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<b>let </b><font color="Maroon" title="c1">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">G</font> be  <a href="graph_1.html#V3" title="GRAPH_1:attr.3">oriented</a> <a href="graph_1.html#V7" title="GRAPH_1:attr.7">finite</a> <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/><b>let </b><font color="Maroon" title="c3">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="c2">G</font>;<br/><b>let </b><font color="Maroon" title="c4">W</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font>, <a href="graph_5.html#K8" title="GRAPH_5:func.8">Real&gt;=0</a> ;<br/><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 <font color="Maroon" title="c2">G</font>;<br/>











<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E19">A1</font></i>: 
( <font color="Maroon" title="c1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font> &amp; <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> &amp; <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">e</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Maroon" title="c1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c7">FS</font> = the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font>;<br/>
<b>set </b><font color="Maroon" title="c8">FT</font> = the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font>;<br/>
<b>set </b><font color="Maroon" title="c9">Eg</font> = the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font>;<br/>
<a NAME="E2:18"/><i><font color="Green" title="E20">A2</font></i>: 
( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v1</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">e</font> <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="graphsp.html#E1:18"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="graph_4.html#D1" title="GRAPH_4:def.1">GRAPH_4:def 1</a></i>;<br/>
<div><i><font color="Green" title="E21">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">S</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="c2">G</font>;<br/><b>assume </b><a NAME="E1:18_1"/><i><font color="Green" title="E22">A4</font></i>: 
<font color="Maroon" title="c10">S</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">v1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><a NAME="E2:18_1"/>
<font color="Maroon" title="c4">W</font> <a href="graph_5.html#R5" title="GRAPH_5:pred.5">is_weight&gt;=0of</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="graph_5.html#D13" title="GRAPH_5:def.13">GRAPH_5:def 13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">s</font> being  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">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><b> such that </b><br/><a NAME="E4:18_1"/><i><font color="Green" title="E23">A5</font></i>: 
<font color="Maroon" title="c11">s</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">v1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c4">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:18_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="graph_5.html#T66" title="GRAPH_5:th.66">GRAPH_5:66</a></i>;<br/><a NAME="E5:18_1"/><i><font color="Green" title="E24">A6</font></i>: 
<font color="Maroon" title="c11">s</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">v1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="graphsp.html#E4:18_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="graph_5.html#D18" title="GRAPH_5:def.18">GRAPH_5:def 18</a></i>;<br/><a NAME="E6:18_1"/><b>then </b><i><font color="Green" title="E25">A7</font></i>: 
<font color="Maroon" title="c11">s</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>
 <b>by </b><i><a class="ref" href="graph_5.html#D4" title="GRAPH_5:def.4">GRAPH_5:def 4</a></i>;<br/><a NAME="E7:18_1"/><b>then </b>
<font color="Maroon" title="c11">s</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="graph_5.html#D3" title="GRAPH_5:def.3">GRAPH_5:def 3</a></i>;<br/><a NAME="E8:18_1"/><b>then </b><i><font color="Green" title="E26">A8</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">s</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T28" title="FINSEQ_1:th.28">FINSEQ_1:28</a></i>;<br/><b>set </b><font color="Maroon" title="c12">Vs</font> =  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c11">s</font>;<br/><a NAME="E9:18_1"/>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c11">s</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="c6">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> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">v1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="graphsp.html#E5:18_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="graph_5.html#D4" title="GRAPH_5:def.4">GRAPH_5:def 4</a></i>;<br/><a NAME="E10:18_1"/><b>then </b>
 <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c11">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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"><font color="Maroon" title="c5">v1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T44" title="XBOOLE_1:th.44">XBOOLE_1:44</a></i>;<br/><a NAME="E11:18_1"/><b>then </b><i><font color="Green" title="E27">A9</font></i>: 
 <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c11">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</a></i>;<br/><a NAME="E12:18_1"/><i><font color="Green" title="E28">A10</font></i>: 
( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">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="c11">s</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">v1</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">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="c11">s</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="c11">s</font></span>)</span></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="graphsp.html#E6:18_1"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="graph_5.html#D3" title="GRAPH_5:def.3">GRAPH_5:def 3</a></i>;<br/><a NAME="E13:18_1"/><b>then </b>
