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

<span class="kw">let </span><font color="Maroon" title="c1">V</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G</font> being   <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a><br/>  for <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">v2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">G</font><br/>  for <font color="Olive" title="b4">p</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Olive" title="b1">G</font>  st <font color="Olive" title="b4">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="b4">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>  &amp;  <a href="graph_5.html#K5" title="GRAPH_5:func.5">AcyclicPaths</a> (<font color="Olive" title="b2">v1</font>,<font color="Olive" title="b3">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>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">G</font> be   <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><br/>  for <font color="Olive" title="b3">p</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font>  st <font color="Olive" title="b3">p</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">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="b3">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>  &amp;  <a href="graph_5.html#K5" title="GRAPH_5:func.5">AcyclicPaths</a> (<font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">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>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">v1</font>, <font color="Maroon" title="c4">v2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font>  st <font color="Olive" title="b1">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> 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>  &amp;  <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>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">p</font> be   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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> implies (  <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>  &amp;  <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>  ) )</span><br/>









<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">p</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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/>
<span class="kw">set </span><font color="Maroon" title="c6">FS</font> =  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c2">G</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">FT</font> =  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c2">G</font>;<br/>
<span class="kw">assume </span><a NAME="E1:57"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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>  &amp;  <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>  )</span><br/>

<a NAME="E2:57"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></span>;<br/>
<a NAME="E3:57"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/>
<a NAME="E4:57"/><span class="kw">then </span>
 <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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T20" title="FINSEQ_1:th.20">FINSEQ_1:20</a></span>;<br/>
<a NAME="E5:57"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A2</font></span>: 
<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="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/>
<a NAME="E6:57"/><span class="lab"><font color="Green" title="E47">A3</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <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="57_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>] implies <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] )</span><br/>

<span class="kw">assume </span><a NAME="E1:57_1"/><span class="lab"><font color="Green" title="E48">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">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]</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">p</font> be   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="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="Olive" title="b1">v1</font>,<font color="Olive" title="b2">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">b<sub>3</sub></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> </span><br/><span class="kw">let </span><font color="Maroon" title="c10">v1</font>, <font color="Maroon" title="c11">v2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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> implies  <a href="graph_5.html#K6" title="GRAPH_5:func.6">AcyclicPaths</a> <font color="Olive">b<sub>1</sub></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>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:57_1_1"/><span class="lab"><font color="Green" title="E49">A5</font></span>: 
 <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
 <span class="kw">and </span><br/><a NAME="E2:57_1_1"/><span class="lab"><font color="Green" title="E50">A6</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="graph_5.html#K6" title="GRAPH_5:func.6">AcyclicPaths</a> <font color="Olive">b<sub>1</sub></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> </span><br/><span class="kw">consider </span><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="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font><span class="kw"> such that </span><br/><a NAME="E4:57_1_1"/><span class="lab"><font color="Green" title="E51">A7</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_4.html#T9" title="GRAPH_4:th.9">GRAPH_4:9</a></span>;<br/><a NAME="E5:57_1_1"/><span class="lab"><font color="Green" title="E52">A8</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57_1_1"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></span>;<br/><a NAME="E6:57_1_1"/><span class="lab"><font color="Green" title="E53">A9</font></span>: 
 <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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1_1"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="ref" href="graph_4.html#D5" title="GRAPH_4:def.5">GRAPH_4:def 5</a></span>;<br/><a NAME="E7:57_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A10</font></span>: 
 <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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E8:57_1_1"/><span class="lab"><font color="Green" title="E55">A11</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57_1_1"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></span>;<br/><a NAME="E9:57_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A12</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/><a NAME="E10:57_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A13</font></span>: 
