<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">V</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">G</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/><b>let </b><font color="Maroon" title="c3">v1</font>, <font color="Maroon" title="c4">v2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">G</font>;<br/><b>let </b><font color="Maroon" title="c5">p</font> be  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font>;<br/>









<b>assume </b><a NAME="E1:59"/><i><font color="Green" title="E47">A1</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font>,<font color="Maroon" title="c1">V</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">p</font> being <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font><br/> for <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">v2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">G</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">p</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Olive" title="b2">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c1">V</font> holds <br/> <a href="graph_5.html#K6" title="GRAPH_5:func.6">AcyclicPaths</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/>
<b>set </b><font color="Maroon" title="c6">FS</font> = the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font>;<br/>
<b>set </b><font color="Maroon" title="c7">FT</font> = the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font>;<br/>
<a NAME="E2:59"/><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">p</font> be  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font>;<br/><b>let </b><font color="Maroon" title="c9">v1</font>, <font color="Maroon" title="c10">v2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">G</font>;<br/>





<b>assume </b><a NAME="E1:59_1"/><i><font color="Green" title="E49">A3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c8">p</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c9">v1</font>,<font color="Maroon" title="c10">v2</font>,<font color="Maroon" title="c1">V</font> )
 ;<br/>

<a NAME="E2:59_1"/><b>then </b>
<font color="Maroon" title="c8">p</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c9">v1</font>,<font color="Maroon" title="c10">v2</font>
 
<b>by </b><i><a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></i>;<br/>
<a NAME="E3:59_1"/><b>then </b><i><font color="Green" title="E50">A4</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></i>;<br/>
<a NAME="E4:59_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T28" title="FINSEQ_1:th.28">FINSEQ_1:28</a></i>;<br/>
<a NAME="E5:59_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:59_1"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">r</font> = <font color="Maroon" title="c8">p</font> as  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a> <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="graph_5.html#T18" target="_self" title="GRAPH_5:th.18">Th18</a></i>;<br/>
<a NAME="E7:59_1"/>
<font color="Maroon" title="c11">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a> <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font> : ( <font color="Olive" title="b1">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">p</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="graph_5.html#E3:59_1"><i><font color="Green" title="E50">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:59_1"/>
 <a href="graph_5.html#K6" title="GRAPH_5:func.6">AcyclicPaths</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:59"/><i><font color="Green" title="E49">A5</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:59_2"/><i><font color="Green" title="E50">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">p</font> be  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font>;<br/><b>let </b><font color="Maroon" title="c10">v1</font>, <font color="Maroon" title="c11">v2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">G</font>;<br/><b>assume </b><a NAME="E1:59_2_1"/><i><font color="Green" title="E51">A7</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">p</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c1">V</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c12">vs</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E3:59_2_1"/><i><font color="Green" title="E52">A8</font></i>: 
<font color="Maroon" title="c12">vs</font> <a href="graph_4.html#R3" title="GRAPH_4:pred.3">is_oriented_vertex_seq_of</a> <font color="Maroon" title="c9">p</font>
 <b>by </b><i><a class="ref" href="graph_4.html#T10" title="GRAPH_4:th.10">GRAPH_4:10</a></i>;<br/><a NAME="E4:59_2_1"/><i><font color="Green" title="E53">A9</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs</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="c9">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font> holds <br/><font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c12">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c12">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green" title="E52">A8</font></i></a>, <a class="ref" href="graph_4.html#D5" title="GRAPH_4:def.5">GRAPH_4:def 5</a></i>;<br/><a NAME="E5:59_2_1"/><i><font color="Green" title="E54">A10</font></i>: 
<font color="Maroon" title="c9">p</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:59_2_1"><i><font color="Green" title="E51">A7</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></i>;<br/><a NAME="E6:59_2_1"/><b>then </b><i><font color="Green" title="E55">A11</font></i>: 
<font color="Maroon" title="c9">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></i>;<br/><a NAME="E7:59_2_1"/><b>then </b><i><font color="Green" title="E56">A12</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T28" title="FINSEQ_1:th.28">FINSEQ_1:28</a></i>;<br/><a NAME="E8:59_2_1"/><i><font color="Green" title="E57">A13</font></i>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:59_2_1"><i><font color="Green" title="E51">A7</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></i>;<br/><a NAME="E9:59_2_1"/><i><font color="Green" title="E58">A14</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E4:59_2_1"><i><font color="Green" title="E53">A9</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E10:59_2_1"/>
<font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c12">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font></span>)</span>,<font color="Maroon" title="c12">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green" title="E52">A8</font></i></a>, <a class="txt" href="graph_5.html#E7:59_2_1"><i><font color="Green" title="E56">A12</font></i></a>, <a class="ref" href="graph_4.html#D5" title="GRAPH_4:def.5">GRAPH_4:def 5</a></i>;<br/><a NAME="E11:59_2_1"/><b>then </b>
