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<b>set </b><font color="Maroon" title="c4">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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">v1'</font>, <font color="Maroon" title="c6">v2'</font> be   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</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>;<br/><b>assume </b><a NAME="E1:63_1_1_1"/><i><font color="Green" title="E51">A1</font></i>: 
<font color="Maroon" title="c5">v1'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">v2'</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c7">v1</font> = <font color="Maroon" title="c5">v1'</font>, <font color="Maroon" title="c8">v2</font> = <font color="Maroon" title="c6">v2'</font> as   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="graph_3.html#D7" target="_self" title="GRAPH_3:def.7">Def7</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">c</font> being    <a href="graph_1.html#M1" title="GRAPH_1:mode.1">Chain</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c10">vs</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:63_1_1_1"/><i><font color="Green" title="E52">A2</font></i>: 
( not <font color="Maroon" title="c9">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c10">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c10">vs</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="c7">v1</font> &amp; <font color="Maroon" title="c10">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="c10">vs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v2</font> )
 <b>by </b><i><a class="txt" href="graph_3.html#E1:63_1_1_1"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="graph_3.html#T23" target="_self" title="GRAPH_3:th.23">Th23</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">c'</font> = <font color="Maroon" title="c9">c</font> as    <a href="graph_1.html#M1" title="GRAPH_1:mode.1">Chain</a> of  <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> <b>by </b><i><a class="ref" href="graph_3.html#T42" target="_self" title="GRAPH_3:th.42">Th42</a></i>;<br/><b>take </b><font color="Maroon" title="c12">c'</font> = <font color="Maroon" title="c11">c'</font>;<br/><b>reconsider </b><font color="Maroon" title="c13">vs'</font> = <font color="Maroon" title="c10">vs</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="c14">vs'</font> = <font color="Maroon" title="c13">vs'</font>;<br/><b>thus </b><a NAME="E7:63_1_1_1"/>
not <font color="Maroon" title="c12">c'</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="graph_3.html#E4:63_1_1_1"><i><font color="Green" title="E52">A2</font></i></a></i>;<br/><b>thus </b><a NAME="E8:63_1_1_1"/>
<font color="Maroon" title="c14">vs'</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c12">c'</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E4:63_1_1_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="graph_3.html#T41" target="_self" title="GRAPH_3:th.41">Th41</a></i>;<br/><b>thus </b><a NAME="E9:63_1_1_1"/>
( <font color="Maroon" title="c14">vs'</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="c5">v1'</font> &amp; <font color="Maroon" title="c14">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="c14">vs'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2'</font> )
 <b>by </b><i><a class="txt" href="graph_3.html#E4:63_1_1_1"><i><font color="Green" title="E52">A2</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:63_1_1"/>
 <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> is <a href="graph_1.html#V6" title="GRAPH_1:attr.6">connected</a>
 <b>by </b><i><a class="ref" href="graph_3.html#T23" target="_self" title="GRAPH_3:th.23">Th23</a></i>;<br/>


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