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<b>let </b><font color="Maroon">V</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">W</font> be   <a href="funct_1.html#NM1">Function</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">v1</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">v2</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font>;<br/>











<b>assume </b><a NAME="E1:95"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">W</font> <a href="graph_5.html#R5" target="_self">is_weight&gt;=0of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">P</font> <a href="graph_5.html#R8" target="_self">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; ( for <font color="Olive">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><br/> for <font color="Olive">v3</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font>  st not <font color="Olive">v3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Olive">Q</font> <a href="graph_5.html#R8" target="_self">is_shortestpath_of</a> <font color="Maroon">v1</font>,<font color="Olive">v3</font>,<font color="Maroon">V</font>,<font color="Maroon">W</font> holds <br/> <a href="graph_5.html#K10" target="_self">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" target="_self">cost</a> <font color="Olive">Q</font>,<font color="Maroon">W</font> ) )
 ;<br/>

<a NAME="E2:95"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">P</font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">V</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:95"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">P</font> <a href="graph_5.html#R1" target="_self">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:95"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">W</font> <a href="graph_5.html#R6" target="_self">is_weight_of</a> <font color="Maroon">G</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:95"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 
<b>by </b><i>, , <a class="ref" href="graph_5.html#T2" target="_self">Th2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</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="E1:95_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">r</font> <a href="graph_5.html#R1" target="_self">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_1"/>
(  not  <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> or  <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1_1"><b>suppose </b></a><a NAME="E1:95_1_1_1"/>
not  <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">s</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>, <font color="Maroon">t</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:95_1_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> &amp; not  <a href="graph_5.html#K1" target="_self">vertices</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">t</font> &amp;  <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 <b>by </b><i/>;<br/><a NAME="E4:95_1_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i><a class="ref" href="graph_5.html#T11" target="_self">Th11</a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E5:95_1_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="graph_5.html#T11" target="_self">Th11</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>then consider </b><font color="Maroon">j</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:95_1_1_1"/><i><font color="Green">E62</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of  <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>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font>, <font color="Maroon">t</font> = <font color="Maroon">t</font> as  <a href="graph_1.html#V8">oriented</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#T11" target="_self">Th11</a>, </i>;<br/><b>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="E11:95_1_1_1"/><i><font color="Green">E63</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">t</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_1.html#M2">Chain</a> of <font color="Maroon">G</font> <b>by </b><i>, </i>;<br/><a NAME="E13:95_1_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">t1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">t2</font>
 <b>by </b><i><a class="ref" href="graph_5.html#T11" target="_self">Th11</a>, , <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><a NAME="E14:95_1_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">r</font>,<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" target="_self">cost</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">t1</font></span>)</span>,<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" target="_self">cost</a> <font color="Maroon">t2</font>,<font color="Maroon">W</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="graph_5.html#E11:95_1_1_1"><i><font color="Green">E63</font></i></a></i>;<br/><a NAME="E15:95_1_1_1"/>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">t2</font>,<font color="Maroon">W</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="graph_5.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E16:95_1_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">r</font>,<font color="Maroon">W</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" target="_self">cost</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">t1</font></span>)</span>,<font color="Maroon">W</font></span>)</span> <a href="real_1.html#K3">+</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>set </b><font color="Maroon">e</font> = <font color="Maroon">r</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E17:95_1_1_1"/><i><font color="Green">E67</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E18:95_1_1_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">r</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="graph_5.html#T11" target="_self">Th11</a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><i><font color="Green">E87</font></i>: <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="graph_5.html#T11" target="_self">Th11</a>, <a class="ref" href="graph_5.html#T12" target="_self">Th12</a>, </i>
<br/>.= 
<font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i>, </i>
;<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/><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>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E21:95_1_1_1"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K1" target="_self">vertices</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="ref" href="graph_5.html#T11" target="_self">Th11</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E22:95_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i/>;<br/><a NAME="E23:95_1_1_1"/><b>then </b><i><font color="Green">E89</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 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">r</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> or <font color="Maroon">x</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">r</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E24:95_1_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t1</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon">v3</font> = <font color="Maroon">x</font> as   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="txt" href="graph_5.html#E23:95_1_1_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graph_5.html#E26:95_1_1_1"><i><font color="Green">E90</font></i></a>, , </i>;<br/><a NAME="E26:95_1_1_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> 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">r</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>
 <b>by </b><i>, </i>;<br/><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_1_1_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:95_1_1_1_2_1_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:95_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> 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">r</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="graph_5.html#E23:95_1_1_1"><i><font color="Green">E89</font></i></a>, </i>;<br/><a NAME="E3:95_1_1_1_2_1_1"/><i><font color="Green">E98</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>, </i>;<br/><a NAME="E4:95_1_1_1_2_1_1"/><i><font color="Green">E107</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> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font>
 <b>by </b><i>, <a class="txt" href="graph_5.html#E26:95_1_1_1"><i><font color="Green">E90</font></i></a>, , </i>;<br/><a NAME="E5:95_1_1_1_2_1_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">t1</font> <a href="graph_5.html#R1" target="_self">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v3</font>
 <b>by </b><i>, , , <a class="txt" href="graph_5.html#E23:95_1_1_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graph_5.html#E26:95_1_1_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graph_5.html#E1:95_1_1_1_2_1_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graph_5.html#E2:95_1_1_1_2_1_1"><i><font color="Green">E92</font></i></a>, , , , <a class="ref" href="graph_5.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E6:95_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:95_1_1_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">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>
 ;<br/>