 <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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T20" title="FINSEQ_1:th.20">FINSEQ_1:20</a></span>;<br/><a NAME="E11:57_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1_1"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="ref" href="graph_4.html#D5" title="GRAPH_4:def.5">GRAPH_4:def 5</a></span>;<br/><a NAME="E12:57_1_1"/><span class="kw">then </span>
 the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_4.html#D1" title="GRAPH_4:def.1">GRAPH_4:def 1</a></span>;<br/><a NAME="E13:57_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A14</font></span>: 
 the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:57_1_1"><span class="lab"><font color="Green" title="E53">A9</font></span></a>, <a class="txt" href="#E7:57_1_1"><span class="lab"><font color="Green" title="E54">A10</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:57_1_1_1"/>
(  for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 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#NR3" title="XXREAL_0:NR.3">&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#M1" title="SUBSET_1:mode.1">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#NR3" title="XXREAL_0:NR.3">&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="57_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:57_1_1_1_1"/>
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 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#NR3" title="XXREAL_0:NR.3">&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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="graph_5.html#K6" title="GRAPH_5:func.6">AcyclicPaths</a> <font color="Olive">b<sub>1</sub></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> </span><br/><div class="add"><a NAME="E2:57_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">p</font> is  <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1_1"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="ref" href="graph_4.html#D7" title="GRAPH_4:def.7">GRAPH_4:def 7</a></span>;<br/><a NAME="E3:57_1_1_1_1"/><span class="kw">then </span>
<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#V7" title="GRAPH_1:attr.7">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#U1" title="GRAPH_1:sel.1">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#U1" title="GRAPH_1:sel.1">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#U2" title="GRAPH_1:sel.2">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#U2" title="GRAPH_1:sel.2">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57_1_1"><span class="lab"><font color="Green" title="E56">A12</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:57_1_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:57_1_1_1_2"/>
 ex <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 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#NR3" title="XXREAL_0:NR.3">&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> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="graph_5.html#K6" title="GRAPH_5:func.6">AcyclicPaths</a> <font color="Olive">b<sub>1</sub></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> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">n</font>, <font color="Maroon" title="c14">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:57_1_1_1_2"/><span class="lab"><font color="Green" title="E59">A15</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font>
 <span class="kw">and </span><br/><a NAME="E4:57_1_1_1_2"/><span class="lab"><font color="Green" title="E60">A16</font></span>: 
<font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">m</font>
 <span class="kw">and </span><br/><a NAME="E5:57_1_1_1_2"/><span class="lab"><font color="Green" title="E61">A17</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E6:57_1_1_1_2"/><span class="lab"><font color="Green" title="E62">A18</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:57_1_1_1_2"/><span class="lab"><font color="Green" title="E63">A19</font></span>: 
(  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/><span class="kw">consider </span><font color="Maroon" title="c15">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E9:57_1_1_1_2"/><span class="lab"><font color="Green" title="E64">A20</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_1_1_1_2"><span class="lab"><font color="Green" title="E59">A15</font></span></a>, <a class="txt" href="#E4:57_1_1_1_2"><span class="lab"><font color="Green" title="E60">A16</font></span></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">i</font> = <font color="Maroon" title="c15">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E11:57_1_1_1_2"/><span class="lab"><font color="Green" title="E65">A21</font></span>: 
<font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_1_1_1_2"><span class="lab"><font color="Green" title="E59">A15</font></span></a>, <a class="txt" href="#E4:57_1_1_1_2"><span class="lab"><font color="Green" title="E60">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:57_1_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="57_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:57_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E66">A22</font></span>: 
<font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="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> </span><br/><div class="add"><a NAME="E2:57_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:57_1_1_1_2"><span class="lab"><font color="Green" title="E61">A17</font></span></a>, <a class="txt" href="#E7:57_1_1_1_2"><span class="lab"><font color="Green" title="E63">A19</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:57_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A23</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:57_1_1"><span class="lab"><font color="Green" title="E53">A9</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:57_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E68">A24</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><a NAME="E6:57_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E69">A25</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1_1"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="txt" href="#E10:57_1_1"><span class="lab"><font color="Green" title="E57">A13</font></span></a>, <a