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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>  <a href="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">W</font> being   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font> holds   <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">W</font> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span><br/><span class="kw">let </span><font color="Maroon" title="c2">W</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">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span><br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="228_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="228_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="E1:228_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="228_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:228_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</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">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span><br/><div class="add"><a NAME="E2:228_1_1_1"/><span class="kw">then </span>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</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> <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">W</font> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D14" target="_self" title="GLIB_001:def.14">Def14</a></span>;<br/><span class="kw">hence </span><a NAME="E3:228_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="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="228_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:228_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</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">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span><br/><div class="add"><a NAME="E2:228_1_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c2">W</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_001.html#E92"><span class="lab"><font color="Green" title="E84">Lm55</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">lenW2</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:228_1_1_2"/><span class="lab"><font color="Green" title="E144">A1</font></span>: 
<font color="Maroon" title="c3">lenW2</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">W</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <span class="kw">and </span><br/><a NAME="E5:228_1_1_2"/><span class="lab"><font color="Green" title="E145">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">lenW2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T133" target="_self" title="GLIB_001:th.133">Th134</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c4">W2</font> = <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>);<br/><span class="kw">set </span><font color="Maroon" title="c5">vs1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a> ;<br/><span class="kw">consider </span><font color="Maroon" title="c6">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E7:228_1_1_2"/><span class="lab"><font color="Green" title="E146">A3</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a>  &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a>  holds <br/><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></span><a NAME="E8:228_1_1_2"/><span class="lab"><font color="Green" title="E147">A4</font></span>: 
<font color="Maroon" title="c3">lenW2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:228_1_1_2"><span class="lab"><font color="Green" title="E144">A1</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/><a NAME="E9:228_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E148">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">lenW2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_001.html#E59"><span class="lab"><font color="Green" title="E51">Lm22</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="228_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x1</font>, <font color="Maroon" title="c8">x2</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="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x2</font> implies <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x2</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:228_1_1_2_1"/><span class="lab"><font color="Green" title="E149">A6</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font>
 <span class="kw">and </span><br/><a NAME="E2:228_1_1_2_1"/><span class="lab"><font color="Green" title="E150">A7</font></span>: 
<font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font>
 <span class="kw">and </span><br/><a NAME="E3:228_1_1_2_1"/><span class="lab"><font color="Green" title="E151">A8</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x2</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="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x2</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c9">a1</font>, <font color="Maroon" title="c10">b1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:228_1_1_2_1"/><span class="lab"><font color="Green" title="E152">A9</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:228_1_1_2"><span class="lab"><font color="Green" title="E146">A3</font></span></a>, <a class="txt" href="#E1:228_1_1_2_1"><span class="lab"><font color="Green" title="E149">A6</font></span></a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/><a NAME="E6:228_1_1_2_1"/><span class="lab"><font color="Green" title="E153">A10</font></span>: 