( <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c11">s</font> &amp; <font color="Maroon" title="c6">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c11">s</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E8:18_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="txt" href="graphsp.html#E15"><i><font color="Green" title="E15">Lm4</font></i></a></i>;<br/><a NAME="E14:18_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">v1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c11">s</font> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">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>  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c11">s</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><a NAME="E15:18_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">v1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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"><font color="Maroon" title="c6">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>  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c11">s</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><a NAME="E16:18_1"/><b>then </b><i><font color="Green" title="E29">A11</font></i>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c11">s</font>
 <b>by </b><i><a class="ref" href="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E18:18_1"/><i><font color="Green" title="E30">A12</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">s</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="c13">i</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E8:18_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">s1</font>, <font color="Maroon" title="c15">s2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E20:18_1"/><i><font color="Green" title="E31">A13</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c15">s2</font> )
 <b>by </b><i><a class="txt" href="graphsp.html#E18:18_1"><i><font color="Green" title="E30">A12</font></i></a>, <a class="ref" href="finseq_2.html#T25" title="FINSEQ_2:th.25">FINSEQ_2:25</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">s1</font> = <font color="Maroon" title="c14">s1</font>, <font color="Maroon" title="c17">s2</font> = <font color="Maroon" title="c15">s2</font> as  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">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> <b>by </b><i><a class="txt" href="graphsp.html#E20:18_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="graph_5.html#T17" title="GRAPH_5:th.17">GRAPH_5:17</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_1"/>
<font color="Maroon" title="c16">s1</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/><a NAME="E2:18_1_1"/><b>then </b><i><font color="Green" title="E32">A14</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">s1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <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="E3:18_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E20:18_1"><i><font color="Green" title="E31">A13</font></i></a></i>;<br/><a NAME="E4:18_1_1"/><b>then </b><i><font color="Green" title="E33">A15</font></i>: 
( the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">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="c11">s</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="c11">s</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">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="c16">s1</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="c16">s1</font></span>)</span></span>)</span> &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">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="c11">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">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="c17">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="graphsp.html#E20:18_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="graphsp.html#E2:18_1_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="ref" href="graphsp.html#T6" target="_self" title="GRAPHSP:th.6">Th6</a></i>;<br/><a NAME="E5:18_1_1"/><i><font color="Green" title="E34">A16</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">s1</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="c11">s</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E18:18_1"><i><font color="Green" title="E30">A12</font></i></a>, <a class="txt" href="graphsp.html#E20:18_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><i><font color="Green" title="E35">A17</font></i>: the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">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="c17">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> = 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">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="c11">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">s1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E20:18_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="graphsp.html#E7"><i><font color="Green" title="E7">Lm2</font></i></a></i>
<br/>.= 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">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="c11">s</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="c16">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E2:18_1_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="txt" href="graphsp.html#E5:18_1_1"><i><font color="Green" title="E34">A16</font></i></a>, <a class="ref" href="graph_1.html#D12" title="GRAPH_1:def.12">GRAPH_1:def 12</a></i>
<br/>.= 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">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="c16">s1</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="c16">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E20:18_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="graphsp.html#E2:18_1_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="txt" href="graphsp.html#E6"><i><font color="Green" title="E6">Lm1</font></i></a></i>
;<br/><a NAME="E7:18_1_1"/>
 <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c17">s2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">s1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c17">s2</font></span>)</span>
 <b>by </b><i><a class="ref" href="graph_5.html#T30" title="GRAPH_5:th.30">GRAPH_5:30</a></i>;<br/><a NAME="E8:18_1_1"/><b>then </b><i><font color="Green" title="E36">A18</font></i>: 