class="ref" href="graph_4.html#D5" title="GRAPH_4:def.5">GRAPH_4:def 5</a></span>;<br/><a NAME="E7:57_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E70">A26</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1_1_1_2"><span class="lab"><font color="Green" title="E60">A16</font></span></a>, <a class="txt" href="#E5:57_1_1_1_2"><span class="lab"><font color="Green" title="E61">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:57_1_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1_1"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="txt" href="#E4:57_1_1_1_2"><span class="lab"><font color="Green" title="E60">A16</font></span></a>, <a class="txt" href="#E1:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E66">A22</font></span></a>, <a class="txt" href="#E3:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E67">A23</font></span></a>, <a class="ref" href="graph_4.html#D5" title="GRAPH_4:def.5">GRAPH_4:def 5</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A27</font></span>:  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_4.html#D1" title="GRAPH_4:def.1">GRAPH_4:def 1</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1_1_1_2"><span class="lab"><font color="Green" title="E60">A16</font></span></a>, <a class="txt" href="#E5:57_1_1_1_2"><span class="lab"><font color="Green" title="E61">A17</font></span></a>, <a class="txt" href="#E1:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E66">A22</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_1_1_1_2"><span class="lab"><font color="Green" title="E59">A15</font></span></a>, <a class="txt" href="#E6:57_1_1_1_2"><span class="lab"><font color="Green" title="E62">A18</font></span></a>, <a class="txt" href="#E7:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E70">A26</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>
<br/>.= 
 the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E66">A22</font></span></a>, <a class="txt" href="#E6:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A25</font></span></a>, <a class="ref" href="graph_4.html#D1" title="GRAPH_4:def.1">GRAPH_4:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c10">v1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57_1_1"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>
;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c17">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E11:57_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E72">A28</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57_1_1_1_2"><span class="lab"><font color="Green" title="E64">A20</font></span></a>, <a class="txt" href="#E5:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E68">A24</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E13:57_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E73">A29</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E14:57_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E74">A30</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E15:57_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E75">A31</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T22" title="FINSEQ_2:th.22">FINSEQ_2:22</a></span>;<br/><a NAME="E16:57_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E76">A32</font></span>: 
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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A30</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E17:57_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A33</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57_1_1_1_2"><span class="lab"><font color="Green" title="E64">A20</font></span></a>, <a class="txt" href="#E13:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E73">A29</font></span></a>, <a class="txt" href="#E15:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E75">A31</font></span></a>, <a class="txt" href="graph_5.html#E7"><span class="lab"><font color="Green" title="E7">Lm2</font></span></a></span>;<br/><a NAME="E18:57_1_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A28</font></span></a>, <a class="txt" href="#E13:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E73">A29</font></span></a>, <a class="txt" href="#E14:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A30</font></span></a>, <a class="txt" href="#E15:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E75">A31</font></span></a>, <a class="txt" href="#E16:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A32</font></span></a>, <a class="txt" href="graph_5.html#E7"><span class="lab"><font color="Green" title="E7">Lm2</font></span></a></span>;<br/><a NAME="E19:57_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A34</font></span>: 
 the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57_1_1"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/><span class="kw">reconsider </span><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#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E75">A31</font></span></a>, <a class="ref" href="graph_5.html#T11" target="_self" title="GRAPH_5:th.11">Th14</a></span>;<br/><a NAME="E21:57_1_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1_1_1_2"><span class="lab"><font color="Green" title="E60">A16</font></span></a>, <a class="txt" href="#E9:57_1_1_1_2"><span class="lab"><font color="Green" title="E64">A20</font></span></a>, <a class="txt" href="#E1:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E66">A22</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E22:57_1_1_1_2_1_1_1"/><span class="kw">then </span>
 <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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A28</font></span></a>, <a class="txt" href="#E14:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A30</font></span></a>, <a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/><a NAME="E23:57_1_1_1_2_1_1_1"/><span class="kw">then </span>
 <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#NR3" title="XXREAL_0:NR.3">&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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1_1"><span class="lab"><font color="Green" title="E49">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E24:57_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A35</font></span>: 