<font color="Maroon" title="c10">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:228_1_1_2"><span class="lab"><font color="Green" title="E146">A3</font></span></a>, <a class="txt" href="#E1:228_1_1_2_1"><span class="lab"><font color="Green" title="E149">A6</font></span></a>, <a class="txt" href="#E5:228_1_1_2_1"><span class="lab"><font color="Green" title="E152">A9</font></span></a>, <a class="ref" href="funct_1.html#T1" title="FUNCT_1:th.1">FUNCT_1:1</a></span>;<br/><span class="lab"><font color="Green" title="E154">A11</font></span>: <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font> = 
<font color="Maroon" title="c7">x1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:228_1_1_2"><span class="lab"><font color="Green" title="E146">A3</font></span></a>, <a class="txt" href="#E1:228_1_1_2_1"><span class="lab"><font color="Green" title="E149">A6</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c10">b1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:228_1_1_2_1"><span class="lab"><font color="Green" title="E152">A9</font></span></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>
;<br/><span class="kw">consider </span><font color="Maroon" title="c11">a2</font>, <font color="Maroon" title="c12">b2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:228_1_1_2_1"/><span class="lab"><font color="Green" title="E155">A12</font></span>: 
<font color="Maroon" title="c8">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">a2</font>,<font color="Maroon" title="c12">b2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:228_1_1_2"><span class="lab"><font color="Green" title="E146">A3</font></span></a>, <a class="txt" href="#E2:228_1_1_2_1"><span class="lab"><font color="Green" title="E150">A7</font></span></a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/><a NAME="E10:228_1_1_2_1"/><span class="lab"><font color="Green" title="E156">A13</font></span>: 
<font color="Maroon" title="c11">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:228_1_1_2"><span class="lab"><font color="Green" title="E146">A3</font></span></a>, <a class="txt" href="#E2:228_1_1_2_1"><span class="lab"><font color="Green" title="E150">A7</font></span></a>, <a class="txt" href="#E9:228_1_1_2_1"><span class="lab"><font color="Green" title="E155">A12</font></span></a>, <a class="ref" href="funct_1.html#T1" title="FUNCT_1:th.1">FUNCT_1:1</a></span>;<br/><a NAME="E11:228_1_1_2_1"/><span class="lab"><font color="Green" title="E157">A14</font></span>: 
<font color="Maroon" title="c9">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:228_1_1_2"><span class="lab"><font color="Green" title="E146">A3</font></span></a>, <a class="txt" href="#E1:228_1_1_2_1"><span class="lab"><font color="Green" title="E149">A6</font></span></a>, <a class="txt" href="#E5:228_1_1_2_1"><span class="lab"><font color="Green" title="E152">A9</font></span></a>, <a class="ref" href="funct_1.html#T1" title="FUNCT_1:th.1">FUNCT_1:1</a></span>;<br/><a NAME="E12:228_1_1_2_1"/><span class="lab"><font color="Green" title="E158">A15</font></span>: 
<font color="Maroon" title="c12">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:228_1_1_2"><span class="lab"><font color="Green" title="E146">A3</font></span></a>, <a class="txt" href="#E2:228_1_1_2_1"><span class="lab"><font color="Green" title="E150">A7</font></span></a>, <a class="txt" href="#E9:228_1_1_2_1"><span class="lab"><font color="Green" title="E155">A12</font></span></a>, <a class="ref" href="funct_1.html#T1" title="FUNCT_1:th.1">FUNCT_1:1</a></span>;<br/><span class="lab"><font color="Green" title="E159">A16</font></span>: <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x2</font> = 
<font color="Maroon" title="c8">x2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:228_1_1_2"><span class="lab"><font color="Green" title="E146">A3</font></span></a>, <a class="txt" href="#E2:228_1_1_2_1"><span class="lab"><font color="Green" title="E150">A7</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c12">b2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:228_1_1_2_1"><span class="lab"><font color="Green" title="E155">A12</font></span></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>
;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">a1</font> = <font color="Maroon" title="c9">a1</font>, <font color="Maroon" title="c14">a2</font> = <font color="Maroon" title="c11">a2</font> as    <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="#E11:228_1_1_2_1"><span class="lab"><font color="Green" title="E157">A14</font></span></a>, <a class="txt" href="#E10:228_1_1_2_1"><span class="lab"><font color="Green" title="E156">A13</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E160">A17</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="228_1_1_2_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">n1</font>, <font color="Maroon" title="c16">n2</font> be    <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="c15">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">n2</font> &amp; <font color="Maroon" title="c15">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span> &amp; <font color="Maroon" title="c16">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span> implies  not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">n2</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:228_1_1_2_1_3"/><span class="lab"><font color="Green" title="E161">A18</font></span>: 
<font color="Maroon" title="c15">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">n2</font>
 <span class="kw">and </span><br/><a NAME="E2:228_1_1_2_1_3"/><span class="lab"><font color="Green" title="E162">A19</font></span>: 
<font color="Maroon" title="c15">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:228_1_1_2_1_3"/><span class="lab"><font color="Green" title="E163">A20</font></span>: 