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="graph_4.html#D1" title="GRAPH_4:def.1">GRAPH_4:def 1</a></i>;<br/><a NAME="E12:59_2_1"/><b>then </b><i><font color="Green" title="E59">A15</font></i>: 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E4:59_2_1"><i><font color="Green" title="E53">A9</font></i></a>, <a class="txt" href="graph_5.html#E9:59_2_1"><i><font color="Green" title="E58">A14</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_2_1_1"/>
( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs</font> &amp; <font color="Maroon" title="c12">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> holds <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs</font> ) or  ex <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs</font> &amp; <font color="Maroon" title="c12">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> &amp; not <!--HUMANRECFAILED-->( <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs</font> ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1"><b>suppose </b></a><a NAME="E1:59_2_1_1_1"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs</font> &amp; <font color="Maroon" title="c12">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> holds <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs</font> )
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">p</font> is <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green" title="E52">A8</font></i></a>, <a class="ref" href="graph_4.html#D7" title="GRAPH_4:def.7">GRAPH_4:def 7</a></i>;<br/><a NAME="E3:59_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a> <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font> : ( <font color="Olive" title="b1">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">w</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">p</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_5.html#E6:59_2_1"><i><font color="Green" title="E55">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E4:59_2_1_1_1"/>
 <a href="graph_5.html#K6" title="GRAPH_5:func.6">AcyclicPaths</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_2"><b>suppose </b></a><a NAME="E1:59_2_1_1_2"/>
 ex <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs</font> &amp; <font color="Maroon" title="c12">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> &amp; not <!--HUMANRECFAILED-->( <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs</font> ) )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c13">n</font>, <font color="Maroon" title="c14">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:59_2_1_1_2"/><i><font color="Green" title="E60">A16</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font> &amp; <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">m</font> &amp; <font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs</font> &amp; <font color="Maroon" title="c12">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font> &amp; (  not <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs</font> ) )
 ;<br/><a NAME="E4:59_2_1_1_2"/><i><font color="Green" title="E61">A17</font></i>: 
<font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1_1_2"><i><font color="Green" title="E60">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:59_2_1_1_2"/><i><font color="Green" title="E62">A18</font></i>: 
<font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">i</font> = <font color="Maroon" title="c15">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_2_1_1_2_1_1"/>
( <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c13">n</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="59_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:59_2_1_1_2_1_1_1"/><i><font color="Green" title="E63">A19</font></i>: 
<font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1_1_2"><i><font color="Green" title="E60">A16</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:59_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E64">A20</font></i>: 
<font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E4:59_2_1"><i><font color="Green" title="E53">A9</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:59_2_1_1_2_1_1_1"/><i><font color="Green" title="E65">A21</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c17">j</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E7:59_2_1_1_2_1_1_1"/><i><font color="Green" title="E66">A22</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E6:59_2_1_1_2"><i><font color="Green" title="E62">A18</font></i></a>, <a class="txt" href="graph_5.html#E5:59_2_1_1_2_1_1_1"><i><font color="Green" title="E65">A21</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c19">p1</font>, <font color="Maroon" title="c20">p2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E9:59_2_1_1_2_1_1_1"/><i><font color="Green" title="E67">A23</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">j</font> &amp; <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">p1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c20">p2</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T25" title="FINSEQ_2:th.25">FINSEQ_2:25</a></i>;<br/><a NAME="E10:59_2_1_1_2_1_1_1"/><i><font color="Green" title="E68">A24</font></i>: 
<font color="Maroon" title="c13">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="c12">vs</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1_1_2"><i><font color="Green" title="E60">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E11:59_2_1_1_2_1_1_1"/><i><font color="Green" title="E69">A25</font></i>: 
<font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c12">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1,<font color="Maroon" title="c12">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green" title="E52">A8</font></i></a>, <a class="txt" href="graph_5.html#E7:59_2_1"><i><font color="Green" title="E56">A12</font></i></a>, <a class="ref" href="graph_4.html#D5" title="GRAPH_4:def.5">GRAPH_4:def 5</a></i>;<br/><a NAME="E12:59_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c12">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">m</font>,<font color="Maroon" title="c12">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green" title="E52">A8</font></i></a>, <a class="txt" href="graph_5.html#E3:59_2_1_1_2"><i><font color="Green" title="E60">A16</font></i></a>, <a class="txt" href="graph_5.html#E1:59_2_1_1_2_1_1_1"><i><font color="Green" title="E63">A19</font></i></a>, <a class="txt" href="graph_5.html#E3:59_2_1_1_2_1_1_1"><i><font color="Green" title="E64">A20</font></i></a>, <a class="ref" href="graph_4.html#D5" title="GRAPH_4:def.5">GRAPH_4:def 5</a></i>;<br/><b>then </b><i><font color="Green" title="E70">A26</font></i>: the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span> = 
