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<b>let </b><font color="Maroon">W</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">V</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">G</font> be  <a href="graph_1.html#V7">finite</a> <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">P</font> be  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">Q</font> be  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font>;<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">v3</font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font>;<br/>















<b>assume </b><a NAME="E1:17"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">W</font> <a href="graph_5.html#R5">is_weight&gt;=0of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">P</font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">V</font>,<font color="Maroon">W</font> &amp; <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v3</font> &amp; <font color="Maroon">Q</font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v3</font>,<font color="Maroon">V</font>,<font color="Maroon">W</font> &amp; ( for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">e</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> or  not <font color="Olive">e</font> <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">v2</font>,<font color="Maroon">v3</font> ) ) &amp; <font color="Maroon">P</font> <a href="graph_5.html#R9">islongestInShortestpath</a> <font color="Maroon">V</font>,<font color="Maroon">v1</font>,<font color="Maroon">W</font> )
 ;<br/>

<b>set </b><font color="Maroon">V'</font> = <font color="Maroon">V</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v2</font></span><a href="tarski.html#K1">}</a></span>;<br/>
<b>set </b><font color="Maroon">FS</font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">FT</font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font>;<br/>
<a NAME="E2:17"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">W</font> <a href="graph_5.html#R6">is_weight_of</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="graph_5.html#T50">GRAPH_5:50</a></i>;<br/>
<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">S</font> be  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E1:17_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">S</font> <a href="graph_5.html#R2">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v3</font>,<font color="Maroon">V</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v2</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><b>then consider </b><font color="Maroon">s</font> being  <a href="graph_1.html#V8">oriented</a> <a href="graph_4.html#V1">Simple</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:17_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">s</font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v3</font>,<font color="Maroon">V</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v2</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="graph_5.html#T66">GRAPH_5:66</a></i>;<br/><a NAME="E4:17_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">s</font> <a href="graph_5.html#R2">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v3</font>,<font color="Maroon">V</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v2</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="graph_5.html#D18">GRAPH_5:def 18</a></i>;<br/><a NAME="E5:17_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">s</font> <a href="graph_5.html#R1">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v3</font>
 <b>by </b><i><a class="ref" href="graph_5.html#D4">GRAPH_5:def 4</a></i>;<br/><a NAME="E6:17_1"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="graph_5.html#D3">GRAPH_5:def 3</a></i>;<br/><a NAME="E7:17_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="graph_5.html#T8">GRAPH_5:8</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:17_1"/><i><font color="Green">E65</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E10:17_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>then consider </b><font color="Maroon">s1</font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">s2</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E12:17_1"/><i><font color="Green">E66</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s2</font> )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T25">FINSEQ_2:25</a></i>;<br/><b>reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">s1</font>, <font color="Maroon">s2</font> = <font color="Maroon">s2</font> as  <a href="graph_1.html#V8">oriented</a> <a href="graph_4.html#V1">Simple</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="ref" href="graph_5.html#T17">GRAPH_5:17</a></i>;<br/><a NAME="E14:17_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E15:17_1"/><i><font color="Green">E68</font></i>: 
( 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">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; 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">s</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v3</font> )
 <b>by </b><i>, <a class="ref" href="graph_5.html#D3">GRAPH_5:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">vx</font> = 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">s2</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> as   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i>, </i>;<br/><div><i><font color="Green">E69</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1_1"/>
<font color="Maroon">vx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font>
 ;<br/><a NAME="E2:17_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> 1 <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">v2</font>,<font color="Maroon">v3</font>