<font color="Maroon" title="c16">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:228_1_1_2_1_3"/><span class="lab"><font color="Green" title="E164">A21</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">n2</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="E5:228_1_1_2_1_3"/><span class="lab"><font color="Green" title="E165">A22</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="c16">n2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:228_1_1_2_1_3"><span class="lab"><font color="Green" title="E163">A20</font></span></a>, <a class="ref" href="glib_001.html#T73" target="_self" title="GLIB_001:th.73">Th74</a></span>;<br/><a NAME="E6:228_1_1_2_1_3"/><span class="lab"><font color="Green" title="E166">A23</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="c15">n1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:228_1_1_2_1_3"><span class="lab"><font color="Green" title="E162">A19</font></span></a>, <a class="ref" href="glib_001.html#T73" target="_self" title="GLIB_001:th.73">Th74</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">a</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c15">n1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1, <font color="Maroon" title="c18">b</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c16">n2</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="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="#E5:228_1_1_2_1_3"><span class="lab"><font color="Green" title="E165">A22</font></span></a></span>;<br/><a NAME="E8:228_1_1_2_1_3"/><span class="lab"><font color="Green" title="E167">A24</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:228_1_1_2"><span class="lab"><font color="Green" title="E147">A4</font></span></a>, <a class="txt" href="#E5:228_1_1_2_1_3"><span class="lab"><font color="Green" title="E165">A22</font></span></a>, <a class="txt" href="glib_001.html#E60"><span class="lab"><font color="Green" title="E52">Lm23</font></span></a></span>;<br/><a NAME="E9:228_1_1_2_1_3"/><span class="lab"><font color="Green" title="E168">A25</font></span>: 
<font color="Maroon" title="c16">n2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:228_1_1_2_1_3"><span class="lab"><font color="Green" title="E163">A20</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E10:228_1_1_2_1_3"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:228_1_1_2_1_3"><span class="lab"><font color="Green" title="E163">A20</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E11:228_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E169">A26</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:228_1_1_2_1_3"><span class="lab"><font color="Green" title="E168">A25</font></span></a>, <a class="ref" href="glib_001.html#D14" target="_self" title="GLIB_001:def.14">Def14</a></span>;<br/><a NAME="E12:228_1_1_2_1_3"/><span class="lab"><font color="Green" title="E170">A27</font></span>: 
<font color="Maroon" title="c15">n1</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:228_1_1_2_1_3"><span class="lab"><font color="Green" title="E162">A19</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E13:228_1_1_2_1_3"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:228_1_1_2_1_3"><span class="lab"><font color="Green" title="E162">A19</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E14:228_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E171">A28</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:228_1_1_2_1_3"><span class="lab"><font color="Green" title="E170">A27</font></span></a>, <a class="ref" href="glib_001.html#D14" target="_self" title="GLIB_001:def.14">Def14</a></span>;<br/><a NAME="E15:228_1_1_2_1_3"/>
<font color="Maroon" title="c18">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">lenW2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:228_1_1_2"><span class="lab"><font color="Green" title="E148">A5</font></span></a>, <a class="txt" href="#E5:228_1_1_2_1_3"><span class="lab"><font color="Green" title="E165">A22</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E16:228_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E172">A29</font></span>: 
<font color="Maroon" title="c18">b</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">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:228_1_1_2"><span class="lab"><font color="Green" title="E147">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E17:228_1_1_2_1_3"/>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c15">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c16">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:228_1_1_2_1_3"><span class="lab"><font color="Green" title="E161">A18</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><a NAME="E18:228_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E173">A30</font></span>: 
<font color="Maroon" title="c17">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T14" title="XREAL_1:th.14">XREAL_1:14</a></span>;<br/><a NAME="E19:228_1_1_2_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:228_1_1_2"><span class="lab"><font color="Green" title="E147">A4</font></span></a>, <a class="txt" href="#E6:228_1_1_2_1_3"><span class="lab"><font color="Green" title="E166">A23</font></span></a>, <a class="txt" href="glib_001.html#E60"><span class="lab"><font color="Green" title="E52">Lm23</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E20:228_1_1_2_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:228_1_1_2_1_3"><span class="lab"><font color="Green" title="E164">A21</font></span></a>, <a class="txt" href="#E14:228_1_1_2_1_3"><span class="lab"><font color="Green" title="E171">A28</font></span></a>, <a class="txt" href="#E11:228_1_1_2_1_3"><span class="lab"><font color="Green" title="E169">A26</font></span></a>, <a class="txt" href="#E8:228_1_1_2_1_3"><span class="lab"><font color="Green" title="E167">A24</font></span></a>, <a class="txt" href="#E18:228_1_1_2_1_3"><span class="lab"><font color="Green" title="E173">A30</font></span></a>, <a class="txt" href="#E16:228_1_1_2_1_3"><span class="lab"><font color="Green" title="E172">A29</font></span></a>, <a class="ref" href="glib_001.html#D28" target="_self" title="GLIB_001:def.28">Def28</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="228_1_1_2_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:228_1_1_2_1_4"/><span class="lab"><font color="Green" title="E161">A31</font></span>: 