<font color="Maroon" title="c12">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">m</font>
<b>by </b><i><a class="ref" href="graph_4.html#D1" title="GRAPH_4:def.1">GRAPH_4:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c12">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font>
<b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1_1_2"><i><font color="Green" title="E60">A16</font></i></a>, <a class="txt" href="graph_5.html#E1:59_2_1_1_2_1_1_1"><i><font color="Green" title="E63">A19</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
<font color="Maroon" title="c12">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">n</font>
<b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1_1_2"><i><font color="Green" title="E60">A16</font></i></a>, <a class="txt" href="graph_5.html#E10:59_2_1_1_2_1_1_1"><i><font color="Green" title="E68">A24</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="graph_5.html#E1:59_2_1_1_2_1_1_1"><i><font color="Green" title="E63">A19</font></i></a>, <a class="txt" href="graph_5.html#E11:59_2_1_1_2_1_1_1"><i><font color="Green" title="E69">A25</font></i></a>, <a class="ref" href="graph_4.html#D1" title="GRAPH_4:def.1">GRAPH_4:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c10">v1</font>
<b>by </b><i><a class="txt" href="graph_5.html#E5:59_2_1"><i><font color="Green" title="E54">A10</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></i>
;<br/><a NAME="E14:59_2_1_1_2_1_1_1"/><i><font color="Green" title="E71">A27</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">p2</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E9:59_2_1_1_2_1_1_1"><i><font color="Green" title="E67">A23</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E15:59_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E72">A28</font></i>: 
<font color="Maroon" title="c20">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E6:59_2_1_1_2"><i><font color="Green" title="E62">A18</font></i></a>, <a class="txt" href="graph_5.html#E9:59_2_1_1_2_1_1_1"><i><font color="Green" title="E67">A23</font></i></a>, <a class="txt" href="graph_5.html#E9"><i><font color="Green" title="E8">Lm2</font></i></a></i>;<br/><a NAME="E16:59_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c20">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E7:59_2_1_1_2_1_1_1"><i><font color="Green" title="E66">A22</font></i></a>, <a class="txt" href="graph_5.html#E9:59_2_1_1_2_1_1_1"><i><font color="Green" title="E67">A23</font></i></a>, <a class="txt" href="graph_5.html#E14:59_2_1_1_2_1_1_1"><i><font color="Green" title="E71">A27</font></i></a>, <a class="txt" href="graph_5.html#E9"><i><font color="Green" title="E8">Lm2</font></i></a></i>;<br/><a NAME="E17:59_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E73">A29</font></i>: 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">v2</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E5:59_2_1"><i><font color="Green" title="E54">A10</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c21">p1</font> = <font color="Maroon" title="c19">p1</font>, <font color="Maroon" title="c22">p2</font> = <font color="Maroon" title="c20">p2</font> as  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="graph_5.html#E9:59_2_1_1_2_1_1_1"><i><font color="Green" title="E67">A23</font></i></a>, <a class="ref" href="graph_5.html#T14" target="_self" title="GRAPH_5:th.14">Th14</a></i>;<br/><a NAME="E19:59_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c22">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="graph_5.html#E14:59_2_1_1_2_1_1_1"><i><font color="Green" title="E71">A27</font></i></a>, <a class="ref" href="finseq_1.html#T28" title="FINSEQ_1:th.28">FINSEQ_1:28</a></i>;<br/><a NAME="E20:59_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E74">A30</font></i>: 
<font color="Maroon" title="c22">p2</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E13:59_2_1_1_2_1_1_1"><i><font color="Green" title="E70">A26</font></i></a>, <a class="txt" href="graph_5.html#E15:59_2_1_1_2_1_1_1"><i><font color="Green" title="E72">A28</font></i></a>, <a class="txt" href="graph_5.html#E17:59_2_1_1_2_1_1_1"><i><font color="Green" title="E73">A29</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></i>;<br/><a NAME="E21:59_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">p1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c22">p2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:59_2_1"><i><font color="Green" title="E51">A7</font></i></a>, <a class="txt" href="graph_5.html#E9:59_2_1_1_2_1_1_1"><i><font color="Green" title="E67">A23</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></i>;<br/><a NAME="E22:59_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c22">p2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="graph_5.html#T26" target="_self" title="GRAPH_5:th.26">Th26</a></i>;<br/><a NAME="E23:59_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E75">A31</font></i>: 
<font color="Maroon" title="c22">p2</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E20:59_2_1_1_2_1_1_1"><i><font color="Green" title="E74">A30</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></i>;<br/><a NAME="E24:59_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1_1_2"><i><font color="Green" title="E60">A16</font></i></a>, <a class="txt" href="graph_5.html#E6:59_2_1_1_2"><i><font color="Green" title="E62">A18</font></i></a>, <a class="txt" href="graph_5.html#E1:59_2_1_1_2_1_1_1"><i><font color="Green" title="E63">A19</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E25:59_2_1_1_2_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">p2</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="c9">p</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E7:59_2_1_1_2_1_1_1"><i><font color="Green" title="E66">A22</font></i></a>, <a class="txt" href="graph_5.html#E9:59_2_1_1_2_1_1_1"><i><font color="Green" title="E67">A23</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E26:59_2_1_1_2_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">p2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E1:59_2_1"><i><font color="Green" title="E51">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E27:59_2_1_1_2_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">p2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E28:59_2_1_1_2_1_1_1"/><b>then </b>
 <a href="graph_5.html#K6" title="GRAPH_5:func.6">AcyclicPaths</a> <font color="Maroon" title="c22">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="graph_5.html#E1:59_2"><i><font color="Green" title="E50">A6</font></i></a>, <a class="txt" href="graph_5.html#E23:59_2_1_1_2_1_1_1"><i><font color="Green" title="E75">A31</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c23">r</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E30:59_2_1_1_2_1_1_1"/><i><font color="Green" title="E76">A32</font></i>: 
