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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/><b>let </b><font color="Maroon" title="c2">v1</font>, <font color="Maroon" title="c3">v2</font> be   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c4">c</font> be    <a href="graph_1.html#M1" title="GRAPH_1:mode.1">Chain</a> of <font color="Maroon" title="c1">G</font>;<br/>







<b>set </b><font color="Maroon" title="c5">G'</font> =  <a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>;<br/>
<a NAME="E1:54"/><i><font color="Green" title="E43">A1</font></i>: 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="graph_3.html#D7" target="_self" title="GRAPH_3:def.7">Def7</a></i>;<br/>
<a NAME="E2:54"/>
<font color="Maroon" title="c4">c</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="msscyc_1.html#D1" title="MSSCYC_1:def.1">MSSCYC_1:def 1</a></i>;<br/>
<a NAME="E3:54"/><b>then </b>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> &amp; the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="graph_3.html#E1:54"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E4:54"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E5:54"/>
<font color="Maroon" title="c4">c</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msscyc_1.html#D1" title="MSSCYC_1:def.1">MSSCYC_1:def 1</a><br/>

<b>consider </b><font color="Maroon" title="c6">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E7:54"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T36" title="GRAPH_2:th.36">GRAPH_2:36</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">p'</font> = <font color="Maroon" title="c6">p</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <span class="p1">(<span class="default"><a href="graph_3.html#K8" title="GRAPH_3:func.8">AddNewEdge</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span>)</span> <b>by </b><i><a class="ref" href="graph_3.html#D7" target="_self" title="GRAPH_3:def.7">Def7</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">p'</font>
;<br/>

<b>thus </b><a NAME="E9:54"/>
<font color="Maroon" title="c7">p'</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E7:54"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="graph_3.html#T41" target="_self" title="GRAPH_3:th.41">Th41</a></i>;<br/>


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