<a NAME="E2:95_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E109</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">t1</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</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> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:95_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K1" target="_self">vertices</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:95_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">t1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">t1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
;<br/>
<a NAME="E5:95_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E110</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 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="finseq_4.html#K4">/.</a> 1</span>)</span> or <font color="Maroon">x</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">t1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E6:95_1_1_1_2_1_1_1"/>
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="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v3</font>
 
<b>by </b><i>, , , <a class="txt" href="graph_5.html#E23:95_1_1_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graph_5.html#E26:95_1_1_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graph_5.html#E1:95_1_1_1_2_1_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graph_5.html#E2:95_1_1_1_2_1_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="graph_5.html#T13" target="_self">Th13</a>, , <a class="ref" href="graph_5.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<b>hence </b><a NAME="E7:95_1_1_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, , , <a class="ref" href="graph_5.html#T14" target="_self">Th14</a>, <a class="ref" href="graph_5.html#T15" target="_self">Th15</a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:95_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">t1</font> <a href="graph_5.html#R2" target="_self">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="txt" href="graph_5.html#E3:95_1_1_1_2_1_1"><i><font color="Green">E98</font></i></a>, </i>;<br/><b>then consider </b><font color="Maroon">q</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="E9:95_1_1_1_2_1_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">q</font> <a href="graph_5.html#R8" target="_self">is_shortestpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v3</font>,<font color="Maroon">V</font>,<font color="Maroon">W</font>
 <b>by </b><i>, <a class="txt" href="graph_5.html#E10"><i><font color="Green">Lemma73</font></i></a></i>;<br/><a NAME="E10:95_1_1_1_2_1_1"/><i><font color="Green">E113</font></i>: 
 <a href="graph_5.html#K10" target="_self">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" target="_self">cost</a> <font color="Maroon">q</font>,<font color="Maroon">W</font>
 <b>by </b><i>, <a class="txt" href="graph_5.html#E1:95_1_1_1_2_1_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="graph_5.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E11:95_1_1_1_2_1_1"/>
 <a href="graph_5.html#K10" target="_self">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" target="_self">cost</a> <font color="Maroon">t1</font>,<font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="graph_5.html#T14" target="_self">Th14</a>, <a class="ref" href="graph_5.html#T15" target="_self">Th15</a>, <a class="txt" href="graph_5.html#E17:95_1_1_1"><i><font color="Green">E67</font></i></a></i>;<br/><a NAME="E12:95_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E114</font></i>: 
 <a href="graph_5.html#K10" target="_self">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" target="_self">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="E13:95_1_1_1_2_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="graph_5.html#T11" target="_self">Th11</a>, , <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E14:95_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><b>hence </b><a NAME="E15:95_1_1_1_2_1_1"/>
 <a href="graph_5.html#K10" target="_self">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" target="_self">cost</a> <font color="Maroon">r</font>,<font color="Maroon">W</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E19:95_1_1_1"><i><font color="Green">E87</font></i></a>, , <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="95_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:95_1_1_1_2_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:95_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E3:95_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E4:95_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:95_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="graph_5.html#T11" target="_self">Th11</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:95_1_1_1_2_1_2"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="graph_5.html#T11" target="_self">Th11</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><i><font color="Green">E118</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_1_1_2_1_2_1"/>
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">r</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v3</font>
 ;<br/><a NAME="E2:95_1_1_1_2_1_2_1"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">v3</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">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="graph_1.html#D12">GRAPH_1:def 12</a></i>;<br/><a NAME="E3:95_1_1_1_2_1_2_1"/>
<font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="graph_5.html#T11" target="_self">Th11</a>, , </i>;<br/><a NAME="E4:95_1_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon">v3</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">s</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E5:95_1_1_1_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="graph_5.html#T11" target="_self">Th11</a>, </i>;<br/></div><b>end;</b></div><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">s</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">t1</font> as  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font> <b>by </b><i>, </i>;<br/><a NAME="E9:95_1_1_1_2_1_2"/><i><font color="Green">E120</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">t1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E10:95_1_1_1_2_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">u</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E11:95_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> 1 = 
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="graph_5.html#T11" target="_self">Th11</a>, , </i>
<br/>.= 
<font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="graph_5.html#T11" target="_self">Th11</a>, , </i>
;<br/><a NAME="E13:95_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E122</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">u</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E14:95_1_1_1_2_1_2"/>
<font color="Maroon">u</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">u</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i>, , </i>;<br/><a NAME="E15:95_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">u</font> <a href="graph_5.html#R1" target="_self">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v3</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E23:95_1_1_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graph_5.html#E26:95_1_1_1"><i><font color="Green">E90</font></i></a>, , , <a class="ref" href="graph_5.html#T14" target="_self">Th14</a>, <a class="ref" href="graph_5.html#T15" target="_self">Th15</a>, </i>;<br/><a NAME="E16:95_1_1_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">u</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_1_2_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:95_1_1_1_2_1_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">u</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>
 ;<br/>