 <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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E25:57_1_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57_1_1"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="txt" href="#E15:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E75">A31</font></span></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></span>;<br/><a NAME="E26:57_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A36</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#T23" target="_self" title="GRAPH_5:th.23">Th26</a></span>;<br/><a NAME="E27:57_1_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A30</font></span></a></span>;<br/><a NAME="E28:57_1_1_1_2_1_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A27</font></span></a>, <a class="txt" href="#E17:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E77">A33</font></span></a>, <a class="txt" href="#E19:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E78">A34</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/><a NAME="E29:57_1_1_1_2_1_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E80">A36</font></span></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></span>;<br/><a NAME="E30:57_1_1_1_2_1_1_1"/><span class="kw">then </span>
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="txt" href="#E24:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E79">A35</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">r</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E32:57_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E81">A37</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c24">q</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">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><span class="kw"> such that </span><br/><a NAME="E34:57_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c23">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q</font>
 <span class="kw">and </span><br/><a NAME="E35:57_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E82">A38</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E36:57_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E83">A39</font></span>: 
(  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">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#U2" title="GRAPH_1:sel.2">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> )
 <span class="kw">and </span><br/><a NAME="E37:57_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E84">A40</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A27</font></span></a>, <a class="txt" href="#E17:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E77">A33</font></span></a>, <a class="txt" href="#E19:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E78">A34</font></span></a>, <a class="txt" href="#E32:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E81">A37</font></span></a></span>;<br/><a NAME="E38:57_1_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E75">A31</font></span></a>, <a class="ref" href="finseq_1.html#T30" title="FINSEQ_1:th.30">FINSEQ_1:30</a></span>;<br/><a NAME="E39:57_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A41</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E84">A40</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E40:57_1_1_1_2_1_1_1"/>
(  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">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#U1" title="GRAPH_1:sel.1">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#U2" title="GRAPH_1:sel.2">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#U2" title="GRAPH_1:sel.2">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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57_1_1"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="txt" href="#E36:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E83">A39</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/><a NAME="E41:57_1_1_1_2_1_1_1"/><span class="kw">then </span>
<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#V7" title="GRAPH_1:attr.7">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#U1" title="GRAPH_1:sel.1">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#U1" title="GRAPH_1:sel.1">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#U2" title="GRAPH_1:sel.2">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#U2" title="GRAPH_1:sel.2">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E82">A38</font></span></a>, <a class="txt" href="#E39:57_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E85">A41</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E42:57_1_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:57_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E66">A42</font></span>: 
<font color="Maroon" title="c13">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="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> </span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c17">n1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:57_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E67">A43</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_1_1_1_2"><span class="lab"><font color="Green" title="E59">A15</font></span></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">n1</font> = <font color="Maroon" title="c17">n1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E5:57_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E68">A44</font></span>: 
<font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1_1_1_2"><span class="lab"><font color="Green" title="E60">A16</font></span></a>, <a class="txt" href="#E5:57_1_1_1_2"><span class="lab"><font color="Green" title="E61">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:57_1_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A45</font></span>: 
<font color="Maroon" title="c18">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:57_1_1"><span class="lab"><font color="Green" title="E53">A9</font></span></a>, <a class="txt" href="#E3:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E67">A43</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:57_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E70">A46</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">j</font> = <font color="Maroon" title="c19">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E11:57_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E71">A47</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E12:57_1_1_1_2_1_1_2"/>
 <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>
 <span class="kw">and </span><br/><a NAME="E13:57_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E72">A48</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E70">A46</font></span></a>, <a class="ref" href="finseq_2.html#T22" title="FINSEQ_2:th.22">FINSEQ_2:22</a></span>;<br/><a NAME="E14:57_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E73">A49</font></span>: 
