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

<b>let </b><font color="Maroon">G</font> be   <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">v1</font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">v2</font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">p'</font> be   <a href="graph_1.html#NM4">Path</a> of  <a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>;<br/>







<b>set </b><font color="Maroon">G'</font> =  <a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>;<br/>
<b>assume </b><a NAME="E1:61"/><i><font color="Green">E30</font></i>: 
not the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p'</font>
 ;<br/>

<b>set </b><font color="Maroon">S</font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">T</font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">E</font> = the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">S'</font> = the <a href="graph_1.html#U3">Source</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">T'</font> = the <a href="graph_1.html#U4">Target</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font></span>)</span>;<br/>
<a NAME="E2:61"/>
the <a href="graph_1.html#U2">Edges</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="graph_3.html#E2"><i><font color="Green">Lemma38</font></i></a></i>;<br/>
<a NAME="E3:61"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p'</font> <a href="tarski.html#R1">c=</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E4:61"/><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p'</font> <a href="tarski.html#R1">c=</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:61_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p'</font>
 ;<br/>

<a NAME="E2:61_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:61_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:61"/>
<font color="Maroon">p'</font> is    <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:61_2"/>
<font color="Maroon">p'</font> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msscyc_1.html#D1">MSSCYC_1:def 1</a><br/>

<b>consider </b><font color="Maroon">vs'</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font></span>)</span><b> such that </b><br/><a NAME="E3:61_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">vs'</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">p'</font>
 <b>by </b><i><a class="ref" href="msscyc_1.html#D1">MSSCYC_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">vs</font> = <font color="Maroon">vs'</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> <b>by </b><i><a class="txt" href="graph_3.html#E2"><i><font color="Green">Lemma38</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">vs</font>
;<br/>

<b>thus </b><a NAME="E5:61_2"/>
<font color="Maroon">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">p'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:61_2_1"/><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="graph_2.html#D7">GRAPH_2:def 7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="graph_2.html#D7">GRAPH_2:def 7</a><br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:61_2_1"/><i><font color="Green">E48</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p'</font> )
 ;<br/>

<a NAME="E3:61_2_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">p'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="graph_1.html#R2">joins</a> <font color="Maroon">vs'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>,<font color="Maroon">vs'</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="graph_2.html#D7">GRAPH_2:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">vn'</font> = <font color="Maroon">vs'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>, <font color="Maroon">vn1'</font> = <font color="Maroon">vs'</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="graph_3.html#NM2">Vertex</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font></span>)</span> ;<br/>
<b>set </b><font color="Maroon">e</font> = <font color="Maroon">p'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>;<br/>
<a NAME="E5:61_2_1"/><i><font color="Green">E73</font></i>: 
( ( the <a href="graph_1.html#U3">Source</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vn'</font> &amp; the <a href="graph_1.html#U4">Target</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vn1'</font> ) or ( the <a href="graph_1.html#U3">Source</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vn1'</font> &amp; the <a href="graph_1.html#U4">Target</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vn'</font> ) )
 
<b>by </b><i>, <a class="ref" href="graph_1.html#D9">GRAPH_1:def 9</a></i>;<br/>
<a NAME="E6:61_2_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p'</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:61_2_1"/><b>then </b>
<font color="Maroon">p'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p'</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:61_2_1"/><b>then </b><i><font color="Green">E75</font></i>: 
( the <a href="graph_1.html#U3">Source</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> &amp; the <a href="graph_1.html#U4">Target</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" target="_self">AddNewEdge</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">vn</font> = <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>, <font color="Maroon">vn1</font> = <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font> ;<br/>
<a NAME="E10:61_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E11:61_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:61_2_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vs</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green">E77</font></i>: <font color="Maroon">vn</font> = 
<font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">vn'</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<a NAME="E14:61_2_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E15:61_2_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vs</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><font color="Maroon">vn1</font> = 
<font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">vn1'</font>
<b>by </b><i><a class="ref" href="graph_3.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E17:61_2_1"/>
<font color="Maroon">p'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="graph_1.html#R2">joins</a> <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>,<font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , , <a class="ref" href="graph_1.html#D9">GRAPH_1:def 9</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">p''</font> = <font color="Maroon">p'</font> as    <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font> ;<br/>
<a NAME="E7:61"/>
<font color="Maroon">p''</font> is <a href="funct_1.html#V2">one-to-one</a>
 
;<br/>
<b>hence </b><a NAME="E8:61"/>
<font color="Maroon">p'</font> is   <a href="graph_1.html#NM4">Path</a> of <font color="Maroon">G</font>
 ;<br/>


</div>