 <b>by </b><i>, , <a class="ref" href="graph_4.html#D1">GRAPH_4:def 1</a></i>;<br/><a NAME="E3:17_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s2</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:17_1_1"/><b>then </b>
<font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> 1 <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="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><b>hence </b><a NAME="E5:17_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E18:17_1"/><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K2">vertices</a> <font color="Maroon">s</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v3</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v2</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="graph_5.html#D4">GRAPH_5:def 4</a></i>;<br/><a NAME="E19:17_1"/><i><font color="Green">E84</font></i>: 
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">s</font>,<font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">S</font>,<font color="Maroon">W</font>
 <b>by </b><i>, , <a class="ref" href="graph_5.html#D18">GRAPH_5:def 18</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_2"/>
(  not <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K2">vertices</a> <font color="Maroon">s</font> or <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v3</font> or ( <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K2">vertices</a> <font color="Maroon">s</font> &amp; <font color="Maroon">v2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v3</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2_1"><b>suppose </b></a><a NAME="E1:17_1_2_1"/><i><font color="Green">E86</font></i>: 
(  not <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K2">vertices</a> <font color="Maroon">s</font> or <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v3</font> )
 ;<br/><div class="add"><b>set </b><font color="Maroon">Vs</font> =  <a href="graph_5.html#K2">vertices</a> <font color="Maroon">s</font>;<br/><a NAME="E2:17_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graph_5.html#K2">vertices</a> <font color="Maroon">s</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v3</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v2</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T43">XBOOLE_1:43</a></i>;<br/><a NAME="E3:17_1_2_1"/><b>then </b><i><font color="Green">E98</font></i>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K2">vertices</a> <font color="Maroon">s</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v3</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v2</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T41">XBOOLE_1:41</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_2_1_1_1"/>
(  not <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K2">vertices</a> <font color="Maroon">s</font> or <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v3</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_2_1_1_1_1"/><i><font color="Green">E99</font></i>: 
not <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K2">vertices</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:17_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graph_5.html#K2">vertices</a> <font color="Maroon">s</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v2</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v3</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="graphsp.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_1.html#T41">XBOOLE_1:41</a></i>;<br/><b>hence </b><a NAME="E3:17_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="graph_5.html#K2">vertices</a> <font color="Maroon">s</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v3</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:17"><i><font color="Green">E34</font></i></a>, <a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_2_1_1_1_2"/>
<font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v3</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_1_2_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="graph_5.html#K2">vertices</a> <font color="Maroon">s</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v3</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="graphsp.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E5:17_1_2_1"/><b>then </b>
<font color="Maroon">s</font> <a href="graph_5.html#R2">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v3</font>,<font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="graph_5.html#D4">GRAPH_5:def 4</a></i>;<br/><a NAME="E6:17_1_2_1"/><b>then </b>
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">Q</font>,<font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">s</font>,<font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="graph_5.html#D18">GRAPH_5:def 18</a></i>;<br/><b>hence </b><a NAME="E7:17_1_2_1"/>
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">Q</font>,<font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">S</font>,<font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2_2"><b>suppose </b></a><a NAME="E1:17_1_2_2"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K2">vertices</a> <font color="Maroon">s</font> &amp; <font color="Maroon">v2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v3</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:17_1_2_2"/><i><font color="Green">E101</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> &amp; <font color="Maroon">v2</font> <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">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="graph_5.html#T32">GRAPH_5:32</a></i>;<br/><a NAME="E4:17_1_2_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, , , , <a class="ref" href="graphsp.html#T1" target="_self">Th1</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:17_1_2_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E6:17_1_2_2"/><b>then </b><i><font color="Green">E102</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E7:17_1_2_2"/><i><font color="Green">E103</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font>