<font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_1_1_1_2"><span class="lab"><font color="Green" title="E59">A15</font></span></a>, <a class="txt" href="#E1:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E66">A42</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E15:57_1_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A50</font></span>: 
<font color="Maroon" title="c18">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E67">A43</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E16:57_1_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A51</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="txt" href="#E13:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A48</font></span></a>, <a class="txt" href="graph_5.html#E6"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>;<br/><a NAME="E17:57_1_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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="txt" href="#E13:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A48</font></span></a>, <a class="txt" href="#E15:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A50</font></span></a>, <a class="txt" href="graph_5.html#E6"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>;<br/><a NAME="E18:57_1_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A52</font></span>: 
 the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57_1_1"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/><a NAME="E19:57_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E77">A53</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A48</font></span></a>, <a class="ref" href="finseq_1.html#T29" title="FINSEQ_1:th.29">FINSEQ_1:29</a></span>;<br/><span class="kw">reconsider </span><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#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A48</font></span></a>, <a class="ref" href="graph_5.html#T11" target="_self" title="GRAPH_5:th.11">Th14</a></span>;<br/><a NAME="E21:57_1_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1_1"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="txt" href="#E6:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E69">A45</font></span></a>, <a class="txt" href="#E15:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A50</font></span></a>, <a class="ref" href="graph_4.html#D5" title="GRAPH_4:def.5">GRAPH_4:def 5</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A54</font></span>:  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E67">A43</font></span></a>, <a class="ref" href="graph_4.html#D1" title="GRAPH_4:def.1">GRAPH_4:def 1</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_1_1_1_2"><span class="lab"><font color="Green" title="E59">A15</font></span></a>, <a class="txt" href="#E5:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E68">A44</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>
;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_1_2_1_1_2_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:57_1_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="57_1_1_1_2_1_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:57_1_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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> </span><br/><div class="add"><a NAME="E2:57_1_1_1_2_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A55</font></span>: 
 the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57_1_1"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="txt" href="#E13:57_1_1"><span class="lab"><font color="Green" title="E58">A14</font></span></a>, <a class="txt" href="#E6:57_1_1_1_2"><span class="lab"><font color="Green" title="E62">A18</font></span></a>, <a class="txt" href="#E22:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A54</font></span></a>, <a class="txt" href="#E16:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A51</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/><a NAME="E3:57_1_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57_1_1"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="txt" href="#E13:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A48</font></span></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></span>;<br/><a NAME="E4:57_1_1_1_2_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A56</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#T23" target="_self" title="GRAPH_5:th.23">Th26</a></span>;<br/><a NAME="E5:57_1_1_1_2_1_1_2_2_1_1"/>
<font color="Maroon" title="c18">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1_1"><span class="lab"><font color="Green" title="E49">A5</font></span></a>, <a class="txt" href="#E6:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E69">A45</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:57_1_1_1_2_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A57</font></span>: 
 <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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E7:57_1_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A49</font></span></a>, <a class="txt" href="#E3:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E67">A43</font></span></a>, <a class="txt" href="#E11:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E8:57_1_1_1_2_1_1_2_2_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E76">A52</font></span></a>, <a class="txt" href="#E2:57_1_1_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E79">A55</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/><a NAME="E9:57_1_1_1_2_1_1_2_2_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1_1_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E80">A56</font></span></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></span>;<br/><a NAME="E10:57_1_1_1_2_1_1_2_2_1_1"/><span class="kw">then </span>
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="txt" href="#E6:57_1_1_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E81">A57</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">r</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E12:57_1_1_1_2_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E82">A58</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c26">q</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">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><span class="kw"> such that </span><br/><a NAME="E14:57_1_1_1_2_1_1_2_2_1_1"/>
