="Green" title="E108">A38</font></span></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></span>;<br/><span class="kw">hence </span><a NAME="E8:100_3_2_1_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:100"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E5:100"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E9:100"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="txt" href="#E13:100"><span class="lab"><font color="Green" title="E90">A9</font></span></a>, <a class="txt" href="#E1:100_3_2"><span class="lab"><font color="Green" title="E103">A31</font></span></a>, <a class="txt" href="#E1:100_3_2_1_1_1_3"><span class="lab"><font color="Green" title="E107">A37</font></span></a>, <a class="txt" href="#E5:100_3_2_1_1_1_3"><span class="lab"><font color="Green" title="E111">A41</font></span></a>, <a class="txt" href="#E3:100_3_2_1_1_1_3"><span class="lab"><font color="Green" title="E109">A39</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a>, <a class="ref" href="glib_001.html#D31" title="GLIB_001:def.31">GLIB_001:def 31</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="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="100_3_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:100_3_2_1_2"/><span class="lab"><font color="Green" title="E104">A42</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</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"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_3_2_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:100_3_2_1_2_1"/>
( <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> or <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> or <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">m</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_3_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:100_3_2_1_2_1_1"/>
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</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">hence </span><a NAME="E2:100_3_2_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:100"><span class="lab"><font color="Green" title="E88">A7</font></span></a>, <a class="txt" href="#E1:100_3_2_1_2"><span class="lab"><font color="Green" title="E104">A42</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</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="100_3_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:100_3_2_1_2_1_2"/>
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</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">hence </span><a NAME="E2:100_3_2_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:100"><span class="lab"><font color="Green" title="E88">A7</font></span></a>, <a class="txt" href="#E1:100_3_2_1_2"><span class="lab"><font color="Green" title="E104">A42</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</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="100_3_2_1_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:100_3_2_1_2_1_3"/><span class="lab"><font color="Green" title="E105">A43</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">m</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="E2:100_3_2_1_2_1_3"/><span class="lab"><font color="Green" title="E106">A44</font></span>: 
<font color="Maroon" title="c2">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span>
 ;<br/><a NAME="E3:100_3_2_1_2_1_3"/><span class="lab"><font color="Green" title="E107">A45</font></span>: 
<font color="Maroon" title="c2">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 ;<br/><a NAME="E4:100_3_2_1_2_1_3"/><span class="lab"><font color="Green" title="E108">A46</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <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="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:100"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="txt" href="#E33:100"><span class="lab"><font color="Green" title="E100">A23</font></span></a>, <a class="txt" href="#E1:100_3_2"><span class="lab"><font color="Green" title="E103">A31</font></span></a>, <a class="txt" href="#E1:100_3_2_1_2_1_3"><span class="lab"><font color="Green" title="E105">A43</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="E5:100_3_2_1_2_1_3"/>
<font color="Maroon" title="c3">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="#E9:100"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="txt" href="#E33:100"><span class="lab"><font color="Green" title="E100">A23</font></span></a>, <a class="txt" href="#E1:100_3_2"><span class="lab"><font color="Green" title="E103">A31</font></span></a>, <a class="txt" href="#E1:100_3_2_1_2_1_3"><span class="lab"><font color="Green" title="E105">A43</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="E6:100_3_2_1_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">P</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100_3_2"><span class="lab"><font color="Green" title="E103">A31</font></span></a>, <a class="txt" href="#E4:100_3_2_1_2_1_3"><span class="lab"><font color="Green" title="E108">A46</font></span></a>, <a class="txt" href="#E3:100_3_2_1_2_1_3"><span class="lab"><font color="Green" title="E107">A45</font></span></a>, <a class="txt" href="#E2:100_3_2_1_2_1_3"><span class="lab"><font color="Green" title="E106">A44</font></span></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></span>;<br/><span class="kw">hence </span><a NAME="E7:100_3_2_1_2_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:100"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E9:100"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="txt" href="#E13:100"><span class="lab"><font color="Green" title="E90">A9</font></span></a>, <a class="txt" href="#E1:100_3_2"><span class="lab"><font color="Green" title="E103">A31</font></span></a>, <a class="txt" href="#E1:100_3_2_1_2"><span class="lab"><font color="Green" title="E104">A42</font></span></a>, <a class="txt" href="#E1:100_3_2_1_2_1_3"><span class="lab"><font color="Green" title="E105">A43</font></span></a>, <a class="txt" href="#E4:100_3_2_1_2_1_3"><span class="lab"><font color="Green" title="E108">A46</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a>, <a class="ref" href="glib_001.html#D31" title="GLIB_001:def.31">GLIB_001:def 31</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_3_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:100_3_2_1_3"/>
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">n</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">hence </span><a NAME="E2:100_3_2_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:100"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="txt" href="#E34:100"><span class="lab"><font color="Green" title="E101">A24</font></span></a>, <a class="txt" href="#E32:100"><span class="lab"><font color="Green" title="E99">A22</font></span></a>, <a class="txt" href="#E1:100_3_2"><span class="lab"><font color="Green" title="E103">A31</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</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>