<font color="Maroon" title="c23">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K6" title="GRAPH_5:func.6">AcyclicPaths</a> <font color="Maroon" title="c22">p2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c24">q</font> being  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a> <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E32:59_2_1_1_2_1_1_1"/><i><font color="Green" title="E77">A33</font></i>: 
( <font color="Maroon" title="c23">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q</font> &amp; <font color="Maroon" title="c24">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v1</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">v2</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c22">p2</font> )
 <b>by </b><i><a class="txt" href="graph_5.html#E13:59_2_1_1_2_1_1_1"><i><font color="Green" title="E70">A26</font></i></a>, <a class="txt" href="graph_5.html#E15:59_2_1_1_2_1_1_1"><i><font color="Green" title="E72">A28</font></i></a>, <a class="txt" href="graph_5.html#E17:59_2_1_1_2_1_1_1"><i><font color="Green" title="E73">A29</font></i></a>, <a class="txt" href="graph_5.html#E30:59_2_1_1_2_1_1_1"><i><font color="Green" title="E76">A32</font></i></a></i>;<br/><a NAME="E33:59_2_1_1_2_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c22">p2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">p</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E9:59_2_1_1_2_1_1_1"><i><font color="Green" title="E67">A23</font></i></a>, <a class="ref" href="finseq_1.html#T43" title="FINSEQ_1:th.43">FINSEQ_1:43</a></i>;<br/><a NAME="E34:59_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E78">A34</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">p</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E32:59_2_1_1_2_1_1_1"><i><font color="Green" title="E77">A33</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E35:59_2_1_1_2_1_1_1"/>
( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_5.html#E5:59_2_1"><i><font color="Green" title="E54">A10</font></i></a>, <a class="txt" href="graph_5.html#E32:59_2_1_1_2_1_1_1"><i><font color="Green" title="E77">A33</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></i>;<br/><a NAME="E36:59_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c24">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a> <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font> : ( <font color="Olive" title="b1">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">w</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">p</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_5.html#E32:59_2_1_1_2_1_1_1"><i><font color="Green" title="E77">A33</font></i></a>, <a class="txt" href="graph_5.html#E34:59_2_1_1_2_1_1_1"><i><font color="Green" title="E78">A34</font></i></a></i>;<br/><b>hence </b><a NAME="E37:59_2_1_1_2_1_1_1"/>
 <a href="graph_5.html#K6" title="GRAPH_5:func.6">AcyclicPaths</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:59_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c13">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E63">A35</font></i>: 
<font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1_1_2"><i><font color="Green" title="E60">A16</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">n1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:59_2_1_1_2_1_1_2"/><i><font color="Green" title="E64">A36</font></i>: 
<font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">n1</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1_1_2"><i><font color="Green" title="E60">A16</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">n1</font> = <font color="Maroon" title="c17">n1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E6:59_2_1_1_2_1_1_2"/><i><font color="Green" title="E65">A37</font></i>: 
<font color="Maroon" title="c13">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="c12">vs</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1_1_2"><i><font color="Green" title="E60">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:59_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E66">A38</font></i>: 
<font color="Maroon" title="c18">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> <font color="Maroon" title="c9">p</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E4:59_2_1"><i><font color="Green" title="E53">A9</font></i></a>, <a class="txt" href="graph_5.html#E4:59_2_1_1_2_1_1_2"><i><font color="Green" title="E64">A36</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><b>then consider </b><font color="Maroon" title="c19">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:59_2_1_1_2_1_1_2"/><i><font color="Green" title="E67">A39</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">j</font> = <font color="Maroon" title="c19">j</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>consider </b><font color="Maroon" title="c21">p1</font>, <font color="Maroon" title="c22">p2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E12:59_2_1_1_2_1_1_2"/><i><font color="Green" title="E68">A40</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">n1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j</font> &amp; <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">p2</font> )
 <b>by </b><i><a class="txt" href="graph_5.html#E9:59_2_1_1_2_1_1_2"><i><font color="Green" title="E67">A39</font></i></a>, <a class="ref" href="finseq_2.html#T25" title="FINSEQ_2:th.25">FINSEQ_2:25</a></i>;<br/><a NAME="E13:59_2_1_1_2_1_1_2"/><i><font color="Green" title="E69">A41</font></i>: 