<font color="Maroon" title="c25">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q</font>
 <span class="kw">and </span><br/><a NAME="E15:57_1_1_1_2_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E83">A59</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E16:57_1_1_1_2_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E84">A60</font></span>: 
 the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">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>
 <span class="kw">and </span><br/><a NAME="E17:57_1_1_1_2_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E85">A61</font></span>: 
 the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">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>
 <span class="kw">and </span><br/><a NAME="E18:57_1_1_1_2_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E86">A62</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E76">A52</font></span></a>, <a class="txt" href="#E2:57_1_1_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E79">A55</font></span></a>, <a class="txt" href="#E12:57_1_1_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E82">A58</font></span></a></span>;<br/><a NAME="E19:57_1_1_1_2_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E87">A63</font></span>: 
 the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">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#U2" title="GRAPH_1:sel.2">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57_1_1"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="txt" href="#E17:57_1_1_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E85">A61</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/><a NAME="E20:57_1_1_1_2_1_1_2_2_1_1"/>
(  <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> &amp;  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">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#U1" title="GRAPH_1:sel.1">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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57_1_1"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="txt" href="#E19:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E77">A53</font></span></a>, <a class="txt" href="#E16:57_1_1_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E84">A60</font></span></a>, <a class="txt" href="#E18:57_1_1_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E86">A62</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E21:57_1_1_1_2_1_1_2_2_1_1"/><span class="kw">then </span>
<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#V7" title="GRAPH_1:attr.7">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#U1" title="GRAPH_1:sel.1">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#U1" title="GRAPH_1:sel.1">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#U2" title="GRAPH_1:sel.2">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#U2" title="GRAPH_1:sel.2">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:57_1_1_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E83">A59</font></span></a>, <a class="txt" href="#E19:57_1_1_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E87">A63</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E22:57_1_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_1_2_1_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:57_1_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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> </span><br/><div class="add"><a NAME="E2:57_1_1_1_2_1_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">vs</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:57_1_1_1_2"><span class="lab"><font color="Green" title="E61">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:57_1_1_1_2_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A64</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:57_1_1"><span class="lab"><font color="Green" title="E53">A9</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">h</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:57_1_1_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E80">A65</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><a NAME="E6:57_1_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1_1"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="txt" href="#E11:57_1_1_1_2"><span class="lab"><font color="Green" title="E65">A21</font></span></a>, <a class="txt" href="#E3:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E79">A64</font></span></a>, <a class="ref" href="graph_4.html#D5" title="GRAPH_4:def.5">GRAPH_4:def 5</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A66</font></span>:  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_4.html#D1" title="GRAPH_4:def.1">GRAPH_4:def 1</a></span>
<br/>.= 
 the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:57_1_1_1_2"><span class="lab"><font color="Green" title="E61">A17</font></span></a>, <a class="txt" href="#E6:57_1_1_1_2"><span class="lab"><font color="Green" title="E62">A18</font></span></a>, <a class="txt" href="#E11:57_1_1_1_2"><span class="lab"><font color="Green" title="E65">A21</font></span></a>, <a class="txt" href="#E22:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A54</font></span></a>, <a class="txt" href="#E16:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A51</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>
;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">h</font> = <font color="Maroon" title="c25">h</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E9:57_1_1_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E82">A67</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57_1_1_1_2"><span class="lab"><font color="Green" title="E64">A20</font></span></a>, <a class="txt" href="#E5:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E80">A65</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E11:57_1_1_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E83">A68</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E12:57_1_1_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E84">A69</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E13:57_1_1_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E85">A70</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T22" title="FINSEQ_2:th.22">FINSEQ_2:22</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c29">q2</font> = <font color="Maroon" title="c28">q2</font> as   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E85">A70</font></span></a>, <a class="ref" href="graph_5.html#T11" target="_self" title="GRAPH_5:th.11">Th14</a></span>;<br/><a NAME="E15:57_1_1_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E86">A71</font></span>: 