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><i><font color="Green">E104</font></i>: <font color="Maroon">vx</font> = 
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">s</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
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">s</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E1:17"><i><font color="Green">E34</font></i></a>, <a class="ref" href="graphsp.html#T2" target="_self">Th2</a>, <a class="ref" href="graph_1.html#D12">GRAPH_1:def 12</a></i>
;<br/><div><i><font color="Green">E105</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1_2_2_2"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font>
 ;<br/><a NAME="E2:17_1_2_2_2"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><b>hence </b><a NAME="E3:17_1_2_2_2"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="graphsp.html#T1" target="_self">Th1</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:17_1_2_2"/><i><font color="Green">E106</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E12:17_1_2_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>then consider </b><font color="Maroon">t1</font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">t2</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E14:17_1_2_2"/><i><font color="Green">E109</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">t2</font> )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T25">FINSEQ_2:25</a></i>;<br/><b>reconsider </b><font color="Maroon">t1</font> = <font color="Maroon">t1</font>, <font color="Maroon">t2</font> = <font color="Maroon">t2</font> as  <a href="graph_1.html#V8">oriented</a> <a href="graph_4.html#V1">Simple</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="ref" href="graph_5.html#T17">GRAPH_5:17</a></i>;<br/><i><font color="Green">E110</font></i>: 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">t1</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> = 
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">s1</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="graphsp.html#T1" target="_self">Th1</a>, , </i>
<br/>.= 
<font color="Maroon">v1</font>
<b>by </b><i>, , <a class="txt" href="graphsp.html#E1:17"><i><font color="Green">E34</font></i></a>, </i>
;<br/><a NAME="E17:17_1_2_2"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">t1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="graphsp.html#T1" target="_self">Th1</a>, , <a class="ref" href="graph_5.html#T8">GRAPH_5:8</a></i>;<br/>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">t1</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">t1</font></span>)</span></span>)</span> = 
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">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="graphsp.html#T1" target="_self">Th1</a>, , </i>
<br/>.= 
<font color="Maroon">v2</font>
<b>by </b><i>, <a class="ref" href="graphsp.html#T1" target="_self">Th1</a>, , </i>
;<br/><a NAME="E19:17_1_2_2"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">t1</font> <a href="graph_5.html#R1">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>
 <b>by </b><i>, , <a class="ref" href="graph_5.html#D3">GRAPH_5:def 3</a></i>;<br/><a NAME="E20:17_1_2_2"/><i><font color="Green">E113</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i/>;<br/><a NAME="E21:17_1_2_2"/><b>then </b><i><font color="Green">E114</font></i>: 
not <font color="Maroon">v3</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K2">vertices</a> <font color="Maroon">s1</font>
 <b>by </b><i>, , , <a class="txt" href="graphsp.html#E1:17"><i><font color="Green">E34</font></i></a>, <a class="ref" href="graph_5.html#T21">GRAPH_5:21</a></i>;<br/><b>set </b><font color="Maroon">Vt</font> =  <a href="graph_5.html#K2">vertices</a> <font color="Maroon">t1</font>;<br/><a NAME="E22:17_1_2_2"/>
 <a href="graph_5.html#K2">vertices</a> <font color="Maroon">t1</font> <a href="tarski.html#R1">c=</a>  <a href="graph_5.html#K2">vertices</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">t2</font></span>)</span>
 <b>by </b><i><a class="ref" href="graph_5.html#T30">GRAPH_5:30</a></i>;<br/><a NAME="E23:17_1_2_2"/><b>then </b><i><font color="Green">E116</font></i>: 
not <font color="Maroon">v3</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K2">vertices</a> <font color="Maroon">t1</font>
 <b>by </b><i>, , , <a class="txt" href="graphsp.html#E1:17"><i><font color="Green">E34</font></i></a>, , , <a class="ref" href="graph_5.html#T21">GRAPH_5:21</a></i>;<br/><a NAME="E24:17_1_2_2"/>
<span class="p1">(<span class="default"><a href="graph_5.html#K2">vertices</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v3</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v2</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , <a class="ref" href="graph_5.html#D4">GRAPH_5:def 4</a></i>;<br/><a NAME="E25:17_1_2_2"/><b>then </b><i><font color="Green">E117</font></i>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K2">vertices</a> <span class="p2">(<span class="default"><font color="Maroon">t1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">t2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v3</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v2</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="graph_5.html#T26">GRAPH_5:26</a></i>;<br/><a NAME="E26:17_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="graph_5.html#K2">vertices</a> <font color="Maroon">t1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v3</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v2</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="graph_5.html#T26">GRAPH_5:26</a></i>;<br/><a NAME="E27:17_1_2_2"/><b>then </b>
