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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">pe</font> be    <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>;<br/><b>let </b><font color="Maroon" title="c3">v1</font>, <font color="Maroon" title="c4">v2</font>, <font color="Maroon" title="c5">v3</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c6">p</font> be  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font>;<br/>











<b>assume </b><a NAME="E1:43"/><i><font color="Green" title="E38">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c6">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> &amp; <font color="Maroon" title="c2">pe</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="c4">v2</font>,<font color="Maroon" title="c5">v3</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">pe</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/>

<b>set </b><font color="Maroon" title="c7">FT</font> = the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">G</font>;<br/>
<b>set </b><font color="Maroon" title="c8">FS</font> = the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">G</font>;<br/>
<a NAME="E2:43"/><i><font color="Green" title="E39">A2</font></i>: 
( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</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="c3">v1</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c6">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v2</font> )
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:43"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></i>;<br/>
<a NAME="E3:43"/><i><font color="Green" title="E40">A3</font></i>: 
( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">pe</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="c4">v2</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">pe</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="c5">v3</font> )
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:43"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="graph_4.html#D1" title="GRAPH_4:def.1">GRAPH_4:def 1</a></i>;<br/>
<a NAME="E4:43"/>
<font color="Maroon" title="c2">pe</font> is  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:43"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="graph_5.html#T18" target="_self" title="GRAPH_5:th.18">Th18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c6">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c2">pe</font> as  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="graph_5.html#E2:43"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="graph_5.html#E3:43"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="graph_5.html#T15" target="_self" title="GRAPH_5:th.15">Th15</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">q</font>
;<br/>

<a NAME="E6:43"/><i><font color="Green" title="E41">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</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="c6">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:43"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E7:43"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E8:43"/><b>then </b><i><font color="Green" title="E42">A5</font></i>: 
<font color="Maroon" title="c9">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> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T28" title="FINSEQ_1:th.28">FINSEQ_1:28</a></i>;<br/>
<a NAME="E9:43"/><i><font color="Green" title="E43">A6</font></i>: 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">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">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="c3">v1</font>
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:43"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="graph_5.html#E2:43"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="graph_5.html#E8"><i><font color="Green" title="E7">Lm1</font></i></a></i>;<br/>
<a NAME="E10:43"/>
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">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">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="c9">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v3</font>
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:43"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="graph_5.html#E3:43"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="graph_5.html#E6:43"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="graph_5.html#E9"><i><font color="Green" title="E8">Lm2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:43"/>
( <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c2">pe</font> &amp; <font color="Maroon" title="c9">q</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="c5">v3</font> )
 <b>by </b><i><a class="txt" href="graph_5.html#E8:43"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="graph_5.html#E9:43"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self" title="GRAPH_5:def.3">Def3</a></i>;<br/>


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