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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E84">A69</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E16:57_1_1_1_2_1_1_2_2_1_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57_1_1_1_2"><span class="lab"><font color="Green" title="E64">A20</font></span></a>, <a class="txt" href="#E11:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E83">A68</font></span></a>, <a class="txt" href="#E13:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E85">A70</font></span></a>, <a class="txt" href="graph_5.html#E7"><span class="lab"><font color="Green" title="E7">Lm2</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E81">A66</font></span></a>, <a class="ref" href="graph_5.html#T12" target="_self" title="GRAPH_5:th.12">Th15</a></span>;<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>
<br/>.= 
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E86">A71</font></span></a>, <a class="txt" href="graph_5.html#E7"><span class="lab"><font color="Green" title="E7">Lm2</font></span></a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E82">A67</font></span></a>, <a class="txt" href="#E11:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E83">A68</font></span></a>, <a class="txt" href="#E12:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E84">A69</font></span></a>, <a class="txt" href="#E13:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E85">A70</font></span></a>, <a class="txt" href="#E15:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E86">A71</font></span></a>, <a class="txt" href="graph_5.html#E7"><span class="lab"><font color="Green" title="E7">Lm2</font></span></a></span>
;<br/><a NAME="E19:57_1_1_1_2_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A72</font></span>: 
 the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57_1_1"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/><a NAME="E20:57_1_1_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E88">A73</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T31" title="FINSEQ_1:th.31">FINSEQ_1:31</a></span>;<br/><a NAME="E21:57_1_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="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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E85">A70</font></span></a>, <a class="ref" href="finseq_1.html#T30" title="FINSEQ_1:th.30">FINSEQ_1:30</a></span>;<br/><a NAME="E22:57_1_1_1_2_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A74</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E77">A53</font></span></a>, <a class="txt" href="#E20:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E88">A73</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/><a NAME="E23:57_1_1_1_2_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A75</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:57_1_1"><span class="lab"><font color="Green" title="E52">A8</font></span></a>, <a class="ref" href="graph_5.html#T22" target="_self" title="GRAPH_5:th.22">Th25</a></span>;<br/><a NAME="E24:57_1_1_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E91">A76</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="txt" href="#E12:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E84">A69</font></span></a>, <a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/><a NAME="E25:57_1_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1_1_1_2"><span class="lab"><font color="Green" title="E60">A16</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E26:57_1_1_1_2_1_1_2_2_1_2"/><span class="kw">then </span>
 <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#NR3" title="XXREAL_0:NR.3">&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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1_1"><span class="lab"><font color="Green" title="E49">A5</font></span></a>, <a class="txt" href="#E3:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E67">A43</font></span></a>, <a class="txt" href="#E5:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E80">A65</font></span></a>, <a class="txt" href="#E24:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E91">A76</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E27:57_1_1_1_2_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A77</font></span>: 
 <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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E28:57_1_1_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E93">A78</font></span>: 
 the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="txt" href="#E15:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A50</font></span></a>, <a class="txt" href="#E18:57_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E76">A52</font></span></a>, <a class="txt" href="graph_5.html#E6"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>;<br/><a NAME="E29:57_1_1_1_2_1_1_2_2_1_2"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E91">A76</font></span></a></span>;<br/><a NAME="E30:57_1_1_1_2_1_1_2_2_1_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E93">A78</font></span></a>, <a class="txt" href="#E19:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E87">A72</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/><a NAME="E31:57_1_1_1_2_1_1_2_2_1_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E90">A75</font></span></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></span>;<br/><a NAME="E32:57_1_1_1_2_1_1_2_2_1_2"/><span class="kw">then </span>
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="txt" href="#E27:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E92">A77</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">r</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E34:57_1_1_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E94">A79</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c32">q</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">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><span class="kw"> such that </span><br/><a NAME="E36:57_1_1_1_2_1_1_2_2_1_2"/>
<font color="Maroon" title="c31">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">q</font>
 <span class="kw">and </span><br/><a NAME="E37:57_1_1_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E95">A80</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E38:57_1_1_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E96">A81</font></span>: 