<font color="Maroon" title="c18">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E2:59_2_1_1_2_1_1_2"><i><font color="Green" title="E63">A35</font></i></a>, <a class="txt" href="graph_5.html#E4:59_2_1_1_2_1_1_2"><i><font color="Green" title="E64">A36</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E14:59_2_1_1_2_1_1_2"/><i><font color="Green" title="E70">A42</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">p1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">p</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E12:59_2_1_1_2_1_1_2"><i><font color="Green" title="E68">A40</font></i></a>, <a class="ref" href="finseq_1.html#T42" title="FINSEQ_1:th.42">FINSEQ_1:42</a></i>;<br/><a NAME="E15:59_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c21">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E12:59_2_1_1_2_1_1_2"><i><font color="Green" title="E68">A40</font></i></a>, <a class="txt" href="graph_5.html#E13:59_2_1_1_2_1_1_2"><i><font color="Green" title="E69">A41</font></i></a>, <a class="txt" href="graph_5.html#E8"><i><font color="Green" title="E7">Lm1</font></i></a></i>;<br/><a NAME="E16:59_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E71">A43</font></i>: 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v1</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E5:59_2_1"><i><font color="Green" title="E54">A10</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></i>;<br/><a NAME="E17:59_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n1</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c12">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n1</font>,<font color="Maroon" title="c12">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green" title="E52">A8</font></i></a>, <a class="txt" href="graph_5.html#E7:59_2_1_1_2_1_1_2"><i><font color="Green" title="E66">A38</font></i></a>, <a class="txt" href="graph_5.html#E13:59_2_1_1_2_1_1_2"><i><font color="Green" title="E69">A41</font></i></a>, <a class="ref" href="graph_4.html#D5" title="GRAPH_4:def.5">GRAPH_4:def 5</a></i>;<br/><b>then </b><i><font color="Green" title="E72">A44</font></i>: the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n1</font></span>)</span> = 
<font color="Maroon" title="c12">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">n</font>
<b>by </b><i><a class="txt" href="graph_5.html#E4:59_2_1_1_2_1_1_2"><i><font color="Green" title="E64">A36</font></i></a>, <a class="ref" href="graph_4.html#D1" title="GRAPH_4:def.1">GRAPH_4:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c12">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font>
<b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1_1_2"><i><font color="Green" title="E60">A16</font></i></a>, <a class="txt" href="graph_5.html#E6:59_2_1_1_2_1_1_2"><i><font color="Green" title="E65">A37</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
;<br/><a NAME="E19:59_2_1_1_2_1_1_2"/><i><font color="Green" title="E73">A45</font></i>: 
<font color="Maroon" title="c21">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">p1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n1</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E12:59_2_1_1_2_1_1_2"><i><font color="Green" title="E68">A40</font></i></a>, <a class="txt" href="graph_5.html#E13:59_2_1_1_2_1_1_2"><i><font color="Green" title="E69">A41</font></i></a>, <a class="txt" href="graph_5.html#E8"><i><font color="Green" title="E7">Lm1</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">p1</font> = <font color="Maroon" title="c21">p1</font>, <font color="Maroon" title="c24">p2</font> = <font color="Maroon" title="c22">p2</font> as  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="graph_5.html#E12:59_2_1_1_2_1_1_2"><i><font color="Green" title="E68">A40</font></i></a>, <a class="ref" href="graph_5.html#T14" target="_self" title="GRAPH_5:th.14">Th14</a></i>;<br/><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_2_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_2_1_1_2_1_1_2_2_1"/>
( <font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs</font> or <font color="Maroon" title="c14">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_2_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:59_2_1_1_2_1_1_2_2_1_1"/>
<font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs</font>
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E74">A46</font></i>: 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">v2</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E5:59_2_1"><i><font color="Green" title="E54">A10</font></i></a>, <a class="txt" href="graph_5.html#E12:59_2_1"><i><font color="Green" title="E59">A15</font></i></a>, <a class="txt" href="graph_5.html#E3:59_2_1_1_2"><i><font color="Green" title="E60">A16</font></i></a>, <a class="txt" href="graph_5.html#E18:59_2_1_1_2_1_1_2"><i><font color="Green" title="E72">A44</font></i></a>, <a class="txt" href="graph_5.html#E19:59_2_1_1_2_1_1_2"><i><font color="Green" title="E73">A45</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></i>;<br/><a NAME="E3:59_2_1_1_2_1_1_2_2_1_1"/>
<font color="Maroon" title="c23">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="graph_5.html#E12:59_2_1_1_2_1_1_2"><i><font color="Green" title="E68">A40</font></i></a>, <a class="txt" href="graph_5.html#E13:59_2_1_1_2_1_1_2"><i><font color="Green" title="E69">A41</font></i></a>, <a class="ref" href="finseq_1.html#T28" title="FINSEQ_1:th.28">FINSEQ_1:28</a></i>;<br/><a NAME="E4:59_2_1_1_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E75">A47</font></i>: 
<font color="Maroon" title="c23">p1</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E16:59_2_1_1_2_1_1_2"><i><font color="Green" title="E71">A43</font></i></a>, <a class="txt" href="graph_5.html#E2:59_2_1_1_2_1_1_2_2_1_1"><i><font color="Green" title="E74">A46</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></i>;<br/><a NAME="E5:59_2_1_1_2_1_1_2_2_1_1"/>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">p1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c24">p2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:59_2_1"><i><font color="Green" title="E51">A7</font></i></a>, <a class="txt" href="graph_5.html#E12:59_2_1_1_2_1_1_2"><i><font color="Green" title="E68">A40</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></i>;<br/><a NAME="E6:59_2_1_1_2_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c23">p1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="graph_5.html#T26" target="_self" title="GRAPH_5:th.26">Th26</a></i>;<br/><a NAME="E7:59_2_1_1_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E76">A48</font></i>: 
<font color="Maroon" title="c23">p1</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E4:59_2_1_1_2_1_1_2_2_1_1"><i><font color="Green" title="E75">A47</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></i>;<br/><a NAME="E8:59_2_1_1_2_1_1_2_2_1_1"/>
<font color="Maroon" title="c18">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E1:59_2_1"><i><font color="Green" title="E51">A7</font></i></a>, <a class="txt" href="graph_5.html#E7:59_2_1_1_2_1_1_2"><i><font color="Green" title="E66">A38</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:59_2_1_1_2_1_1_2_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E12:59_2_1_1_2_1_1_2"><i><font color="Green" title="E68">A40</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E10:59_2_1_1_2_1_1_2_2_1_1"/><b>then </b>