<font color="Maroon" title="c13">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">a2</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><a class="txt" onclick="hs2(this)" href="javascript:()" title="228_1_1_2_1_4_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="228_1_1_2_1_4_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="E1:228_1_1_2_1_4_1_1"/>
( <font color="Maroon" title="c13">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">a2</font> or <font color="Maroon" title="c13">a1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">a2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="228_1_1_2_1_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:228_1_1_2_1_4_1_1_1"/>
<font color="Maroon" title="c13">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">a2</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="E2:228_1_1_2_1_4_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:228_1_1_2_1_4"><span class="lab"><font color="Green" title="E161">A31</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="E3:228_1_1_2_1_4_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:228_1_1_2_1"><span class="lab"><font color="Green" title="E151">A8</font></span></a>, <a class="txt" href="#E7:228_1_1_2_1"><span class="lab"><font color="Green" title="E154">A11</font></span></a>, <a class="txt" href="#E13:228_1_1_2_1"><span class="lab"><font color="Green" title="E159">A16</font></span></a>, <a class="txt" href="#E11:228_1_1_2_1"><span class="lab"><font color="Green" title="E157">A14</font></span></a>, <a class="txt" href="#E6:228_1_1_2_1"><span class="lab"><font color="Green" title="E153">A10</font></span></a>, <a class="txt" href="#E10:228_1_1_2_1"><span class="lab"><font color="Green" title="E156">A13</font></span></a>, <a class="txt" href="#E12:228_1_1_2_1"><span class="lab"><font color="Green" title="E158">A15</font></span></a>, <a class="txt" href="#E15:228_1_1_2_1"><span class="lab"><font color="Green" title="E160">A17</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="228_1_1_2_1_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:228_1_1_2_1_4_1_1_2"/>
<font color="Maroon" title="c13">a1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">a2</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="E2:228_1_1_2_1_4_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:228_1_1_2_1"><span class="lab"><font color="Green" title="E151">A8</font></span></a>, <a class="txt" href="#E7:228_1_1_2_1"><span class="lab"><font color="Green" title="E154">A11</font></span></a>, <a class="txt" href="#E13:228_1_1_2_1"><span class="lab"><font color="Green" title="E159">A16</font></span></a>, <a class="txt" href="#E11:228_1_1_2_1"><span class="lab"><font color="Green" title="E157">A14</font></span></a>, <a class="txt" href="#E6:228_1_1_2_1"><span class="lab"><font color="Green" title="E153">A10</font></span></a>, <a class="txt" href="#E10:228_1_1_2_1"><span class="lab"><font color="Green" title="E156">A13</font></span></a>, <a class="txt" href="#E12:228_1_1_2_1"><span class="lab"><font color="Green" title="E158">A15</font></span></a>, <a class="txt" href="#E15:228_1_1_2_1"><span class="lab"><font color="Green" title="E160">A17</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:228_1_1_2_1_4"/>
contradiction
 ; <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">hence </span><a NAME="E17:228_1_1_2_1"/>
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:228_1_1_2_1"><span class="lab"><font color="Green" title="E151">A8</font></span></a>, <a class="txt" href="#E5:228_1_1_2_1"><span class="lab"><font color="Green" title="E152">A9</font></span></a>, <a class="txt" href="#E9:228_1_1_2_1"><span class="lab"><font color="Green" title="E155">A12</font></span></a>, <a class="txt" href="#E7:228_1_1_2_1"><span class="lab"><font color="Green" title="E154">A11</font></span></a>, <a class="txt" href="#E13:228_1_1_2_1"><span class="lab"><font color="Green" title="E159">A16</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="E11:228_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E149">A32</font></span>: 
<font color="Maroon" title="c6">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="228_1_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">y</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="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> implies <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> )</span><br/><span class="kw">assume </span><a NAME="E1:228_1_1_2_2"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</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="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:228_1_1_2_2"/><span class="lab"><font color="Green" title="E150">A33</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font>