(  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">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#U2" title="GRAPH_1:sel.2">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> )
 <span class="kw">and </span><br/><a NAME="E39:57_1_1_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E97">A82</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E93">A78</font></span></a>, <a class="txt" href="#E19:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E87">A72</font></span></a>, <a class="txt" href="#E34:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E94">A79</font></span></a></span>;<br/><a NAME="E40:57_1_1_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E98">A83</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E89">A74</font></span></a>, <a class="txt" href="#E39:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E97">A82</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E41:57_1_1_1_2_1_1_2_2_1_2"/>
(  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">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#U1" title="GRAPH_1:sel.1">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#U2" title="GRAPH_1:sel.2">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#U2" title="GRAPH_1:sel.2">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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57_1_1"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="txt" href="#E38:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E96">A81</font></span></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/><a NAME="E42:57_1_1_1_2_1_1_2_2_1_2"/><span class="kw">then </span>
<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#V7" title="GRAPH_1:attr.7">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#U1" title="GRAPH_1:sel.1">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#U1" title="GRAPH_1:sel.1">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#U2" title="GRAPH_1:sel.2">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#U2" title="GRAPH_1:sel.2">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E95">A80</font></span></a>, <a class="txt" href="#E40:57_1_1_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E98">A83</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E43:57_1_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:57_1"/>
<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]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:57"/><span class="lab"><font color="Green" title="E48">A84</font></span>: 
<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="57_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">p</font> be   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="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="Olive" title="b1">v1</font>,<font color="Olive" title="b2">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="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> </span><br/><span class="kw">let </span><font color="Maroon" title="c9">v1</font>, <font color="Maroon" title="c10">v2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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> implies  <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>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:57_2"/><span class="lab"><font color="Green" title="E49">A85</font></span>: 
 <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
 <span class="kw">and </span><br/><a NAME="E2:57_2"/><span class="lab"><font color="Green" title="E50">A86</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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> </span><br/>

<a NAME="E3:57_2"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57_2"><span class="lab"><font color="Green" title="E50">A86</font></span></a>, <a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></span>;<br/>
<a NAME="E4:57_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A87</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></span>;<br/>
<a NAME="E5:57_2"/><span class="kw">then </span>
 <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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T20" title="FINSEQ_1:th.20">FINSEQ_1:20</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">r</font> = <font color="Maroon" title="c8">p</font> as   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_2"><span class="lab"><font color="Green" title="E49">A85</font></span></a>, <a class="ref" href="graph_5.html#T15" target="_self" title="GRAPH_5:th.15">Th18</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E7:57_2"/>
<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#V7" title="GRAPH_1:attr.7">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#U1" title="GRAPH_1:sel.1">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#U1" title="GRAPH_1:sel.1">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#U2" title="GRAPH_1:sel.2">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#U2" title="GRAPH_1:sel.2">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_2"><span class="lab"><font color="Green" title="E51">A87</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:57_2"/>
 <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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:57"/>
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E7:57"><span class="lab"><font color="Green" title="E48">A84</font></span></a>, <a class="txt" href="#E6:57"><span class="lab"><font color="Green" title="E47">A3</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E9:57"/>
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E5:57"><span class="lab"><font color="Green" title="E46">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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> </span><br/>

<a NAME="E10:57"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A88</font></span>: 
 ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K6" title="GRAPH_5:func.6">AcyclicPaths</a> <font color="Maroon" title="c5">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E11:57"/>
 <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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="graph_5.html#T41" target="_self" title="GRAPH_5:th.41">Th45</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:57"/>
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:57"><span class="lab"><font color="Green" title="E49">A88</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