 <a href="graph_5.html#K6" title="GRAPH_5:func.6">AcyclicPaths</a> <font color="Maroon" title="c23">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="graph_5.html#E1:59_2"><i><font color="Green" title="E50">A6</font></i></a>, <a class="txt" href="graph_5.html#E7:59_2_1_1_2_1_1_2_2_1_1"><i><font color="Green" title="E76">A48</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c25">r</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:59_2_1_1_2_1_1_2_2_1_1"/><i><font color="Green" title="E77">A49</font></i>: 
<font color="Maroon" title="c25">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K6" title="GRAPH_5:func.6">AcyclicPaths</a> <font color="Maroon" title="c23">p1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c26">q</font> being  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a> <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E14:59_2_1_1_2_1_1_2_2_1_1"/><i><font color="Green" title="E78">A50</font></i>: 
( <font color="Maroon" title="c25">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q</font> &amp; <font color="Maroon" title="c26">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v1</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">v2</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c26">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c23">p1</font> )
 <b>by </b><i><a class="txt" href="graph_5.html#E16:59_2_1_1_2_1_1_2"><i><font color="Green" title="E71">A43</font></i></a>, <a class="txt" href="graph_5.html#E2:59_2_1_1_2_1_1_2_2_1_1"><i><font color="Green" title="E74">A46</font></i></a>, <a class="txt" href="graph_5.html#E12:59_2_1_1_2_1_1_2_2_1_1"><i><font color="Green" title="E77">A49</font></i></a></i>;<br/><a NAME="E15:59_2_1_1_2_1_1_2_2_1_1"/><i><font color="Green" title="E79">A51</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c26">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">p</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E14:59_2_1_1_2_1_1_2"><i><font color="Green" title="E70">A42</font></i></a>, <a class="txt" href="graph_5.html#E14:59_2_1_1_2_1_1_2_2_1_1"><i><font color="Green" title="E78">A50</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E16:59_2_1_1_2_1_1_2_2_1_1"/>
( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_5.html#E5:59_2_1"><i><font color="Green" title="E54">A10</font></i></a>, <a class="txt" href="graph_5.html#E14:59_2_1_1_2_1_1_2_2_1_1"><i><font color="Green" title="E78">A50</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></i>;<br/><a NAME="E17:59_2_1_1_2_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c26">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a> <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font> : ( <font color="Olive" title="b1">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">w</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">p</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_5.html#E14:59_2_1_1_2_1_1_2_2_1_1"><i><font color="Green" title="E78">A50</font></i></a>, <a class="txt" href="graph_5.html#E15:59_2_1_1_2_1_1_2_2_1_1"><i><font color="Green" title="E79">A51</font></i></a></i>;<br/><b>hence </b><a NAME="E18:59_2_1_1_2_1_1_2_2_1_1"/>
 <a href="graph_5.html#K6" title="GRAPH_5:func.6">AcyclicPaths</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_2_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:59_2_1_1_2_1_1_2_2_1_2"/>
<font color="Maroon" title="c14">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs</font>
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_2_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1_1_2"><i><font color="Green" title="E60">A16</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:59_2_1_1_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E74">A52</font></i>: 
<font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E4:59_2_1"><i><font color="Green" title="E53">A9</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c25">h</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:59_2_1_1_2_1_1_2_2_1_2"/><i><font color="Green" title="E75">A53</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c25">h</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">h</font> = <font color="Maroon" title="c25">h</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E7:59_2_1_1_2_1_1_2_2_1_2"/><i><font color="Green" title="E76">A54</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">h</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E6:59_2_1_1_2"><i><font color="Green" title="E62">A18</font></i></a>, <a class="txt" href="graph_5.html#E5:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E75">A53</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c27">q1</font>, <font color="Maroon" title="c28">q2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E9:59_2_1_1_2_1_1_2_2_1_2"/><i><font color="Green" title="E77">A55</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c28">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">h</font> &amp; <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c28">q2</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T25" title="FINSEQ_2:th.25">FINSEQ_2:25</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c29">q2</font> = <font color="Maroon" title="c28">q2</font> as  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="graph_5.html#E9:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E77">A55</font></i></a>, <a class="ref" href="graph_5.html#T14" target="_self" title="GRAPH_5:th.14">Th14</a></i>;<br/><a NAME="E11:59_2_1_1_2_1_1_2_2_1_2"/>
<font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c12">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">m</font>,<font color="Maroon" title="c12">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green" title="E52">A8</font></i></a>, <a class="txt" href="graph_5.html#E4:59_2_1_1_2"><i><font color="Green" title="E61">A17</font></i></a>, <a class="txt" href="graph_5.html#E3:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E74">A52</font></i></a>, <a class="ref" href="graph_4.html#D5" title="GRAPH_4:def.5">GRAPH_4:def 5</a></i>;<br/><b>then </b><i><font color="Green" title="E78">A56</font></i>: the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span> = 
<font color="Maroon" title="c12">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">m</font>
<b>by </b><i><a class="ref" href="graph_4.html#D1" title="GRAPH_4:def.1">GRAPH_4:def 1</a></i>
<br/>.= 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1_1_2"><i><font color="Green" title="E60">A16</font></i></a>, <a class="txt" href="graph_5.html#E4:59_2_1_1_2"><i><font color="Green" title="E61">A17</font></i></a>, <a class="txt" href="graph_5.html#E18:59_2_1_1_2_1_1_2"><i><font color="Green" title="E72">A44</font></i></a>, <a class="txt" href="graph_5.html#E19:59_2_1_1_2_1_1_2"><i><font color="Green" title="E73">A45</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