 <span class="kw">and </span><br/><a NAME="E4:228_1_1_2_2"/><span class="lab"><font color="Green" title="E151">A34</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:228_1_1_2_2"/><span class="lab"><font color="Green" title="E152">A35</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:228_1_1_2"><span class="lab"><font color="Green" title="E146">A3</font></span></a>, <a class="txt" href="#E3:228_1_1_2_2"><span class="lab"><font color="Green" title="E150">A33</font></span></a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/><a NAME="E7:228_1_1_2_2"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:228_1_1_2"><span class="lab"><font color="Green" title="E146">A3</font></span></a>, <a class="txt" href="#E3:228_1_1_2_2"><span class="lab"><font color="Green" title="E150">A33</font></span></a>, <a class="txt" href="#E4:228_1_1_2_2"><span class="lab"><font color="Green" title="E151">A34</font></span></a></span>;<br/><a NAME="E8:228_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E153">A36</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:228_1_1_2_2"><span class="lab"><font color="Green" title="E152">A35</font></span></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>;<br/><a NAME="E9:228_1_1_2_2"/><span class="lab"><font color="Green" title="E154">A37</font></span>: 
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:228_1_1_2"><span class="lab"><font color="Green" title="E146">A3</font></span></a>, <a class="txt" href="#E3:228_1_1_2_2"><span class="lab"><font color="Green" title="E150">A33</font></span></a>, <a class="txt" href="#E6:228_1_1_2_2"><span class="lab"><font color="Green" title="E152">A35</font></span></a>, <a class="ref" href="funct_1.html#T1" title="FUNCT_1:th.1">FUNCT_1:1</a></span>;<br/><a NAME="E10:228_1_1_2_2"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:228_1_1_2"><span class="lab"><font color="Green" title="E146">A3</font></span></a>, <a class="txt" href="#E3:228_1_1_2_2"><span class="lab"><font color="Green" title="E150">A33</font></span></a>, <a class="txt" href="#E6:228_1_1_2_2"><span class="lab"><font color="Green" title="E152">A35</font></span></a>, <a class="ref" href="funct_1.html#T1" title="FUNCT_1:th.1">FUNCT_1:1</a></span>;<br/><a NAME="E11:228_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:228_1_1_2_2"><span class="lab"><font color="Green" title="E153">A36</font></span></a>, <a class="txt" href="#E9:228_1_1_2_2"><span class="lab"><font color="Green" title="E154">A37</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E12:228_1_1_2_2"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 ; <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="E13:228_1_1_2"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E14:228_1_1_2"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:228_1_1_2"><span class="lab"><font color="Green" title="E146">A3</font></span></a>, <a class="txt" href="#E11:228_1_1_2"><span class="lab"><font color="Green" title="E149">A32</font></span></a>, <a class="ref" href="card_1.html#T10" title="CARD_1:th.10">CARD_1:10</a></span>;<br/><a NAME="E15:228_1_1_2"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>;<br/><a NAME="E16:228_1_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D24" title="GLIB_000:def.24">GLIB_000:def 24</a></span>;<br/><a NAME="E17:228_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E150">A38</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><a NAME="E18:228_1_1_2"/><span class="lab"><font color="Green" title="E151">A39</font></span>: 
<font color="Maroon" title="c3">lenW2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:228_1_1_2"><span class="lab"><font color="Green" title="E144">A1</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/><a NAME="E19:228_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E152">A40</font></span>: 
<font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">lenW2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">lenW2</font>,<font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">lenW2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D3" target="_self" title="GLIB_001:def.3">Def3</a></span>;<br/><a NAME="E20:228_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">lenW2</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="E21:228_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">lenW2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:228_1_1_2"><span class="lab"><font color="Green" title="E151">A39</font></span></a>, <a class="txt" href="glib_001.html#E53"><span class="lab"><font color="Green" title="E45">Lm16</font></span></a>, <a class="ref" href="jordan12.html#T2" title="JORDAN12:th.2">JORDAN12:2</a></span>;<br/><a NAME="E22:228_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">W</font> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">lenW2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:228_1_1_2"><span class="lab"><font color="Green" title="E145">A2</font></span></a>, <a class="txt" href="#E19:228_1_1_2"><span class="lab"><font color="Green" title="E152">A40</font></span></a>, <a class="ref" href="glib_001.html#T75" target="_self" title="GLIB_001:th.75">Th76</a></span>;<br/><a NAME="E23:228_1_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c3">lenW2</font>)</span>)</span> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">lenW2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/><span class="kw">hence </span><a NAME="E24:228_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:228_1_1_2"><span class="lab"><font color="Green" title="E150">A38</font></span></a>, <a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</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>
<span class="kw">hence </span><a NAME="E2:228"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K11" title="GLIB_001:func.11">.vertexSeq()</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