 <a href="graph_5.html#K2">vertices</a> <font color="Maroon">t1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v2</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/><a NAME="E28:17_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="graph_5.html#K2">vertices</a> <font color="Maroon">t1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v2</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T43">XBOOLE_1:43</a></i>;<br/><a NAME="E29:17_1_2_2"/><b>then </b>
<font color="Maroon">t1</font> <a href="graph_5.html#R2">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="graph_5.html#D4">GRAPH_5:def 4</a></i>;<br/><a NAME="E30:17_1_2_2"/><b>then </b><i><font color="Green">E118</font></i>: 
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">P</font>,<font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">t1</font>,<font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="graph_5.html#D18">GRAPH_5:def 18</a></i>;<br/><a NAME="E31:17_1_2_2"/>
<font color="Maroon">vx</font> <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">s1</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="graphsp.html#E1:17"><i><font color="Green">E34</font></i></a>, <a class="ref" href="graphsp.html#T3" target="_self">Th3</a>, </i>;<br/><a NAME="E32:17_1_2_2"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">vx</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K2">vertices</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:17"><i><font color="Green">E34</font></i></a>, </i>;<br/><a NAME="E33:17_1_2_2"/>
not <font color="Maroon">vx</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v2</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E34:17_1_2_2"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">vx</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="graph_5.html#K2">vertices</a> <font color="Maroon">s1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v2</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E35:17_1_2_2"/>
 <a href="graph_5.html#K2">vertices</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v2</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/><a NAME="E36:17_1_2_2"/><b>then </b><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K2">vertices</a> <font color="Maroon">s1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v2</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T43">XBOOLE_1:43</a></i>;<br/><a NAME="E37:17_1_2_2"/>
not <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K2">vertices</a> <font color="Maroon">s2</font>
 <b>by </b><i>, , , <a class="txt" href="graphsp.html#E1:17"><i><font color="Green">E34</font></i></a>, <a class="ref" href="graph_5.html#T21">GRAPH_5:21</a></i>;<br/><a NAME="E38:17_1_2_2"/><b>then </b>
<font color="Maroon">vx</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v1</font>
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">q</font> being  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E40:17_1_2_2"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">q</font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">vx</font>,<font color="Maroon">V</font>,<font color="Maroon">W</font> &amp;  <a href="graph_5.html#K10">cost</a> <font color="Maroon">q</font>,<font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">P</font>,<font color="Maroon">W</font> )
 <b>by </b><i>, , <a class="ref" href="graphsp.html#T4" target="_self">Th4</a>, <a class="ref" href="graph_5.html#D19">GRAPH_5:def 19</a></i>;<br/><a NAME="E41:17_1_2_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">q</font> <a href="graph_5.html#R2">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">vx</font>,<font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="graph_5.html#D18">GRAPH_5:def 18</a></i>;<br/><a NAME="E42:17_1_2_2"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> 1 <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">vx</font>,<font color="Maroon">v3</font>
 <b>by </b><i>, , <a class="ref" href="graph_4.html#D1">GRAPH_4:def 1</a></i>;<br/><a NAME="E43:17_1_2_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">q</font> <a href="graph_5.html#R1">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">vx</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E2:17"><i><font color="Green">E52</font></i></a>, <a class="ref" href="graph_5.html#D4">GRAPH_5:def 4</a></i>;<br/><a NAME="E44:17_1_2_2"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="graph_5.html#D3">GRAPH_5:def 3</a></i>;<br/><a NAME="E45:17_1_2_2"/><b>then </b><i><font color="Green">E126</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="graph_5.html#T8">GRAPH_5:8</a></i>;<br/><b>then consider </b><font color="Maroon">p</font> being  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E47:17_1_2_2"/><i><font color="Green">E127</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">p</font> <a href="graph_5.html#R1">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v3</font> )
 <b>by </b><i>, <a class="txt" href="graphsp.html#E3:17"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graphsp.html#E1:17_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="graph_5.html#T37">GRAPH_5:37</a></i>;<br/><a NAME="E48:17_1_2_2"/>