;<br/><a NAME="E13:59_2_1_1_2_1_1_2_2_1_2"/><i><font color="Green" title="E79">A57</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">q2</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E9:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E77">A55</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E14:59_2_1_1_2_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c29">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E6:59_2_1_1_2"><i><font color="Green" title="E62">A18</font></i></a>, <a class="txt" href="graph_5.html#E9:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E77">A55</font></i></a>, <a class="txt" href="graph_5.html#E9"><i><font color="Green" title="E8">Lm2</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c30">pq</font> = <font color="Maroon" title="c23">p1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">q2</font> as  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="graph_5.html#E12:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E78">A56</font></i></a>, <a class="ref" href="graph_5.html#T15" target="_self" title="GRAPH_5:th.15">Th15</a></i>;<br/><a NAME="E16:59_2_1_1_2_1_1_2_2_1_2"/><i><font color="Green" title="E80">A58</font></i>: 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">pq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v1</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E12:59_2_1_1_2_1_1_2"><i><font color="Green" title="E68">A40</font></i></a>, <a class="txt" href="graph_5.html#E13:59_2_1_1_2_1_1_2"><i><font color="Green" title="E69">A41</font></i></a>, <a class="txt" href="graph_5.html#E16:59_2_1_1_2_1_1_2"><i><font color="Green" title="E71">A43</font></i></a>, <a class="txt" href="graph_5.html#E8"><i><font color="Green" title="E7">Lm1</font></i></a></i>;<br/><font color="Maroon" title="c30">pq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">pq</font></span>)</span> = 
<font color="Maroon" title="c30">pq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">q2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c29">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">q2</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_5.html#E13:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E79">A57</font></i></a>, <a class="txt" href="graph_5.html#E9"><i><font color="Green" title="E8">Lm2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_5.html#E7:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E76">A54</font></i></a>, <a class="txt" href="graph_5.html#E9:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E77">A55</font></i></a>, <a class="txt" href="graph_5.html#E13:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E79">A57</font></i></a>, <a class="txt" href="graph_5.html#E9"><i><font color="Green" title="E8">Lm2</font></i></a></i>
;<br/><a NAME="E18:59_2_1_1_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E81">A59</font></i>: 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">pq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">pq</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">v2</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E5:59_2_1"><i><font color="Green" title="E54">A10</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></i>;<br/><a NAME="E19:59_2_1_1_2_1_1_2_2_1_2"/><i><font color="Green" title="E82">A60</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">pq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">h</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E12:59_2_1_1_2_1_1_2"><i><font color="Green" title="E68">A40</font></i></a>, <a class="txt" href="graph_5.html#E9:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E77">A55</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><a NAME="E20:59_2_1_1_2_1_1_2_2_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">h</font> &amp; 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">h</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">h</font></span>)</span> )
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E21:59_2_1_1_2_1_1_2_2_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">pq</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E19:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E82">A60</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E22:59_2_1_1_2_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c30">pq</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T28" title="FINSEQ_1:th.28">FINSEQ_1:28</a></i>;<br/><a NAME="E23:59_2_1_1_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E83">A61</font></i>: 
<font color="Maroon" title="c30">pq</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E16:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E80">A58</font></i></a>, <a class="txt" href="graph_5.html#E18:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E81">A59</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></i>;<br/><a NAME="E24:59_2_1_1_2_1_1_2_2_1_2"/><i><font color="Green" title="E84">A62</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">p</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E9:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E77">A55</font></i></a>, <a class="ref" href="finseq_1.html#T43" title="FINSEQ_1:th.43">FINSEQ_1:43</a></i>;<br/><a NAME="E25:59_2_1_1_2_1_1_2_2_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c30">pq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c23">p1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">q2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>;<br/><a NAME="E26:59_2_1_1_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E85">A63</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c30">pq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">p</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E14:59_2_1_1_2_1_1_2"><i><font color="Green" title="E70">A42</font></i></a>, <a class="txt" href="graph_5.html#E24:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E84">A62</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><a NAME="E27:59_2_1_1_2_1_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c30">pq</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E8:59_2_1"><i><font color="Green" title="E57">A13</font></i></a>, <a class="ref" href="graph_5.html#T25" target="_self" title="GRAPH_5:th.25">Th25</a></i>;<br/><a NAME="E28:59_2_1_1_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E86">A64</font></i>: 
<font color="Maroon" title="c30">pq</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E23:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E83">A61</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></i>;<br/><a NAME="E29:59_2_1_1_2_1_1_2_2_1_2"/>
