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<div class="add">

<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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_1_1"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
for <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>  holds not <font color="Olive" title="b1">W</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="glib_002.html#D2" target="_self" title="GLIB_002:def.2">GLIB_002:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><a NAME="E2:7_1_1_1"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
<font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">and </span><br/><a NAME="E3:7_1_1_1"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">set </span><font color="Maroon" title="c3">v1</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">e</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c4">v1</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">e</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7_1_1_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c5">W</font> = <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c4">v1</font>;<br/><a NAME="E5:7_1_1_1"/>
<font color="Maroon" title="c2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7_1_1_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="txt" href="#E3:7_1_1_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="glib_000.html#D15" title="GLIB_000:def.15">GLIB_000:def 15</a></span>;<br/><a NAME="E6:7_1_1_1"/><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="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c4">v1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T15" title="GLIB_001:th.15">GLIB_001:15</a></span>;<br/><a NAME="E7:7_1_1_1"/><span class="kw">then </span>
not <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c4">v1</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/><a NAME="E8:7_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c4">v1</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D31" title="GLIB_001:def.31">GLIB_001:def 31</a></span>;<br/><span class="kw">hence </span><a NAME="E9:7_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A4</font></span>: 
<font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D20" title="GLIB_000:def.20">GLIB_000:def 20</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">e1</font>, <font color="Maroon" title="c3">e2</font>, <font color="Maroon" title="c4">v1</font>, <font color="Maroon" title="c5">v2</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c3">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>,<font color="Maroon" title="c1">G</font> implies <font color="Maroon" title="c2">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e2</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:7_1_1_2"/><span class="lab"><font color="Green" title="E7">A5</font></span>: 
<font color="Maroon" title="c2">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">and </span><br/><a NAME="E2:7_1_1_2"/><span class="lab"><font color="Green" title="E8">A6</font></span>: 
<font color="Maroon" title="c3">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>,<font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e2</font></span><br/><a NAME="E3:7_1_1_2"/><span class="lab"><font color="Green" title="E9">A7</font></span>: 
<font color="Maroon" title="c3">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v2</font>,<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7_1_1_2"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="glib_000.html#T17" title="GLIB_000:th.17">GLIB_000:17</a></span>;<br/><a NAME="E4:7_1_1_2"/><span class="lab"><font color="Green" title="E10">A8</font></span>: 
<font color="Maroon" title="c4">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_1_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E1:7_1_1_2"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="ref" href="glib_000.html#T21" title="GLIB_000:th.21">GLIB_000:21</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c6">W1</font> = <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c5">v2</font>;<br/><span class="kw">set </span><font color="Maroon" title="c7">W</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c5">v2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c3">e2</font>;<br/><span class="kw">assume </span><a NAME="E1:7_1_1_2_1"/><span class="lab"><font color="Green" title="E11">A9</font></span>: 
<font color="Maroon" title="c2">e1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">e2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c5">v2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c3">e2</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  &amp; contradiction )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c8">W1</font> = <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c5">v2</font> as   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font> ;<br/><a NAME="E3:7_1_1_2_1"/><span class="lab"><font color="Green" title="E12">A10</font></span>: 
<font color="Maroon" title="c8">W1</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="c5">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_1_2"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="ref" href="glib_001.html#T16" title="GLIB_001:th.16">GLIB_001:16</a></span>;<br/><a NAME="E4:7_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A11</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c5">v2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c3">e2</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="c4">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_1_1_2"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="glib_001.html#T64" title="GLIB_001:th.64">GLIB_001:64</a></span>;<br/><a NAME="E5:7_1_1_2_1"/><span class="lab"><font color="Green" title="E14">A12</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_1_2"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="ref" href="glib_001.html#T15" title="GLIB_001:th.15">GLIB_001:15</a></span>;<br/><div><span class="lab"><font color="Green" title="E15">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">n</font> be   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Maroon" title="c9">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="c8">W1</font> implies <font color="Maroon" title="c8">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v1</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:7_1_1_2_1_1"/><span class="lab"><font color="Green" title="E16">A14</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">and </span><br/><a NAME="E2:7_1_1_2_1_1"/><span class="lab"><font color="Green" title="E17">A15</font></span>: 