<font color="Maroon">p</font> <a href="graph_5.html#R2">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v3</font>,<font color="Maroon">V</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">vx</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="txt" href="graphsp.html#E2:17"><i><font color="Green">E52</font></i></a>, <a class="txt" href="graphsp.html#E3:17"><i><font color="Green">E53</font></i></a>, <a class="ref" href="graphsp.html#T5" target="_self">Th5</a>, <a class="txt" href="graphsp.html#E6"><i><font color="Green">Lemma56</font></i></a>, <a class="ref" href="graph_5.html#T38">GRAPH_5:38</a></i>;<br/><a NAME="E49:17_1_2_2"/><b>then </b>
<font color="Maroon">p</font> <a href="graph_5.html#R2">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v3</font>,<font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="graphsp.html#T4" target="_self">Th4</a>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>;<br/><a NAME="E50:17_1_2_2"/><b>then </b>
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">Q</font>,<font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">p</font>,<font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="graph_5.html#D18">GRAPH_5:def 18</a></i>;<br/><a NAME="E51:17_1_2_2"/><b>then </b><i><font color="Green">E128</font></i>: 
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">Q</font>,<font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10">cost</a> <font color="Maroon">q</font>,<font color="Maroon">W</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10">cost</a> <font color="Maroon">s2</font>,<font color="Maroon">W</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="graphsp.html#E6"><i><font color="Green">Lemma56</font></i></a>, <a class="ref" href="graph_5.html#T58">GRAPH_5:58</a></i>;<br/><a NAME="E52:17_1_2_2"/><i><font color="Green">E129</font></i>: 
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">q</font>,<font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">t1</font>,<font color="Maroon">W</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E53:17_1_2_2"/><i><font color="Green">E130</font></i>: 
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">s1</font>,<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10">cost</a> <font color="Maroon">t1</font>,<font color="Maroon">W</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10">cost</a> <font color="Maroon">t2</font>,<font color="Maroon">W</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="graph_5.html#T58">GRAPH_5:58</a></i>;<br/><a NAME="E54:17_1_2_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">t2</font>,<font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="graph_5.html#T54">GRAPH_5:54</a></i>;<br/><a NAME="E55:17_1_2_2"/><b>then </b>
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10">cost</a> <font color="Maroon">t1</font>,<font color="Maroon">W</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">s1</font>,<font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E56:17_1_2_2"/><b>then </b><i><font color="Green">E131</font></i>: 
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">q</font>,<font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">s1</font>,<font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="graphsp.html#T6" target="_self">Th6</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E57:17_1_2_2"/>
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">s</font>,<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10">cost</a> <font color="Maroon">s1</font>,<font color="Maroon">W</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10">cost</a> <font color="Maroon">s2</font>,<font color="Maroon">W</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="graph_5.html#T58">GRAPH_5:58</a></i>;<br/><a NAME="E58:17_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="graph_5.html#K10">cost</a> <font color="Maroon">q</font>,<font color="Maroon">W</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10">cost</a> <font color="Maroon">s2</font>,<font color="Maroon">W</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">s</font>,<font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E59:17_1_2_2"/><b>then </b>
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">Q</font>,<font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">s</font>,<font color="Maroon">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E7"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E60:17_1_2_2"/>
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">Q</font>,<font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">S</font>,<font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E4:17"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">Q</font> <a href="graph_5.html#R2">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v3</font>,<font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="graph_5.html#D18">GRAPH_5:def 18</a></i>;<br/>
<a NAME="E5:17"/>
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v2</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E6:17"/><b>then </b>
<font color="Maroon">Q</font> <a href="graph_5.html#R2">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v3</font>,<font color="Maroon">V</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v2</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="graph_5.html#T36">GRAPH_5:36</a></i>;<br/>
<b>hence </b><a NAME="E7:17"/>
<font color="Maroon">Q</font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v3</font>,<font color="Maroon">V</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v2</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="graph_5.html#D18">GRAPH_5:def 18</a></i>;<br/>


</div>