 <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c17">s2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="graphsp.html#E11:18_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="graphsp.html#E16:18_1"><i><font color="Green" title="E29">A11</font></i></a>, <a class="txt" href="graphsp.html#E20:18_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">vx</font> = the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">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="c17">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> as   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="graphsp.html#E20:18_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="graphsp.html#E13"><i><font color="Green" title="E13">Lm3</font></i></a></i>;<br/><a NAME="E10:18_1_1"/>
<font color="Maroon" title="c18">vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c17">s2</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E20:18_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="graphsp.html#E15"><i><font color="Green" title="E15">Lm4</font></i></a></i>;<br/><b>hence </b><a NAME="E11:18_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="graphsp.html#E12:18_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="graphsp.html#E4:18_1_1"><i><font color="Green" title="E33">A15</font></i></a>, <a class="txt" href="graphsp.html#E6:18_1_1"><i><font color="Green" title="E35">A17</font></i></a>, <a class="txt" href="graphsp.html#E8:18_1_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><a NAME="E23:18_1"/><b>then </b><i><font color="Green" title="E32">A19</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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
 <b>by </b><i><a class="txt" href="graphsp.html#E18:18_1"><i><font color="Green" title="E30">A12</font></i></a>, <a class="txt" href="graphsp.html#E20:18_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>;<br/><a NAME="E24:18_1"/>
<font color="Maroon" title="c11">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:18"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/><a NAME="E25:18_1"/><b>then </b>
<font color="Maroon" title="c11">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E23:18_1"><i><font color="Green" title="E32">A19</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E26:18_1"/><b>then </b>
<font color="Maroon" title="c1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E1:18"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="graphsp.html#E2:18"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="graphsp.html#E12:18_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="graphsp.html#E23:18_1"><i><font color="Green" title="E32">A19</font></i></a>, <a class="ref" href="graph_1.html#D4" title="GRAPH_1:def.4">GRAPH_1:def 4</a></i>;<br/><a NAME="E27:18_1"/><b>then </b>
<font color="Maroon" title="c11">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:18"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="graphsp.html#E23:18_1"><i><font color="Green" title="E32">A19</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><b>hence </b><a NAME="E28:18_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c10">S</font>,<font color="Maroon" title="c4">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:18_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="graphsp.html#E4:18_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="graph_5.html#D18" title="GRAPH_5:def.18">GRAPH_5:def 18</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:18"/><i><font color="Green" title="E22">A20</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">P</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="c1">e</font> )
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:18"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E5:18"/><b>then </b><i><font color="Green" title="E23">A21</font></i>: 
<font color="Maroon" title="c3">P</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:18"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="graphsp.html#E2:18"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="graph_5.html#D3" title="GRAPH_5:def.3">GRAPH_5:def 3</a></i>;<br/>
<a NAME="E6:18"/><i><font color="Green" title="E24">A22</font></i>: 
<font color="Maroon" title="c3">P</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">e</font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E4:18"><i><font color="Green" title="E22">A20</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E7:18"/>
 <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K1" title="GRAPH_5:func.1">vertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E4:18"><i><font color="Green" title="E22">A20</font></i></a>, <a class="ref" href="graph_5.html#T29" title="GRAPH_5:th.29">GRAPH_5:29</a></i>;<br/>
<a NAME="E8:18"/><b>then </b>
 <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E2:18"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="graphsp.html#E6:18"><i><font color="Green" title="E24">A22</font></i></a>, <a class="ref" href="graph_5.html#D1" title="GRAPH_5:def.1">GRAPH_5:def 1</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c3">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="c6">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">v1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">v1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T40" title="XBOOLE_1:th.40">XBOOLE_1:40</a></i>
;<br/>
<a NAME="E10:18"/><b>then </b>
<font color="Maroon" title="c3">P</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">v1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="graphsp.html#E5:18"><i><font color="Green" title="E23">A21</font></i></a>, <a class="ref" href="graph_5.html#D4" title="GRAPH_5:def.4">GRAPH_5:def 4</a></i>;<br/>
<b>hence </b><a NAME="E11:18"/>
<font color="Maroon" title="c3">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">v1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c4">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E3:18"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="graph_5.html#D18" title="GRAPH_5:def.18">GRAPH_5:def 18</a></i>;<br/>


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