<font color="Maroon" title="c14">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">h</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">h</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1_1_2"><i><font color="Green" title="E60">A16</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E30:59_2_1_1_2_1_1_2_2_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">pq</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E1:59_2_1"><i><font color="Green" title="E51">A7</font></i></a>, <a class="txt" href="graph_5.html#E4:59_2_1_1_2_1_1_2"><i><font color="Green" title="E64">A36</font></i></a>, <a class="txt" href="graph_5.html#E5:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E75">A53</font></i></a>, <a class="txt" href="graph_5.html#E19:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E82">A60</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E31:59_2_1_1_2_1_1_2_2_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">pq</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E32:59_2_1_1_2_1_1_2_2_1_2"/><b>then </b>
 <a href="graph_5.html#K6" title="GRAPH_5:func.6">AcyclicPaths</a> <font color="Maroon" title="c30">pq</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="graph_5.html#E1:59_2"><i><font color="Green" title="E50">A6</font></i></a>, <a class="txt" href="graph_5.html#E28:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E86">A64</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c31">r</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E34:59_2_1_1_2_1_1_2_2_1_2"/><i><font color="Green" title="E87">A65</font></i>: 
<font color="Maroon" title="c31">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K6" title="GRAPH_5:func.6">AcyclicPaths</a> <font color="Maroon" title="c30">pq</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c32">q</font> being  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a> <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E36:59_2_1_1_2_1_1_2_2_1_2"/><i><font color="Green" title="E88">A66</font></i>: 
( <font color="Maroon" title="c31">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">q</font> &amp; <font color="Maroon" title="c32">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v1</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">v2</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c32">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c30">pq</font> )
 <b>by </b><i><a class="txt" href="graph_5.html#E16:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E80">A58</font></i></a>, <a class="txt" href="graph_5.html#E18:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E81">A59</font></i></a>, <a class="txt" href="graph_5.html#E34:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E87">A65</font></i></a></i>;<br/><a NAME="E37:59_2_1_1_2_1_1_2_2_1_2"/><i><font color="Green" title="E89">A67</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c32">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">p</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E26:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E85">A63</font></i></a>, <a class="txt" href="graph_5.html#E36:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E88">A66</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E38:59_2_1_1_2_1_1_2_2_1_2"/>
( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_5.html#E5:59_2_1"><i><font color="Green" title="E54">A10</font></i></a>, <a class="txt" href="graph_5.html#E36:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E88">A66</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></i>;<br/><a NAME="E39:59_2_1_1_2_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c32">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a> <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font> : ( <font color="Olive" title="b1">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">w</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">p</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_5.html#E36:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E88">A66</font></i></a>, <a class="txt" href="graph_5.html#E37:59_2_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E89">A67</font></i></a></i>;<br/><b>hence </b><a NAME="E40:59_2_1_1_2_1_1_2_2_1_2"/>
 <a href="graph_5.html#K6" title="GRAPH_5:func.6">AcyclicPaths</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:59_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:59"/><i><font color="Green" title="E50">A68</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="graph_5.html#E2:59"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="graph_5.html#E3:59"><i><font color="Green" title="E49">A5</font></i></a>);<br/></i>
<a NAME="E5:59"/>
<font color="Maroon" title="c5">p</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font>
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:59"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></i>;<br/>
<a NAME="E6:59"/><b>then </b>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></i>;<br/>
<a NAME="E7:59"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T28" title="FINSEQ_1:th.28">FINSEQ_1:28</a></i>;<br/>
<a NAME="E8:59"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font>
 
<b>by </b><i><a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/>
<b>hence </b><a NAME="E9:59"/>
 <a href="graph_5.html#K6" title="GRAPH_5:func.6">AcyclicPaths</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="graph_5.html#E1:59"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="graph_5.html#E4:59"><i><font color="Green" title="E50">A68</font></i></a></i>;<br/>

<b>then consider </b><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:59"/><i><font color="Green" title="E51">A69</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K6" title="GRAPH_5:func.6">AcyclicPaths</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E12:59"/>
 <a href="graph_5.html#K6" title="GRAPH_5:func.6">AcyclicPaths</a> <font color="Maroon" title="c5">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="graph_5.html#K5" title="GRAPH_5:func.5">AcyclicPaths</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font>,<font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:59"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="graph_5.html#T45" target="_self" title="GRAPH_5:th.45">Th45</a></i>;<br/>
<b>hence </b><a NAME="E13:59"/>
 <a href="graph_5.html#K5" title="GRAPH_5:func.5">AcyclicPaths</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font>,<font color="Maroon" title="c1">V</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="graph_5.html#E11:59"><i><font color="Green" title="E51">A69</font></i></a></i>;<br/>


</div>