<a NAME="E2:95_1_1_1_2_1_2_3"/><b>then </b><i><font color="Green">E124</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">u</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</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> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">V3</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>;<br/>
<a NAME="E3:95_1_1_1_2_1_2_3"/>
 <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">s</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="graph_5.html#T11" target="_self">Th11</a>, , , </i>;<br/>
<a NAME="E4:95_1_1_1_2_1_2_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">s</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:95_1_1_1_2_1_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="graph_5.html#T11" target="_self">Th11</a>, <a class="txt" href="graph_5.html#E21:95_1_1_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="graph_5.html#E26:95_1_1_1"><i><font color="Green">E90</font></i></a>, , , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:95_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">u</font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v3</font>,<font color="Maroon">V</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_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="E19:95_1_1_1_2_1_2"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">q</font> <a href="graph_5.html#R8" target="_self">is_shortestpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v3</font>,<font color="Maroon">V</font>,<font color="Maroon">W</font>
 <b>by </b><i>, <a class="txt" href="graph_5.html#E10"><i><font color="Green">Lemma73</font></i></a></i>;<br/><a NAME="E20:95_1_1_1_2_1_2"/><i><font color="Green">E138</font></i>: 
 <a href="graph_5.html#K10" target="_self">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" target="_self">cost</a> <font color="Maroon">q</font>,<font color="Maroon">W</font>
 <b>by </b><i>, , <a class="ref" href="graph_5.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E21:95_1_1_1_2_1_2"/>
 <a href="graph_5.html#K10" target="_self">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" target="_self">cost</a> <font color="Maroon">u</font>,<font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="graph_5.html#T16" target="_self">Th16</a>, <a class="txt" href="graph_5.html#E17:95_1_1_1"><i><font color="Green">E67</font></i></a></i>;<br/><a NAME="E22:95_1_1_1_2_1_2"/><b>then </b>
 <a href="graph_5.html#K10" target="_self">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" target="_self">cost</a> <font color="Maroon">u</font>,<font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E23:95_1_1_1_2_1_2"/>
 <a href="graph_5.html#K10" target="_self">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" target="_self">cost</a> <font color="Maroon">r</font>,<font color="Maroon">W</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E19:95_1_1_1"><i><font color="Green">E87</font></i></a>, <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></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1_2"><b>suppose </b></a><a NAME="E1:95_1_1_2"/>
 <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 ;<br/><div class="add"><a NAME="E2:95_1_1_2"/><b>then </b><i><font color="Green">E139</font></i>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">r</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> <a href="subset_1.html#K6">\</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#T33">XBOOLE_1:33</a></i>;<br/><a NAME="E3:95_1_1_2"/>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">r</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="graph_5.html#T11" target="_self">Th11</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E4:95_1_1_2"/><b>then </b>
<font color="Maroon">r</font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">V</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:95_1_1_2"/>
 <a href="graph_5.html#K10" target="_self">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" target="_self">cost</a> <font color="Maroon">r</font>,<font color="Maroon">W</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:95"/>
<font color="Maroon">P</font> <a href="graph_5.html#R7" target="_self">is_shortestpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="graph_5.html#T10" target="_self">Th10</a>, </i>;<br/>


</div>