<font color="Maroon" title="c9">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="c8">W1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v1</font></span><br/><a NAME="E3:7_1_1_2_1_1"/>
1 <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> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_1_2_1_1"><span class="lab"><font color="Green" title="E16">A14</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:7_1_1_2_1_1"/><span class="kw">then </span>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">n</font>
 <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/><a NAME="E5:7_1_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</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> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:7_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A16</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_1_1_2_1"><span class="lab"><font color="Green" title="E14">A12</font></span></a>, <a class="txt" href="#E2:7_1_1_2_1_1"><span class="lab"><font color="Green" title="E17">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E7:7_1_1_2_1_1"/>
<font color="Maroon" title="c8">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c5">v2</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_1_2"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="ref" href="glib_001.html#D5" title="GLIB_001:def.5">GLIB_001:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E8:7_1_1_2_1_1"/>
<font color="Maroon" title="c8">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7_1_1_2"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E6:7_1_1_2_1_1"><span class="lab"><font color="Green" title="E18">A16</font></span></a>, <a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E7:7_1_1_2_1"/>
( <font color="Maroon" title="c8">W1</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="c4">v1</font> &amp; <font color="Maroon" title="c8">W1</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="c5">v2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_1_2"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="ref" href="glib_001.html#T16" title="GLIB_001:th.16">GLIB_001:16</a></span>;<br/><a NAME="E8:7_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A17</font></span>: 
not <font color="Maroon" title="c8">W1</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="#E4:7_1_1_2"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></span>;<br/><a NAME="E9:7_1_1_2_1"/>
<font color="Maroon" title="c8">W1</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="c4">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_1_2"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="ref" href="glib_001.html#T16" title="GLIB_001:th.16">GLIB_001:16</a></span>;<br/><a NAME="E10:7_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c5">v2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c3">e2</font></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_1_1_2"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E3:7_1_1_2_1"><span class="lab"><font color="Green" title="E12">A10</font></span></a>, <a class="ref" href="glib_001.html#T64" title="GLIB_001:th.64">GLIB_001:64</a></span>;<br/><a NAME="E11:7_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A18</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c5">v2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c3">e2</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="#E4:7_1_1_2_1"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></span>;<br/><a NAME="E12:7_1_1_2_1"/><span class="lab"><font color="Green" title="E18">A19</font></span>: 
<font color="Maroon" title="c3">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">W1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_1_2"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E3:7_1_1_2"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="glib_001.html#T16" title="GLIB_001:th.16">GLIB_001:16</a></span>;<br/><a NAME="E13:7_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A20</font></span>: 
not <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c5">v2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c3">e2</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#T133" title="GLIB_001:th.133">GLIB_001:133</a></span>;<br/><a NAME="E14:7_1_1_2_1"/>
<font color="Maroon" title="c8">W1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">e1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_1_2"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="ref" href="glib_001.html#T109" title="GLIB_001:th.109">GLIB_001:109</a></span>;<br/><a NAME="E15:7_1_1_2_1"/><span class="kw">then </span>
not <font color="Maroon" title="c3">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">W1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_1_2_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E16:7_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c5">v2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c3">e2</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:7_1_1_2_1"><span class="lab"><font color="Green" title="E18">A19</font></span></a>, <a class="txt" href="#E8:7_1_1_2_1"><span class="lab"><font color="Green" title="E16">A17</font></span></a>, <a class="txt" href="#E6:7_1_1_2_1"><span class="lab"><font color="Green" title="E15">A13</font></span></a>, <a class="ref" href="glib_001.html#T151" title="GLIB_001:th.151">GLIB_001:151</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E17:7_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c5">v2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c3">e2</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:7_1_1_2_1"><span class="lab"><font color="Green" title="E17">A18</font></span></a>, <a class="txt" href="#E13:7_1_1_2_1"><span class="lab"><font color="Green" title="E19">A20</font></span></a>, <a class="ref" href="glib_001.html#D31" title="GLIB_001:def.31">GLIB_001:def 31</a></span>;<br/><span class="kw">hence </span><a NAME="E18:7_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:7_1_1_2"/>
<font color="Maroon" title="c2">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E5:7_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D22" title="GLIB_000:def.22">GLIB_000:def 22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:7_1_1"/>
<font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_1_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
